5394:
5038:
6275:
5389:{\displaystyle {\begin{aligned}S(f,g)&=\left(\mathrm {lcm} -{\frac {1}{\operatorname {lc} (g)}}\,{\frac {\mathrm {lcm} }{\operatorname {lm} (g)}}\,g\right)-\left(\mathrm {lcm} -{\frac {1}{\operatorname {lc} (f)}}\,{\frac {\mathrm {lcm} }{\operatorname {lm} (f)}}\,f\right)\\&={\frac {1}{\operatorname {lc} (f)}}\,{\frac {\mathrm {lcm} }{\operatorname {lm} (f)}}\,f-{\frac {1}{\operatorname {lc} (g)}}\,{\frac {\mathrm {lcm} }{\operatorname {lm} (g)}}\,g\\\end{aligned}}.}
11326:). Tuning F5 for a general use is difficult, since its performances depend on an order on the input polynomials and a balance between the incrementation of the working polynomial degree and of the number of the input polynomials that are considered. To date (2022), there is no distributed implementation that is significantly more efficient than F4, but, over modular integers F5 has been used successfully for several
5578:
9468:, many properties become false if the triangle degenerates to a line segment, i.e. the length of one side is equal to the sum of the lengths of the other sides. In such situations, one cannot deduce relevant information from the polynomial system unless the degenerate solutions are ignored. More precisely, the system of equations defines an
4724:
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Gröbner basis has been applied in the theory of error-correcting codes for algebraic decoding. By using Gröbner basis computation on various forms of error-correcting equations, decoding methods were developed for correcting errors of cyclic codes, affine variety codes, algebraic-geometric codes and
1141:
Gröbner basis theory was initially introduced for the lexicographical ordering. It was soon realised that the Gröbner basis for degrevlex is almost always much easier to compute, and that it is almost always easier to compute a lex Gröbner basis by first computing the degrevlex basis and then using a
7477:
of polynomials of the basis that depend only on the first variable. After substituting this root in the basis, the second coordinate of this solution is a root of the greatest common divisor of the resulting polynomials that depend only on the second variable, and so on. This solving process is only
2886:
may be reduced. However, the reduction of a term amounts to removing this term at the cost of adding new lower terms; if it is not the first reducible term that is reduced, it may occur that a further reduction adds a similar term, which must be reduced again. It is therefore always better to reduce
6153:
by removing the polynomials whose leading monomials are multiple of the leading monomial of another element of the Gröbner basis. However, if two polynomials of the basis have the same leading monomial, only one must be removed. So, every Gröbner basis contains a minimal Gröbner basis as a subset.
7258:
It is a common misconception that the lexicographical order is needed for some of these results. On the contrary, the lexicographical order is, almost always, the most difficult to compute, and using it makes impractical many computations that are relatively easy with graded reverse lexicographic
6184:
by first removing the polynomials that are lead-reducible by other elements of the basis (for getting a minimal basis); then replacing each element of the basis by the result of the complete reduction by the other elements of the basis; and, finally, by dividing each element of the basis by its
5920:
There are many characterizing properties, which can each be taken as an equivalent definition of Gröbner bases. For conciseness, in the following list, the notation "one-word/another word" means that one can take either "one-word" or "another word" for having two different characterizations of
7411:
by equating the polynomials to zero. The set of the solutions of such a system depends only on the generated ideal, and, therefore does not change when the given generating set is replaced by the Gröbner basis, for any ordering, of the generated ideal. Such a solution, with coordinates in an
7193:
8995:
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1206:
Most polynomial operations related to Gröbner bases involve the leading terms. So, the representation of polynomials as sorted lists make these operations particularly efficient (reading the first element of a list takes a constant time, independently of the length of the list).
1159:
Once a monomial ordering is fixed, the terms of a polynomial (product of a monomial with its nonzero coefficient) are naturally ordered by decreasing monomials (for this order). This makes the representation of a polynomial as a sorted list of pairs coefficient–exponent vector a
3535:
10661:
9495:
consists in adjoining to it the formal inverses of some elements. This section concerns only the case of a single element, or equivalently a finite number of elements (adjoining the inverses of several elements is equivalent to adjoining the inverse of their product). The
2607:
In the example that follows, there are exactly two complete lead-reductions that produce two very different results. The fact that the results are irreducible (not only lead-irreducible) is specific to the example, although this is rather common with such small examples.
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7478:
theoretical, because it implies GCD computation and root-finding of polynomials with approximate coefficients, which are not practicable because of numeric instability. Therefore, other methods have been developed to solve polynomial systems through Gröbner bases (see
2263:
6124:
The above method is an algorithm for computing Gröbner bases; however, it is very inefficient. Many improvements of the original
Buchberger's algorithm, and several other algorithms have been proposed and implemented, which dramatically improve the efficiency. See
4426:
10486:
of the field of the coefficients, if and only if 1 belongs to the generated ideal. This is easily tested with a Gröbner basis computation, because 1 belongs to an ideal if and only if 1 belongs to the Gröbner basis of the ideal, for any monomial ordering.
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2030:
is a special case of reduction that is easier to compute. It is fundamental for Gröbner basis computation, since general reduction is needed only at the end of a Gröbner basis computation, for getting a reduced Gröbner basis from a non-reduced one.
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11368:
have implementations of one or several algorithms for Gröbner bases, often also embedded in other functions, such as for solving systems of polynomial equations or for simplifying trigonometric functions; this is the case, for example, of
6224:, it is useful to work only with polynomials with integer coefficients. In this case, the condition on the leading coefficients in the definition of a reduced basis may be replaced by the condition that all elements of the basis are
3175:
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Counting the above definition, this provides 12 characterizations of Gröbner bases. The fact that so many characterizations are possible makes Gröbner bases very useful. For example, condition 3 provides an algorithm for testing
776:
formed by an exponent vector and the corresponding coefficient. This representation of polynomials is especially efficient for Gröbner basis computation in computers, although it is less convenient for other computations such as
11405:. When F4 is available, it is generally much more efficient than Buchberger's algorithm. The implementation techniques and algorithmic variants are not always documented, although they may have a dramatic effect on efficiency.
9869:
12071:
7462:(without any other variable appearing in the leading term). If this is the case, then the number of zeros, counted with multiplicity, is equal to the number of monomials that are not multiples of any leading monomial of
6918:
7243:. So, it can be applied mechanically to any similar example, although, in general, there are many polynomials and S-polynomials to consider, and the computation is generally too large for being done without a computer.
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All minimal Gröbner bases of a given ideal (for a fixed monomial ordering) have the same number of elements, and the same leading monomials, and the non-minimal Gröbner bases have more elements than the minimal ones.
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can be used. This solves partially issue 4., as reductions to zero in
Buchberger's algorithm correspond to relations between rows of the matrix to be reduced, and the zero rows of the reduced matrix correspond to a
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In the worst case, the main parameter of the complexity is the maximal degree of the elements of the resulting reduced Gröbner basis. More precisely, if the Gröbner basis contains an element of a large degree
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1994:
127:
had introduced a similar notion in 1913, published in various
Russian mathematical journals. These papers were largely ignored by the mathematical community until their rediscovery in 1987 by Bodo Renschuch
10140:. Which method is most efficient depends on the problem. However, if the saturation does not remove any component, that is if the ideal is equal to its saturated ideal, computing first the Gröbner basis of
7271:
for any given ideal and any monomial ordering. Thus two ideals are equal if and only if they have the same (reduced) Gröbner basis (usually a Gröbner basis software always produces reduced Gröbner bases).
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Thus a Gröbner basis for this ordering carries much more information than usually necessary. This may explain why Gröbner bases for the lexicographical ordering are usually the most difficult to compute.
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10978:
One could guess that it suffices to eliminate the parameters to obtain the implicit equations of the variety, as it has been done in the case of curves. Unfortunately this is not always the case. If the
10540:
5573:{\displaystyle S(f,g)={\frac {1}{\operatorname {lc} (f)}}\,{\frac {\operatorname {lm} (g)}{\mathrm {gcd} }}\,f-{\frac {1}{\operatorname {lc} (g)}}\,{\frac {\operatorname {lm} (f)}{\mathrm {gcd} }}\,g;}
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together with the Gröbner basis theory. It is straightforward to implement, but it appeared soon that raw implementations can solve only trivial problems. The main issues are the following ones:
1052:
11337:
Issue 5. has been solved by the discovery of basis conversion algorithms that start from the Gröbner basis for one monomial ordering for computing a Gröbner basis for another monomial ordering.
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10441:
4719:{\displaystyle g_{3}=\left(x^{2}y-{\frac {x^{2}y}{lt(g_{2})}}g_{2}\right)-\left(x^{2}y-{\frac {x^{2}y}{lt(g_{1})}}g_{1}\right)={\frac {x^{2}y}{lt(g_{1})}}g_{1}-{\frac {x^{2}y}{lt(g_{2})}}g_{2}}
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11261:. As in many computational problems, heuristics cannot detect most hidden simplifications, and if heuristic choices are avoided, one may get a dramatic improvement of the algorithm efficiency.
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F5 algorithm improves F4 by introducing a criterion that allows reducing the size of the matrices to be reduced. This criterion is almost optimal, since the matrices to be reduced have
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When the number of zeros is finite, the Gröbner basis for a lexicographical monomial ordering provides, theoretically, a solution: the first coordinate of a solution is a root of the
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2716:
9948:
9450:
6541:
6251:(the whole polynomial ring). Conversely, every Gröbner basis of the unit ideal contains a nonzero constant. The reduced Gröbner basis of the unit is formed by the single polynomial
1215:
The other polynomial operations involved by Gröbner basis computations are also compatible with the monomial ordering; that is, they can be performed without reordering the result:
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Although Gröbner basis theory does not depend on a particular choice of an admissible monomial ordering, three monomial orderings are specially important for the applications:
11838:
6188:
All reduced Gröbner bases of an ideal (for a fixed monomial ordering) are equal. It follows that two ideals are equal if and only if they have the same reduced Gröbner basis.
5765:{\displaystyle S(f,g)=\operatorname {lc} (g)\,{\frac {\operatorname {lm} (g)}{\mathrm {gcd} }}\,f-\operatorname {lc} (f)\,{\frac {\operatorname {lm} (f)}{\mathrm {gcd} }}\,g;}
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When modeling a problem by polynomial equations, it is often assumed that some quantities are non-zero, so as to avoid degenerate cases. For example, when dealing with
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is needed, degrevlex is not convenient; both lex and lexdeg may be used but, again, many computations are relatively easy with lexdeg and almost impossible with lex.
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12635:. 2021 International Symposium on Symbolic and Algebraic Computation. 46th International Symposium on Symbolic and Algebraic Computation. Saint Petersburg, Russia.
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Sometimes, reduced Gröbner bases are defined without the condition on the leading coefficients. In this case, the uniqueness of reduced Gröbner bases is true only
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there are three ways to proceed which give the same result but may have very different computation times (it depends on the problem which is the most efficient).
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1503:
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244:
12683:
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is such a basis conversion algorithm that works only in the zero-dimensional case (where the polynomials have a finite number of complex common zeros) and has a
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of the ideal and of its associated algebraic set is the number of points of this finite intersection, counted with multiplicity. In particular, the degree of a
6316:
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In the case of polynomials in a single variable, there is a unique admissible monomial ordering, the ordering by the degree. The minimal Gröbner bases are the
3025:
10867:
1223:
of the two corresponding lists of terms, with a special treatment in the case of a conflict (that is, when the same monomial appears in the two polynomials).
8509:
12678:
12353:
is a polynomial ring, this reduces the theory and the algorithms of Gröbner bases of modules to the theory and the algorithms of Gröbner bases of ideals.
147:
The theory of Gröbner bases has been extended by many authors in various directions. It has been generalized to other structures such as polynomials over
9781:
2354:
consists of iterating one-step reductions (respect. one-step lead reductions) until getting a polynomial that is irreducible (resp. lead-irreducible) by
1226:
The multiplication of a polynomial by a scalar consists of multiplying each coefficient by this scalar, without any other change in the representation.
12356:
The concept and algorithms of Gröbner bases have also been generalized to ideals over various rings, commutative or not, like polynomial rings over a
11271:
that is most often needed for the applications (pure lexicographic) is not the ordering that leads to the easiest computation, generally the ordering
7188:{\displaystyle {\begin{aligned}yf-xk&=y(x^{2}-y)-x(xy-x)=f-h\\yk-xh&=y(xy-x)-x(y^{2}-y)=0\\y^{2}f-x^{2}h&=y(yf-xk)+x(yk-xh)\end{aligned}}}
13131:
11954:
11219:
Even when the resulting Gröbner basis is small, the intermediate polynomials can be huge. It results that most of the computing time may be spent in
7869:
that provide functions to compute Gröbner bases provide also functions for computing the
Hilbert series, and thus also the dimension and the degree.
2887:
first the largest (for the monomial order) reducible term; that is, in particular, to lead-reduce first until getting a lead-irreducible polynomial.
10490:
The second version asserts that the set of common zeros (in an algebraic closure of the field of the coefficients) of an ideal is contained in the
12898:"Bruno Buchberger's PhD thesis 1965: An algorithm for finding the basis elements of the residue class ring of a zero dimensional polynomial ideal"
12467:"Contributions to constructive polynomial ideal theory XXIII: forgotten works of Leningrad mathematician N. M. Gjunter on polynomial ideal theory"
12664:
8990:{\displaystyle {\begin{aligned}x_{1}&={\frac {f_{1}(t)}{g_{1}(t)}}\\&\;\;\vdots \\x_{n}&={\frac {f_{n}(t)}{g_{n}(t)}},\end{aligned}}}
9298:
7662:
3775:{\displaystyle f\;\xrightarrow {-2xg_{1}} \;f_{1}\;\xrightarrow {yg_{1}} \;-2xy^{2}+2x+2y^{3}+2y\;\xrightarrow {2yg_{2}} \;f_{3}=2x+2y^{3}-2y.}
2499:
1611:
89:
13179:
7439:), or, equivalently, if its Gröbner basis (for any monomial ordering) contains 1, or, also, if the corresponding reduced Gröbner basis is .
2008:
of a polynomial by other polynomials with respect to a monomial ordering is central to Gröbner basis theory. It is a generalization of both
13054:
1778:
9180:
13213:
Computer Aided
Systems Theory — EUROCAST 2001: A Selection of Papers from the 8th International Workshop on Computer Aided Systems Theory
8318:
1923:
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Both degree and dimension depend only on the set of the leading monomials of the Gröbner basis of the ideal for any monomial ordering.
7259:
order (grevlex), or, when elimination is needed, the elimination order (lexdeg) which restricts to grevlex on each block of variables.
2595:
is the quotient. Moreover, the division algorithm is exactly the process of lead-reduction. For this reason, some authors use the term
120:
3530:{\displaystyle f\;\xrightarrow {-2xg_{1}} \;f_{1}\;\xrightarrow {xg_{2}} \;-2xy^{2}+y^{3}+3y\;\xrightarrow {2yg_{2}} \;f_{2}=y^{3}-y.}
10656:{\displaystyle {\begin{aligned}x_{1}&={\frac {p_{1}}{p_{0}}}\\&\;\;\vdots \\x_{n}&={\frac {p_{n}}{p_{0}}},\end{aligned}}}
10144:
is usually faster. On the other hand, if the saturation removes some components, the direct computation may be dramatically faster.
12378:
even general linear block codes. Applying Gröbner basis in algebraic decoding is still a research area of channel coding theory.
7833:
Although the dimension and the degree do not depend on the choice of the monomial ordering, the
Hilbert series and the polynomial
12717:
11883:
elements. As every algorithm for computing a Gröbner basis must write its result, this provides a lower bound of the complexity.
9687:
is not efficient because of the need to manage the denominators. Therefore, localization is usually replaced by the operation of
12779:
Bulygin, S.; Pellikaan, R. (2009). "Decoding linear error-correcting codes up to half the minimum distance with Gröbner bases".
7307:. This allows to test the membership of an element in an ideal. Another method consists in verifying that the Gröbner basis of
6145:
if all leading monomials of its elements are irreducible by the other elements of the basis. Given a Gröbner basis of an ideal
11264:
In most cases most S-polynomials that are computed are reduced to zero; that is, most computing time is spent to compute zero.
7656:
that are not multiple of any leading monomial in the Gröbner basis. The
Hilbert series may be summed into a rational fraction
685:
13300:
13221:
12978:
12852:
12792:
12449:
2258:{\displaystyle \operatorname {red} _{1}(f,g)=f-{\frac {c}{\operatorname {lc} (g)}}\,{\frac {m}{\operatorname {lm} (g)}}\,g.}
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12929:
12885:
11284:
1013:
2034:
Let an admissible monomial ordering be fixed, to which refers every monomial comparison that will occur in this section.
1319:
1249:
116:
10365:
471:
13480:
11498:
7474:
6225:
6023:; condition 4 provides an algorithm for testing whether a set of polynomials is a Gröbner basis and forms the basis of
782:
93:
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12829:
12523:
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as leading coefficient. So, every reduced Gröbner basis is minimal, but a minimal Gröbner basis need not be reduced.
4188:
2017:
66:
9961:
The important property of the saturation, which ensures that it removes from the algebraic set defined by the ideal
332:
69:
and the number of zeros when it is finite. Gröbner basis computation is one of the main practical tools for solving
12784:
11378:
5868:
851:
40:
12902:
12817:
11420:
11350:
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2399:
12973:. London Mathematical Society Lecture Note Series. Vol. 251. Cambridge University Press. pp. 535–545.
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7408:
7402:
926:
791:
70:
12516:
Ideals, Varieties, and
Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra
6228:
with integer coefficients, with positive leading coefficients. This restores the uniqueness of reduced bases.
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2647:
13195:
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11382:
9483:
the equations by the degeneracy conditions, which may be done via the elimination property of Gröbner bases.
6247:
For every monomial ordering, a set of polynomials that contains a nonzero constant is a Gröbner basis of the
12887:
An
Algorithm for Finding the Basis Elements of the Residue Class Ring of a Zero Dimensional Polynomial Ideal
11423:
that outperforms dramatically the other software for this problem (Maple and Magma). Msolve is available on
9905:
9403:
6505:
4982:{\displaystyle S(f,g)=\operatorname {red} _{1}(\mathrm {lcm} ,g)-\operatorname {red} _{1}(\mathrm {lcm} ,f)}
13536:
13475:
13459:
11431:, Maple and SageMath; this means that Msolve can be used directly from within these software environments.
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10479:
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5614:-polynomials. This is a fundamental fact for Gröbner basis theory and all algorithms for computing them.
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which are needed to have an intersection with the algebraic set, which is a finite number of points. The
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11223:. So, specialized memory management algorithms may be a fundamental part of an efficient implementation.
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995:
on the monomials, with the following properties of compatibility with multiplication. For all monomials
13551:
13340:
12387:
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6114:
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of the polynomials (that is, two polynomials are equal if and only they have the same representation).
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On the other hand, examples have been given of reduced Gröbner bases containing polynomials of degree
11257:
The choice of the S-polynomials to reduce and of the polynomials used for reducing them is devoted to
13546:
13541:
13293:
13232:
11797:
11247:
11208:
10039:
9511:
7240:
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6024:
4282:-polynomial by Buchberger, is the difference of the one-step reductions of the least common multiple
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is a Gröbner basis (with respect to the monomial ordering), or, more precisely, a Gröbner basis of
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1220:
778:
78:
11345:
in the number of common zeros. A basis conversion algorithm that works is the general case is the
6548:
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3953:
3330:
2370:. In general this form is not uniquely defined because there are, in general, several elements of
13424:
11745:
11365:
7866:
6259:
4138:
4040:
3996:
12572:
12432:(1983). "Gröbner bases, Gaussian elimination and resolution of systems of algebraic equations".
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11288:
1717:
1584:
253:. Although the theory works for any field, most Gröbner basis computations are done either when
13454:
13449:
13444:
13322:
13169:(on infinite dimensional Gröbner bases for polynomial rings in infinitely many indeterminates).
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11308:
10014:
consists of the components of the primary decomposition of I that do not contain any power of f
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has two versions. The first one asserts that a set of polynomials has no common zeros over an
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for this ordering and the corresponding monomial and coefficient are respectively called the
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and the other block contains all the other variables (this means that a monomial containing
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of the variables such that there is no leading monomial depending only on the variables in
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2013:
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Gröbner basis computation can be seen as a multivariate, non-linear generalization of both
11298:
Roughly speaking, F4 algorithm solves 3. by replacing many S-polynomial reductions by the
8463:}, then a single Gröbner basis computation produces a Gröbner basis of their intersection
8011:, that is a monomial ordering for which two monomials are compared by comparing first the
7891:
7836:
7804:
7775:
6082:
This algorithm uses condition 4, and proceeds roughly as follows: for any two elements of
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8:
13313:
11416:
11409:
10531:
8821:
7255:(see that article for the definitions of the different orders that are mentioned below).
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5872:
5790:
5610:, one can deal with all cases of non-uniqueness of the reduction by considering only the
3170:{\displaystyle f\;\xrightarrow {-2xg_{1}} \;f_{1}=f-2xg_{1}=-x^{2}y-2xy^{2}+2x+y^{3}+3y.}
2873:{\displaystyle {\begin{aligned}g_{1}&=x^{2}+y^{2}-1,\\g_{2}&=xy-2.\end{aligned}}}
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11291:. As these algorithms are designed for integer coefficients or with coefficients in the
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12744:
Fitzgerald, J.; Lax, R.F. (1998). "Decoding affine variety codes using Gröbner bases".
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12489:
12465:
Renschuch, Bodo; Roloff, Hartmut; Rasputin, Georgij G.; Abramson, Michael (June 2003).
12176:
11331:
11243:
11235:
11182:
11133:
11086:
10968:{\displaystyle I=\left\langle p_{0}x_{1}-p_{1},\ldots ,p_{0}x_{n}-p_{n}\right\rangle .}
8202:
7882:
6365:
be the ring of bivariate polynomials with rational coefficients and consider the ideal
6301:
6281:
6106:
6056:
5776:
2573:
258:
28:
8628:{\displaystyle K=\langle tf_{1},\ldots ,tf_{m},(1-t)g_{1},\ldots ,(1-t)g_{k}\rangle .}
124:
13335:
13252:
13217:
13028:
12974:
12958:
12867:
12848:
12825:
12788:
12519:
12445:
11739:
11557:
11292:
11268:
11220:
10483:
9163:
7252:
6169:
if every polynomial in it is irreducible by the other elements of the basis, and has
765:
74:
62:
13269:
13109:
12713:"Use of Gröbner bases to decode binary cyclic codes up to the true minimum distance"
6274:
5921:
Gröbner bases. All the following assertions are characterizations of Gröbner bases:
5617:
For avoiding fractions when dealing with polynomials with integer coefficients, the
13388:
13381:
13376:
13255:
13164:
13150:
13097:
13009:
12946:
12925:
12911:
12881:
12844:
12765:
12753:
12726:
12692:
12646:
12598:
12552:
12493:
12481:
12437:
11386:
11323:
11212:
10523:
9864:{\displaystyle I:f^{\infty }=\{g\in R\mid (\exists k\in \mathbb {N} )f^{k}g\in I\}}
7527:
6263:
5856:. In this section, we suppose that an admissible monomial ordering has been fixed.
3932:
137:
104:
61:. A Gröbner basis allows many important properties of the ideal and the associated
24:
13155:
12630:
12612:
11697:
The worst-case complexity of a Gröbner basis computation is doubly exponential in
6117:). Condition 4 ensures that the result is a Gröbner basis, and the definitions of
6075:
and generates the same ideal. Moreover, such a Gröbner basis may be computed with
13398:
13393:
13345:
13330:
13042:
12967:"An Algorithmic Criterion for the Solvability of a System of Algebraic Equations"
12930:"An Algorithmic Criterion for the Solvability of a System of Algebraic Equations"
12511:
9159:
8817:
8218:
7503:
7421:
6221:
6110:
5849:
183:
173:
152:
44:
13239:
11322:
in sufficiently regular cases (in particular, when the input polynomials form a
9456:. In the other case, the resultant is a power of the result of the elimination.
6318:
form the three blue vertical lines. Their intersection consists of three points.
13369:
13364:
12066:{\displaystyle R/\left\langle \{e_{i}e_{j}|1\leq i\leq j\leq l\}\right\rangle }
11565:
11338:
11303:
11251:
11199:
to get a Gröbner basis of the ideal (of the implicit equations) of the variety.
8813:
8477:, and one uses an elimination ordering such that the first block contains only
7564:
7297:
1161:
986:
13046:
13014:
12989:
12916:
12897:
12757:
12712:
12603:
12576:
11443:
of the Gröbner basis computations is commonly evaluated in term of the number
1237:. This does not change the term ordering by definition of a monomial ordering.
13525:
12441:
12429:
12403:
11419:, and combines all these functions in an algorithm for the real solutions of
11354:
11299:
11239:
9469:
7425:
2009:
773:
682:
of the variables is fixed, the notation of monomials is often abbreviated as
12650:
11211:
is the oldest algorithm for computing Gröbner bases. It has been devised by
2319:
if it is reducible or lead-reducible, respectively, by at least one element
107:
in his 1965 Ph.D. thesis, which also included an algorithm to compute them (
13505:
13273:
12697:
12557:
12540:
12507:
12393:
12361:
11569:
11327:
11312:
10491:
7535:
6121:-polynomials and reduction ensure that the generated ideal is not changed.
5775:
This does not changes anything to the theory since the two polynomials are
133:
82:
13244:
13101:
12485:
9389:{\displaystyle \langle g_{1}x_{1}-f_{1},\ldots ,g_{n}x_{n}-f_{n}\rangle .}
7758:{\displaystyle \sum _{i=0}^{\infty }d_{i}t^{i}={\frac {P(t)}{(1-t)^{d}}},}
7450:, it has only a finite number of zeros, if and only if, for each variable
1707:{\textstyle {\frac {N}{M}}=x_{1}^{b_{1}-a_{1}}\cdots x_{n}^{b_{n}-a_{n}}.}
12628:
12398:
11903:
11390:
2553:{\displaystyle \operatorname {lm} (q_{g}\,g)\leq \operatorname {lm} (f).}
992:
156:
20:
12629:
Berthomieu \first1=Jérémy; Eder, Christian; Safey El Din, Mohab (2021).
9958:
produces a Gröbner basis of the saturation of an ideal by a polynomial.
2611:
In this two variable example, the monomial ordering that is used is the
2327:. In this case, a one-step reduction (resp. one-step lead-reduction) of
144:. This term has been used by some authors to also denote Gröbner bases.
13309:
12950:
11915:
11295:, Buchberger's algorithm remains useful for more general coefficients.
7523:
7251:
Unless explicitly stated, all the results that follow are true for any
6248:
6241:
4178:
Here
Buchberger's algorithm for Gröbner bases would begin by adding to
3574:
One gets a different result with the other choice for the second step:
1096:
265:
12730:
12436:. Lecture Notes in Computer Science. Vol. 162. pp. 146–156.
9287:
Elimination with Gröbner bases allows to implicitize for any value of
3297:
So, one has two choices for the second reduction step. If one chooses
1876:{\textstyle x_{1}^{\min(a_{1},b_{1})}\cdots x_{n}^{\min(a_{n},b_{n})}}
1099:, that is, every strictly decreasing sequence of monomials is finite.
764:
Monomials are uniquely defined by their exponent vectors, and, when a
13439:
13260:
13145:
11899:
11898:
The concept and algorithms of Gröbner bases have been generalized to
11701:. More precisely, the complexity is upper bounded by a polynomial in
11556:
On the other hand, if all polynomials in the reduced Gröbner basis a
11319:
11280:
11258:
9171:
6262:
consisting of a single polynomial. The reduced Gröbner bases are the
6237:
2882:
For the first reduction step, either the first or the second term of
2378:; this non-uniqueness is the starting point of Gröbner basis theory.
11230:
occurring during a computation may be sufficiently large for making
10502:
belongs to the ideal. This may be tested by saturating the ideal by
9277:{\displaystyle {\text{Res}}_{t}(g_{1}x_{1}-f_{1},g_{2}x_{2}-f_{2}).}
8007:. Let us also choose an elimination monomial ordering "eliminating"
7556:
there is a leading monomial in the Gröbner basis that is a power of
3931:
It is for dealing with the problems set by this non-uniqueness that
3705:
3630:
3592:
3475:
3412:
3374:
3037:
155:, and also some classes of non-commutative rings and algebras, like
13429:
12681:(September 1997), "Some Complexity Results for Polynomial Ideals",
12641:
11887:
11398:
11394:
9465:
6027:
for computing Gröbner bases; conditions 5 and 6 allow computing in
5890:
the ideal generated by the leading monomials of the polynomials in
769:
461:
12593:
11349:. In its original form, FGLM may be the critical step for solving
8407:{\displaystyle \{x_{1},\ldots ,x_{k}\},\{x_{k+1},\ldots ,x_{n}\}.}
6098:
if it is not zero; repeat this operation with the new elements of
1989:{\displaystyle \operatorname {lcm} (M,N)={\frac {MN}{\gcd(M,N)}}.}
768:(see below) is fixed, a polynomial is uniquely represented by the
11415:. Beside Gröbner algorithms, Msolve contains fast algorithms for
11227:
11079:>1, two Gröbner basis computations are needed to implicitize:
9153:
7458:
contains a polynomial with a leading monomial that is a power of
7366:. One may also test the equality of the reduced Gröbner bases of
5934:, if and only if some/every complete lead-reduction/reduction of
13114:(translated from Sibirsk. Mat. Zh. Siberian Mathematics Journal
11238:
useful. For this reason, most optimized implementations use the
8217:
into a subspace of the ambient space: with above notation, the (
5587:
denotes the greatest common divisor of the leading monomials of
1088:
A total order satisfying these condition is sometimes called an
991:
All operations related to Gröbner bases require the choice of a
12464:
11424:
11045:. It follows that, in this case, the direct elimination of the
10028:
of a polynomial ideal generated by a finite set of polynomials
7424:
coefficients, this algebraically closed field is chosen as the
7396:
7246:
12864:
Gröbner Bases: A Computational Approach to Commutative Algebra
9954:
is a polynomial ring, a Gröbner basis computation eliminating
9400:= 2, the result is the same as with the resultant, if the map
4997:
denotes the least common multiple of the leading monomials of
2288:
produces a polynomial all of whose monomials are smaller than
12539:
Collart, Stéphane; Kalkbrener, Michael; Mall, Daniel (1997).
11402:
11370:
10147:
If one wants to saturate with respect to several polynomials
9170:= 2, that is for plane curves, this may be computed with the
8221:
of) the projection of the algebraic set defined by the ideal
7552:. Thus, if the ideal has dimension 0, then for each variable
6192:
2335:
is any one-step reduction (resp. one-step lead-reduction) of
2020:. When completed as much as possible, it is sometimes called
11408:
Implementations of F4 and (sparse)-FGLM are included in the
11041:
is an irreducible component of the algebraic set defined by
8198:
has many applications, some described in the next sections.
7431:
An ideal does not have any zero (the system of equations is
7416:
containing the coefficients of the polynomials, is called a
6067:
For every admissible monomial ordering and every finite set
5911:
is a multiple of the leading monomial of some polynomial in
1741:
is the componentwise subtraction of the exponent vectors of
13205:"Gröbner Bases: A Short Introduction for Systems Theorists"
2381:
The definition of the reduction shows immediately that, if
1233:
consists of multiplying each monomial of the polynomial by
12866:. Graduate Texts in Mathematics. Vol. 141. Springer.
12711:
Chen, X.; Reed, I.S.; Helleseth, T.; Truong, T.K. (1994).
10193:
or with respect to a single polynomial which is a product
6102:
included until, eventually, all reductions produce zero.
13210:. In Moreno-Diaz, R.; Buchberger, B.; Freire, J. (eds.).
12963:(This is the journal publication of Buchberger's thesis.)
6195:
the multiplication of polynomials by a nonzero constant.
3812:
is irreducible, although only lead reductions were done.
1154:
13308:
12390:, an extension of Gröbner bases to non-commutative rings
8023:-variable is greater than every monomial independent of
2276:
without changing the terms with a monomial greater than
755:{\displaystyle x_{1}^{a_{1}}\cdots x_{n}^{a_{n}}=X^{A}.}
12965:
Burchberger, B.; Winkler, F., eds. (26 February 1998).
12710:
11302:
of a single large matrix for which advanced methods of
8015:-parts, and, in case of equality only, considering the
7485:
6784:, and none of them is reducible by the others, none of
6071:
of polynomials, there is a Gröbner basis that contains
5894:
equals the ideal generated by the leading monomials of
13250:
11846:
11800:
11748:
11707:
10517:
10510:
belongs to the ideal if and only if the saturation by
8807:
7573:
4175:
and this restores the uniqueness of the reduced form.
1883:
whose exponent vector is the componentwise minimum of
1781:
1720:
1614:
1587:
871:
846:
is a finite set of polynomials in the polynomial ring
12538:
12317:
12280:
12234:
12179:
12133:
12079:
11957:
11943:
as a ring by defining the product of two elements of
11923:
11624:
11585:
11501:
11465:
11185:
11158:
11136:
11109:
11089:
11051:
11020:
10985:
10870:
10821:
10775:
10726:
10672:
10543:
10449:
10368:
10331:
10301:
10252:
10199:
10153:
10090:
10042:
9986:
9908:
9784:
9754:
9720:
9666:
9632:
9602:
9514:
9406:
9301:
9183:
9122:
9086:
9042:
9006:
8833:
8776:
8736:
8710:
8669:
8512:
8485:
is greater than every monomial that does not contain
8321:
8275:
8235:
8191:, as only the leading monomials need to be checked).
8162:
8115:
8076:
8041:
7894:
7839:
7807:
7778:
7665:
7631:
7538:
is equal to the degree of its definition polynomial.
7275:
6921:
6859:
6824:
6790:
6705:
6630:
6551:
6508:
6453:
6412:
6371:
6328:
6304:
6284:
6204:
6033:
5989:
5793:
5630:
5420:
5041:
5011:
4882:
4823:
4796:
4736:
4429:
4399:
4372:
4345:
4318:
4288:
4191:
4141:
4114:
4087:
4043:
3999:
3956:
3871:
3825:
3791:
3583:
3546:
3365:
3333:
3303:
3273:
3246:
3219:
3186:
3028:
2987:
2957:
2930:
2896:
2779:
2724:
2650:
2621:
2502:
2475:
2402:
2161:
1926:
1535:
1450:
1392:
1322:
1252:
1061:
1016:
929:
865:; that is the set of polynomials that can be written
794:
688:
630:
568:
541:
474:
468:
by Buchberger and some of his followers) of the form
439:
404:
335:
274:
191:
10861:
of the points of the variety are zeros of the ideal
9174:. The implicit equation is the following resultant:
2981:
So the first reduction step consists of multiplying
1047:{\displaystyle M\leq N\Longleftrightarrow MP\leq NP}
13240:
Comparative Timings Page for Gröbner Bases Software
13121:
12506:
11495:nonzero terms whose computation requires a time of
9580:is a new indeterminate representing the inverse of
8489:). With this monomial ordering, a Gröbner basis of
8019:-parts. This implies that a monomial containing an
6126:
3327:one gets a polynomial that can be reduced again by
2346:The (complete) reduction (resp. lead-reduction) of
1379:{\displaystyle N=x_{1}^{b_{1}}\cdots x_{n}^{b_{n}}}
1309:{\displaystyle M=x_{1}^{a_{1}}\cdots x_{n}^{a_{n}}}
12964:
12861:
12341:
12303:
12266:
12220:
12165:
12111:
12065:
11935:
11875:
11832:
11786:
11730:
11687:
11610:
11548:
11487:
11191:
11171:
11142:
11122:
11095:
11064:
11033:
10998:
10967:
10853:
10807:
10761:
10704:
10655:
10462:
10436:{\displaystyle 1-t_{1}f_{1},\ldots ,1-t_{k}f_{k},}
10435:
10347:
10317:
10284:
10234:
10185:
10117:
10080:that is by keeping the polynomials independent of
10072:
10005:
9942:
9863:
9770:
9736:
9679:
9648:
9618:
9568:
9444:
9388:
9276:
9144:
9108:
9064:
9028:
8989:
8797:
8748:
8722:
8696:
8627:
8406:
8307:
8259:
8183:
8136:
8097:
8062:
7999:in which the variables are split into two subsets
7991:
7854:
7822:
7793:
7757:
7644:
7617:
7563:Both dimension and degree may be deduced from the
7187:
6877:
6845:
6811:
6717:
6688:
6590:
6535:
6478:
6437:
6395:
6357:
6310:
6290:
6240:of polynomials is the unique Gröbner basis of the
6212:
6047:
6006:
5957:, some/every complete lead-reduction/reduction of
5840:
5764:
5572:
5388:
5024:
4981:
4839:
4809:
4786:This does not complete Buchberger's algorithm, as
4778:
4718:
4412:
4385:
4358:
4331:
4304:
4267:
4167:
4127:
4100:
4081:, and this allows more reductions. In particular,
4069:
4025:
3985:
3921:
3857:
3804:
3774:
3559:
3529:
3349:
3319:
3289:
3259:
3232:
3205:
3169:
3000:
2973:
2943:
2912:
2872:
2765:
2710:
2636:
2552:
2488:
2450:
2257:
1988:
1875:
1733:
1706:
1600:
1561:
1497:
1436:
1378:
1308:
1076:
1046:
970:
915:
838:
754:
674:
612:
554:
528:{\displaystyle x_{1}^{a_{1}}\cdots x_{n}^{a_{n}},}
527:
452:
417:
390:
321:
238:
123:for this work. However, the Russian mathematician
13132:Transactions of the American Mathematical Society
11549:{\displaystyle \Omega (D^{n})>D^{\Omega (n)}.}
11203:
8659:. This may be proven by remarking that the ideal
7435:) if and only if 1 belongs to the ideal (this is
1229:The multiplication of a polynomial by a monomial
857:is the set of linear combinations of elements of
13523:
13180:Faugère's own implementation of his F4 algorithm
12987:
12812:Adams, William W.; Loustaunau, Philippe (1994).
12778:
12632:Msolve: a library for solving polynomial systems
12406:, an alternative way to represent algebraic sets
10019:
8499:consists in the polynomials that do not contain
2101:may be reducible without being lead-reducible.)
1962:
1839:
1792:
13083:"Certain algorithmic problems for Lie algebras"
12811:
11279:For solving 3. many improvements, variants and
10720:variables (parameters of the parameterization)
9486:
8473:. For this, one introduces a new indeterminate
8315:is an elimination ordering for every partition
7862:change when one changes the monomial ordering.
7215:The method that has been used here for finding
4268:{\displaystyle g_{3}=yg_{1}-xg_{2}=2x+y^{3}-y.}
3240:is a multiple of the leading monomials of both
13331:Zero polynomial (degree undefined or −1 or −∞)
13230:
11283:have been proposed before the introduction of
11014:of the non-empty algebraic set defined by the
7544:The dimension is the maximal size of a subset
7239:is a Gröbner basis is a direct application of
2602:
2280:(for the monomial ordering). In particular, a
1219:The addition of two polynomials consists in a
391:{\displaystyle c_{1}M_{1}+\cdots +c_{m}M_{m},}
13294:
12862:Becker, Thomas; Weispfenning, Volker (1998).
12743:
12571:
11357:. This has been fixed by the introduction of
10136:, one may also start from a Gröbner basis of
6269:
6198:When working with polynomials over the field
2024:although its result is not uniquely defined.
1121:Total degree reverse lexicographical ordering
13055:Notices of the American Mathematical Society
12910:(3–4). Translated by Abramson, M.: 471–511.
12055:
12006:
11353:because FGML does not take into account the
10474:
10112:
10097:
10064:
10049:
9858:
9804:
9380:
9302:
9156:(they have no non-constant common factors).
8619:
8519:
8398:
8360:
8354:
8322:
7618:{\textstyle \sum _{i=0}^{\infty }d_{i}t^{i}}
7397:Solutions of a system of algebraic equations
7247:Properties and applications of Gröbner bases
6872:
6860:
6837:
6825:
6803:
6791:
6527:
6515:
6390:
6378:
5907:the leading monomial of every polynomial in
5598:As the monomials that are reducible by both
2757:
2731:
2018:Euclidean division of univariate polynomials
1241:
833:
801:
12663:: CS1 maint: numeric names: authors list (
6105:The algorithm terminates always because of
2451:{\displaystyle f=h+\sum _{g\in G}q_{g}\,g,}
1095:These conditions imply that the order is a
264:In the context of Gröbner bases, a nonzero
13301:
13287:
13202:
12988:Buchberger, Bruno; Kauers, Manuel (2010).
12924:
12895:
12880:
12123:. This allows identifying a submodule of
11618:So, the complexity of this computation is
10597:
10596:
8909:
8908:
8431:are two ideals generated respectively by {
7885:. This is based on the following theorem.
7407:Any set of polynomials may be viewed as a
4037:. So, this ideal is not changed by adding
3724:
3700:
3646:
3625:
3614:
3587:
3494:
3470:
3428:
3407:
3396:
3369:
3059:
3032:
2560:In the case of univariate polynomials, if
1167:The first (greatest) term of a polynomial
121:Paris Kanellakis Theory and Practice Award
16:Mathematical construct in computer algebra
13154:
13144:
13041:
13013:
12915:
12696:
12640:
12602:
12592:
12556:
12372:
11688:{\displaystyle O(D^{n})^{O(1)}=D^{O(n)}.}
9832:
8308:{\displaystyle x_{1}>\cdots >x_{n}}
8156:(this makes very easy the computation of
6336:
6206:
5968:all complete reductions of an element of
5755:
5722:
5700:
5667:
5563:
5530:
5499:
5466:
5375:
5342:
5311:
5278:
5235:
5202:
5147:
5114:
4790:gives different results, when reduced by
2522:
2441:
2248:
2223:
971:{\displaystyle g_{1},\ldots ,g_{k}\in R.}
839:{\displaystyle F=\{f_{1},\ldots ,f_{k}\}}
13080:
12624:
12622:
12541:"Converting bases with the Gröbner walk"
12367:
10362:or to its Gröbner basis the polynomials
10125:for an elimination ordering eliminating
7877:The computation of Gröbner bases for an
6273:
6132:
5025:{\displaystyle \operatorname {red} _{1}}
2711:{\displaystyle f=2x^{3}-x^{2}y+y^{3}+3y}
1386:be two monomials, with exponent vectors
1210:
178:Gröbner bases are primarily defined for
13127:"Finite generation of symmetric ideals"
13022:
12781:Gröbner Bases, Coding, and Cryptography
12718:IEEE Transactions on Information Theory
11564:, the Gröbner basis can be computed by
10514:provides a Gröbner basis containing 1.
8269:The lexicographical ordering such that
8144:consists exactly of the polynomials of
7881:monomial ordering allows computational
1714:In other words, the exponent vector of
261:or the integers modulo a prime number.
13524:
12428:
11072:provides an empty set of polynomials.
10470:in a single Gröbner basis computation.
10292:in a single Gröbner basis computation.
9943:{\displaystyle I:f^{\infty }=J\cap R.}
9445:{\displaystyle t\mapsto (x_{1},x_{2})}
8756:. Such a polynomial is independent of
8418:
6536:{\displaystyle I=\langle f,k\rangle :}
6180:, one gets a reduced Gröbner basis of
6149:, one gets a minimal Gröbner basis of
5975:the monomials that are irreducible by
3815:In summary, the complete reduction of
1155:Leading term, coefficient and monomial
916:{\textstyle \sum _{i=1}^{k}g_{i}f_{i}}
132:An analogous concept for multivariate
13282:
13251:
13062:(10): 1199–1200, a brief introduction
12619:
12424:
12422:
12420:
11254:are used in optimized implementations
11006:have a common zero (sometimes called
10024:A Gröbner basis of the saturation by
7652:is the number of monomials of degree
7262:
6396:{\displaystyle I=\langle f,g\rangle }
6298:form the red parabola; the zeroes of
6127:§ Algorithms and implementations
3540:No further reduction is possible, so
1142:"change of ordering algorithm". When
562:are nonnegative integers. The vector
12677:
11906:over a polynomial ring. In fact, if
11447:of variables and the maximal degree
10762:{\displaystyle t_{1},\ldots ,t_{k}.}
10235:{\displaystyle f=f_{1}\cdots f_{k},}
8503:, in the Gröbner basis of the ideal
8209:realizes the geometric operation of
7486:Dimension, degree and Hilbert series
6689:{\displaystyle h=xk-(y-1)f=y^{2}-y.}
6617:, which gives another polynomial in
6086:, compute the complete reduction by
5411:-polynomial can also be written as:
5399:Using the property that relates the
2268:This operation removes the monomial
980:
12342:{\displaystyle 1\leq i\leq j\leq l}
12267:{\displaystyle g_{1},\ldots ,g_{k}}
12166:{\displaystyle g_{1},\ldots ,g_{k}}
12112:{\displaystyle e_{1},\ldots ,e_{l}}
11951:. This ring may be identified with
11572:of polynomials of degree less than
10854:{\displaystyle x_{1},\ldots ,x_{n}}
10808:{\displaystyle t_{1},\ldots ,t_{k}}
10705:{\displaystyle p_{0},\ldots ,p_{n}}
10518:Implicitization in higher dimension
10285:{\displaystyle f=f_{1}\cdots f_{k}}
10186:{\displaystyle f_{1},\ldots ,f_{k}}
9660:is a polynomial ring, computing in
9072:are univariate polynomials for 1 ≤
8808:Implicitization of a rational curve
6094:-polynomial, and add the result to
3922:{\displaystyle f_{3}=2x+2y^{3}-2y.}
1149:
117:Association for Computing Machinery
115:. In 2007, Buchberger received the
111:). He named them after his advisor
94:polynomial greatest common divisors
13:
12838:
12805:
12671:
12417:
11893:
11857:
11811:
11529:
11502:
11466:
9998:
9920:
9822:
9796:
9080:. One may (and will) suppose that
8229:-subspace is defined by the ideal
7772:is the dimension of the ideal and
7682:
7590:
7567:of the ideal, which is the series
7276:Membership and inclusion of ideals
6699:Under lexicographic ordering with
6176:Given a Gröbner basis of an ideal
5863:be a finite set of polynomials in
5749:
5746:
5743:
5694:
5691:
5688:
5557:
5554:
5551:
5493:
5490:
5487:
5352:
5349:
5346:
5288:
5285:
5282:
5212:
5209:
5206:
5171:
5168:
5165:
5124:
5121:
5118:
5083:
5080:
5077:
4966:
4963:
4960:
4927:
4924:
4921:
4779:{\displaystyle g_{3}=f_{3}-f_{2}.}
4033:belongs to the ideal generated by
2766:{\displaystyle G=\{g_{1},g_{2}\},}
1577:is componentwise not greater than
783:polynomial greatest common divisor
167:
65:to be deduced easily, such as the
14:
13563:
13173:
10032:, may be obtained by eliminating
9875:whose product with some power of
8035:for this monomial ordering, then
7351:, it suffices to test that every
6236:For every monomial ordering, the
6055:in a way that is very similar to
2644:and we consider the reduction of
329:is commonly represented as a sum
103:Gröbner bases were introduced by
13025:An Introduction to Gröbner Bases
12814:An Introduction to Gröbner Bases
11833:{\textstyle d^{2^{\Omega (n)}},}
8105:(this ideal is often called the
7522:. It is also equal to number of
6231:
6109:or because polynomial rings are
1143:
12903:Journal of Symbolic Computation
12893:(PhD). University of Innsbruck.
12818:Graduate Studies in Mathematics
12772:
12746:Designs, Codes and Cryptography
12581:Journal of Symbolic Computation
12545:Journal of Symbolic Computation
11876:{\textstyle d^{2^{\Omega (n)}}}
11421:systems of polynomial equations
11351:systems of polynomial equations
10073:{\displaystyle F\cup \{1-tf\},}
9569:{\displaystyle R_{f}=R/(1-ft),}
8031:is a Gröbner basis of an ideal
6609:is reducible by the other, but
6498:, one obtains a new polynomial
5621:polynomial is often defined as
4849:
1562:{\displaystyle a_{i}\leq b_{i}}
624:of the monomial. When the list
136:was developed independently by
71:systems of polynomial equations
13125:; Hillar, Christopher (2007).
13081:Shirshov, Anatoliĭ I. (1999).
13047:"What is ... a Gröbner Basis?"
12971:Gröbner Bases and Applications
12737:
12704:
12565:
12532:
12500:
12458:
12215:
12183:
12030:
11993:
11961:
11866:
11860:
11820:
11814:
11774:
11768:
11677:
11671:
11655:
11649:
11642:
11628:
11602:
11589:
11538:
11532:
11518:
11505:
11488:{\displaystyle \Omega (D^{n})}
11482:
11469:
11293:integers modulo a prime number
11232:fast multiplication algorithms
11204:Algorithms and implementations
10325:then saturating the result by
10118:{\displaystyle F\cup \{1-tf\}}
9871:consisting in all elements of
9836:
9819:
9560:
9545:
9537:
9531:
9452:is injective for almost every
9439:
9413:
9410:
9268:
9196:
9139:
9133:
9103:
9097:
9059:
9053:
9023:
9017:
8974:
8968:
8953:
8947:
8895:
8889:
8874:
8868:
8682:
8670:
8606:
8594:
8572:
8560:
8251:
8245:
8178:
8172:
8150:whose leading terms belong to
8131:
8125:
8092:
8086:
8057:
8051:
7983:
7971:
7962:
7898:
7872:
7849:
7843:
7817:
7811:
7788:
7782:
7740:
7727:
7722:
7716:
7516:dimension of the algebraic set
7480:System of polynomial equations
7409:system of polynomial equations
7403:System of polynomial equations
7281:
7178:
7160:
7151:
7133:
7081:
7062:
7053:
7038:
6994:
6979:
6970:
6951:
6658:
6646:
6358:{\displaystyle R=\mathbb {Q} }
6352:
6340:
5835:
5803:
5738:
5732:
5719:
5713:
5683:
5677:
5664:
5658:
5646:
5634:
5546:
5540:
5524:
5518:
5482:
5476:
5460:
5454:
5436:
5424:
5369:
5363:
5336:
5330:
5305:
5299:
5272:
5266:
5229:
5223:
5196:
5190:
5141:
5135:
5108:
5102:
5061:
5049:
4976:
4956:
4937:
4917:
4898:
4886:
4700:
4687:
4644:
4631:
4583:
4570:
4501:
4488:
2544:
2538:
2526:
2509:
2374:that can be used for reducing
2303:of polynomials, one says that
2242:
2236:
2217:
2211:
2187:
2175:
1977:
1965:
1945:
1933:
1868:
1842:
1821:
1795:
1489:
1457:
1431:
1399:
1183:and denoted, in this article,
1026:
669:
637:
607:
575:
316:
284:
233:
201:
1:
13491:Horner's method of evaluation
13231:Buchberger, B.; Zapletal, A.
13156:10.1090/S0002-9947-07-04116-5
12822:American Mathematical Society
12410:
11910:is a free module over a ring
11742:, it is therefore bounded by
11434:
11355:sparsity of involved matrices
10494:of the zeros of a polynomial
10020:Computation of the saturation
10006:{\displaystyle I:f^{\infty }}
9744:under the canonical map from
9459:
8798:{\displaystyle b\in I\cap J.}
6403:generated by the polynomials
5942:produces the zero polynomial;
5782:
5606:are exactly the multiples of
4857:Given monomial ordering, the
3935:introduced Gröbner bases and
3858:{\displaystyle f_{2}=y^{3}-y}
2564:consists of a single element
1581:. In this case, the quotient
13233:"Gröbner Bases Bibliography"
12841:Gröbner Bases in Ring Theory
11914:, then one may consider the
11787:{\textstyle d^{2^{n+o(n)}}.}
11330:; for example, for breaking
10498:, if and only if a power of
9487:Definition of the saturation
9166:of such a curve. In case of
8663:consists of the polynomials
7830:is the degree of the ideal.
7466:. This number is called the
6591:{\displaystyle k=g-xf=xy-x.}
6213:{\displaystyle \mathbb {Q} }
6062:
4312:of the leading monomials of
3986:{\displaystyle f_{2}-f_{3}.}
3350:{\displaystyle g_{2}\colon }
2097:, it is also reducible, but
1999:
1117:(for pure lexical ordering).
73:and computing the images of
7:
13496:Polynomial identity testing
13216:. Springer. pp. 1–19.
13191:Encyclopedia of Mathematics
12381:
11459:, this element may contain
9973:is zero, is the following:
7888:Consider a polynomial ring
4168:{\displaystyle f_{3}-f_{2}}
4070:{\displaystyle f_{3}-f_{2}}
4026:{\displaystyle f_{2}-f_{3}}
3939:-polynomials. Intuitively,
3567:is a complete reduction of
2603:Non uniqueness of reduction
2358:. It is sometimes called a
2016:and division steps of the
1734:{\textstyle {\frac {N}{M}}}
1601:{\textstyle {\frac {N}{M}}}
23:, and more specifically in
10:
13568:
13270:Gröbner basis introduction
13072:: CS1 maint: postscript (
12551:(3–4). Elsevier: 465–469.
12304:{\displaystyle e_{i}e_{j}}
11451:of the input polynomials.
9890:is the ideal generated by
9162:consists in computing the
7801:is a polynomial such that
7414:algebraically closed field
7400:
7347:is contained in the ideal
7267:Reduced Gröbner bases are
7198:can be reduced to zero by
6912:, since the S-polynomials
6270:Example and counterexample
5005:. Using the definition of
2496:are polynomials such that
1906:is defined similarly with
984:
171:
41:generating set of an ideal
13468:
13407:
13320:
13015:10.4249/scholarpedia.7763
12917:10.1016/j.jsc.2005.09.007
12604:10.1016/j.jsc.2016.07.025
11936:{\displaystyle R\oplus L}
11611:{\displaystyle O(D^{n}).}
11560:have a degree of at most
11427:, and is interfaced with
11248:Chinese remainder theorem
10522:By definition, an affine
10480:Hilbert's Nullstellensatz
10475:Effective Nullstellensatz
7437:Hilbert's Nullstellensatz
6479:{\displaystyle g=x^{3}-x}
6438:{\displaystyle f=x^{2}-y}
4730:In this example, one has
3015:and adding the result to
1242:Divisibility of monomials
12938:Aequationes Mathematicae
12577:"Sparse FGLM algorithms"
12442:10.1007/3-540-12868-9_99
11579:, which has a dimension
11366:computer algebra systems
11328:cryptographic challenges
10084:in the Gröbner basis of
9969:on which the polynomial
9714:is the inverse image of
9291:, simply by eliminating
9145:{\displaystyle g_{i}(t)}
9109:{\displaystyle f_{i}(t)}
9065:{\displaystyle g_{i}(t)}
9029:{\displaystyle f_{i}(t)}
8697:{\displaystyle (a-b)t+b}
8260:{\displaystyle I\cap K.}
8201:Another application, in
7867:computer algebra systems
7442:Given the Gröbner basis
6846:{\displaystyle \{f,h\},}
6812:{\displaystyle \{f,k\},}
5972:produce the same result;
4278:This polynomial, called
2572:is the remainder of the
2049:if the leading monomial
1162:canonical representation
1107:Lexicographical ordering
1077:{\displaystyle M\leq MP}
779:polynomial factorization
425:are nonzero elements of
162:
140:in 1964, who named them
39:is a particular kind of
13481:Greatest common divisor
13203:Buchberger, B. (2003).
13123:Aschenbrenner, Matthias
12758:10.1023/A:1008274212057
12651:10.1145/3452143.3465545
12388:Bergman's diamond lemma
11731:{\textstyle d^{2^{n}}.}
11236:multimodular arithmetic
11130:to get a Gröbner basis
9472:which may have several
8184:{\displaystyle G\cap K}
8137:{\displaystyle G\cap K}
8098:{\displaystyle I\cap K}
8063:{\displaystyle G\cap K}
7475:greatest common divisor
7420:. In the usual case of
6878:{\displaystyle \{h,k\}}
6115:Hilbert's basis theorem
5924:
5905:
3206:{\displaystyle -x^{2}y}
2637:{\displaystyle x>y,}
2282:one step lead-reduction
2124:such that the monomial
1754:greatest common divisor
13353:Quadratic function (2)
13245:Prof. Bruno Buchberger
13023:Fröberg, Ralf (1997).
12698:10.1006/jcom.1997.0447
12575:; Chenqi, Mou (2017).
12558:10.1006/jsco.1996.0145
12373:Error-Correcting Codes
12343:
12305:
12268:
12222:
12167:
12113:
12067:
11937:
11877:
11834:
11788:
11732:
11689:
11612:
11550:
11489:
11359:sparse FGLM algorithms
11347:Gröbner walk algorithm
11209:Buchberger's algorithm
11193:
11173:
11144:
11124:
11097:
11066:
11035:
11000:
10969:
10855:
10809:
10763:
10716:+1 polynomials in the
10706:
10657:
10506:; in fact, a power of
10464:
10437:
10349:
10348:{\displaystyle f_{2},}
10319:
10318:{\displaystyle f_{1},}
10286:
10236:
10187:
10119:
10074:
10007:
9967:irreducible components
9944:
9865:
9772:
9771:{\displaystyle R_{f}.}
9738:
9737:{\displaystyle R_{f}I}
9681:
9650:
9649:{\displaystyle I_{f}.}
9620:
9619:{\displaystyle R_{f}I}
9570:
9493:localization of a ring
9474:irreducible components
9446:
9390:
9278:
9146:
9110:
9066:
9030:
8991:
8799:
8750:
8749:{\displaystyle b\in J}
8724:
8723:{\displaystyle a\in I}
8698:
8629:
8408:
8309:
8261:
8185:
8138:
8099:
8070:is a Gröbner basis of
8064:
7993:
7856:
7824:
7795:
7759:
7686:
7646:
7619:
7594:
7241:Buchberger's algorithm
7189:
6908:is a Gröbner basis of
6885:is a Gröbner basis of
6879:
6847:
6813:
6719:
6718:{\displaystyle x>y}
6690:
6592:
6537:
6480:
6439:
6397:
6359:
6319:
6312:
6292:
6214:
6077:Buchberger's algorithm
6049:
6025:Buchberger's algorithm
6008:
5842:
5766:
5574:
5390:
5032:, this translates to:
5026:
4983:
4841:
4840:{\displaystyle g_{3}.}
4811:
4780:
4720:
4414:
4387:
4360:
4333:
4306:
4305:{\displaystyle x^{2}y}
4269:
4169:
4129:
4102:
4071:
4027:
3987:
3923:
3859:
3806:
3776:
3561:
3531:
3351:
3321:
3320:{\displaystyle g_{2},}
3291:
3290:{\displaystyle g_{2},}
3261:
3234:
3207:
3171:
3002:
2975:
2974:{\displaystyle g_{2}.}
2945:
2914:
2913:{\displaystyle 2x^{3}}
2874:
2767:
2712:
2638:
2599:instead of reduction.
2554:
2490:
2452:
2259:
2148:consists of replacing
2045:by another polynomial
1990:
1877:
1735:
1708:
1602:
1563:
1499:
1438:
1380:
1310:
1078:
1048:
972:
917:
892:
840:
756:
676:
614:
556:
529:
454:
419:
392:
323:
240:
109:Buchberger's algorithm
13336:Constant function (0)
13102:10.1145/334714.334715
12684:Journal of Complexity
12587:. Elsevier: 538–569.
12573:Faugère, Jean-Charles
12486:10.1145/944567.944569
12368:Areas of applications
12344:
12306:
12269:
12223:
12168:
12114:
12068:
11938:
11878:
11835:
11789:
11733:
11690:
11613:
11551:
11490:
11364:Most general-purpose
11343:polynomial complexity
11194:
11174:
11172:{\displaystyle t_{i}}
11145:
11125:
11123:{\displaystyle p_{0}}
11098:
11067:
11065:{\displaystyle t_{i}}
11036:
11034:{\displaystyle p_{i}}
11012:irreducible component
11001:
10999:{\displaystyle p_{i}}
10970:
10856:
10810:
10764:
10707:
10658:
10465:
10463:{\displaystyle t_{i}}
10438:
10350:
10320:
10287:
10237:
10188:
10120:
10075:
10008:
9977:primary decomposition
9945:
9866:
9773:
9739:
9682:
9680:{\displaystyle R_{f}}
9651:
9621:
9571:
9447:
9391:
9279:
9147:
9111:
9067:
9031:
8992:
8800:
8751:
8725:
8699:
8630:
8409:
8310:
8262:
8186:
8139:
8100:
8065:
7994:
7857:
7825:
7796:
7760:
7666:
7647:
7645:{\displaystyle d_{i}}
7620:
7574:
7498:in a polynomial ring
7325:To test if the ideal
7288:by the Gröbner basis
7190:
6880:
6848:
6814:
6720:
6691:
6593:
6538:
6481:
6440:
6398:
6360:
6313:
6293:
6277:
6226:primitive polynomials
6215:
6185:leading coefficient.
6133:Reduced Gröbner bases
6050:
6009:
5843:
5767:
5575:
5391:
5027:
4984:
4842:
4812:
4810:{\displaystyle g_{2}}
4781:
4721:
4415:
4413:{\displaystyle g_{1}}
4388:
4386:{\displaystyle g_{2}}
4361:
4359:{\displaystyle g_{2}}
4334:
4332:{\displaystyle g_{1}}
4307:
4270:
4170:
4130:
4128:{\displaystyle f_{2}}
4103:
4101:{\displaystyle f_{3}}
4072:
4028:
3988:
3924:
3860:
3819:can result in either
3807:
3805:{\displaystyle f_{3}}
3777:
3562:
3560:{\displaystyle f_{2}}
3532:
3352:
3322:
3292:
3262:
3260:{\displaystyle g_{1}}
3235:
3233:{\displaystyle f_{1}}
3208:
3172:
3003:
3001:{\displaystyle g_{1}}
2976:
2946:
2944:{\displaystyle g_{1}}
2915:
2875:
2768:
2713:
2639:
2597:multivariate division
2555:
2491:
2489:{\displaystyle q_{g}}
2453:
2260:
2093:is lead-reducible by
2022:multivariate division
1991:
1893:least common multiple
1878:
1736:
1709:
1603:
1564:
1500:
1439:
1381:
1311:
1211:Polynomial operations
1079:
1049:
973:
918:
872:
861:with coefficients in
841:
757:
677:
615:
557:
555:{\displaystyle a_{i}}
530:
455:
453:{\displaystyle M_{i}}
420:
418:{\displaystyle c_{i}}
393:
324:
241:
149:principal ideal rings
13469:Tools and algorithms
13389:Quintic function (5)
13377:Quartic function (4)
13314:polynomial functions
12358:principal ideal ring
12315:
12278:
12232:
12177:
12131:
12077:
11955:
11921:
11844:
11798:
11746:
11705:
11622:
11583:
11499:
11463:
11315:of these relations.
11289:Jean-Charles Faugère
11285:F4 and F5 algorithms
11183:
11156:
11134:
11107:
11087:
11049:
11018:
10983:
10868:
10819:
10815:and the coordinates
10773:
10769:Thus the parameters
10724:
10670:
10541:
10532:parametric equations
10530:may be described by
10447:
10443:and eliminating the
10366:
10329:
10299:
10250:
10197:
10151:
10088:
10040:
9984:
9950:It follows that, if
9906:
9782:
9752:
9718:
9664:
9630:
9600:
9512:
9404:
9299:
9181:
9120:
9084:
9040:
9004:
8831:
8822:parametric equations
8774:
8734:
8708:
8667:
8510:
8319:
8273:
8233:
8215:affine algebraic set
8196:elimination property
8160:
8113:
8074:
8039:
7992:{\displaystyle K=K,}
7892:
7855:{\displaystyle P(t)}
7837:
7823:{\displaystyle P(1)}
7805:
7794:{\displaystyle P(t)}
7776:
7663:
7629:
7571:
7514:and is equal to the
6919:
6857:
6822:
6788:
6703:
6628:
6549:
6506:
6451:
6410:
6369:
6326:
6302:
6282:
6202:
6031:
6007:{\displaystyle R/I.}
5987:
5979:form a basis of the
5791:
5628:
5418:
5039:
5009:
4880:
4821:
4794:
4734:
4427:
4397:
4370:
4343:
4316:
4286:
4189:
4139:
4112:
4085:
4041:
3997:
3954:
3869:
3823:
3789:
3581:
3544:
3363:
3331:
3301:
3271:
3244:
3217:
3184:
3026:
2985:
2955:
2928:
2894:
2777:
2722:
2648:
2619:
2500:
2473:
2400:
2385:is a normal form of
2159:
2077:if some monomial of
2014:Gaussian elimination
1924:
1779:
1718:
1612:
1585:
1533:
1448:
1390:
1320:
1250:
1131:Elimination ordering
1059:
1014:
927:
869:
792:
686:
628:
566:
539:
472:
437:
402:
333:
272:
189:
100:for linear systems.
98:Gaussian elimination
31:, and computational
13537:Commutative algebra
13399:Septic equation (7)
13394:Sextic equation (6)
13341:Linear function (1)
13090:ACM SIGSAM Bulletin
13006:2010SchpJ...5.7763B
12839:Li, Huishi (2011).
11417:real-root isolation
8770:, which means that
8419:Intersecting ideals
7482:for more details).
7223:, and proving that
6892:On the other hand,
6161:A Gröbner basis is
6137:A Gröbner basis is
6048:{\displaystyle R/I}
5841:{\displaystyle R=F}
4865:of two polynomials
3722:
3644:
3612:
3492:
3426:
3394:
3057:
2613:lexicographic order
2299:Given a finite set
1872:
1825:
1700:
1662:
1498:{\displaystyle B=.}
1375:
1350:
1305:
1280:
1181:leading coefficient
1090:admissible ordering
735:
710:
521:
496:
322:{\displaystyle R=K}
239:{\displaystyle R=K}
75:algebraic varieties
33:commutative algebra
13532:Algebraic geometry
13365:Cubic function (3)
13358:Quadratic equation
13253:Weisstein, Eric W.
12951:10.1007/BF01844169
12787:. pp. 361–5.
12339:
12301:
12264:
12218:
12173:with the ideal of
12163:
12109:
12063:
11933:
11873:
11830:
11784:
11728:
11685:
11608:
11546:
11485:
11244:modular arithmetic
11189:
11169:
11140:
11120:
11093:
11062:
11031:
10996:
10965:
10851:
10805:
10759:
10702:
10653:
10651:
10460:
10433:
10345:
10315:
10282:
10232:
10183:
10115:
10070:
10003:
9940:
9861:
9768:
9734:
9677:
9646:
9616:
9566:
9442:
9386:
9274:
9164:implicit equations
9142:
9106:
9062:
9026:
8987:
8985:
8820:that has a set of
8795:
8746:
8720:
8694:
8625:
8404:
8305:
8257:
8203:algebraic geometry
8181:
8134:
8095:
8060:
7989:
7883:elimination theory
7852:
7820:
7791:
7755:
7642:
7615:
7263:Equality of ideals
7185:
7183:
6875:
6843:
6809:
6715:
6686:
6588:
6533:
6476:
6435:
6393:
6355:
6320:
6308:
6288:
6210:
6057:modular arithmetic
6045:
6004:
5902:or, equivalently,
5838:
5762:
5570:
5386:
5381:
5022:
4979:
4873:is the polynomial
4837:
4807:
4776:
4716:
4410:
4383:
4356:
4329:
4302:
4265:
4165:
4125:
4108:can be reduced to
4098:
4067:
4023:
3993:This implies that
3983:
3950:may be reduced to
3919:
3855:
3802:
3785:Again, the result
3772:
3557:
3527:
3347:
3317:
3287:
3257:
3230:
3203:
3167:
2998:
2971:
2941:
2910:
2870:
2868:
2763:
2708:
2634:
2574:Euclidean division
2550:
2486:
2465:is irreducible by
2448:
2430:
2255:
2138:one-step reduction
1986:
1873:
1829:
1782:
1731:
1704:
1666:
1628:
1598:
1559:
1495:
1437:{\displaystyle A=}
1434:
1376:
1354:
1329:
1306:
1284:
1259:
1123:, commonly called
1109:, commonly called
1074:
1044:
968:
913:
852:ideal generated by
836:
752:
714:
689:
675:{\displaystyle X=}
672:
613:{\displaystyle A=}
610:
552:
525:
500:
475:
450:
415:
388:
319:
259:field of rationals
236:
90:Euclid's algorithm
29:algebraic geometry
13552:Rewriting systems
13519:
13518:
13460:Quasi-homogeneous
13223:978-3-540-45654-4
13118:(1962), 292–296).
13045:(November 2005).
12980:978-0-521-63298-0
12926:Buchberger, Bruno
12882:Buchberger, Bruno
12854:978-981-4365-13-0
12794:978-3-540-93805-7
12731:10.1109/18.333885
12451:978-3-540-12868-7
12274:and the products
12221:{\displaystyle R}
11888:EXPSPACE-complete
11886:Gröbner basis is
11740:little o notation
11558:homogeneous ideal
11269:monomial ordering
11221:memory management
11192:{\displaystyle G}
11143:{\displaystyle G}
11096:{\displaystyle I}
10644:
10587:
10484:algebraic closure
10132:Instead of using
9188:
8978:
8899:
8107:elimination ideal
7750:
7418:zero of the ideal
7253:monomial ordering
6311:{\displaystyle g}
6291:{\displaystyle f}
6264:monic polynomials
5753:
5698:
5561:
5528:
5497:
5464:
5373:
5340:
5309:
5276:
5233:
5200:
5145:
5112:
4704:
4648:
4587:
4505:
3723:
3645:
3613:
3493:
3427:
3395:
3180:The leading term
3058:
2890:The leading term
2415:
2339:by an element of
2246:
2221:
2128:is a multiple of
2065:. The polynomial
2057:is a multiple of
1981:
1729:
1623:
1596:
981:Monomial ordering
766:monomial ordering
63:algebraic variety
13559:
13547:Invariant theory
13542:Computer algebra
13382:Quartic equation
13303:
13296:
13289:
13280:
13279:
13266:
13265:
13247:Bruno Buchberger
13236:
13227:
13209:
13199:
13168:
13158:
13148:
13113:
13087:
13077:
13071:
13063:
13051:
13043:Sturmfels, Bernd
13038:
13019:
13017:
12984:
12962:
12934:
12921:
12919:
12896:— (2006).
12894:
12892:
12877:
12858:
12845:World Scientific
12835:
12799:
12798:
12776:
12770:
12769:
12741:
12735:
12734:
12708:
12702:
12701:
12700:
12675:
12669:
12668:
12662:
12654:
12644:
12626:
12617:
12616:
12606:
12596:
12569:
12563:
12562:
12560:
12536:
12530:
12529:
12510:; Little, John;
12504:
12498:
12497:
12471:
12462:
12456:
12455:
12434:Computer Algebra
12426:
12352:
12348:
12346:
12345:
12340:
12310:
12308:
12307:
12302:
12300:
12299:
12290:
12289:
12273:
12271:
12270:
12265:
12263:
12262:
12244:
12243:
12227:
12225:
12224:
12219:
12214:
12213:
12195:
12194:
12172:
12170:
12169:
12164:
12162:
12161:
12143:
12142:
12126:
12118:
12116:
12115:
12110:
12108:
12107:
12089:
12088:
12072:
12070:
12069:
12064:
12062:
12058:
12033:
12028:
12027:
12018:
12017:
12000:
11992:
11991:
11973:
11972:
11950:
11946:
11942:
11940:
11939:
11934:
11913:
11909:
11882:
11880:
11879:
11874:
11872:
11871:
11870:
11869:
11839:
11837:
11836:
11831:
11826:
11825:
11824:
11823:
11793:
11791:
11790:
11785:
11780:
11779:
11778:
11777:
11737:
11735:
11734:
11729:
11724:
11723:
11722:
11721:
11700:
11694:
11692:
11691:
11686:
11681:
11680:
11659:
11658:
11640:
11639:
11617:
11615:
11614:
11609:
11601:
11600:
11578:
11563:
11555:
11553:
11552:
11547:
11542:
11541:
11517:
11516:
11494:
11492:
11491:
11486:
11481:
11480:
11458:
11450:
11446:
11324:regular sequence
11213:Bruno Buchberger
11198:
11196:
11195:
11190:
11178:
11176:
11175:
11170:
11168:
11167:
11149:
11147:
11146:
11141:
11129:
11127:
11126:
11121:
11119:
11118:
11102:
11100:
11099:
11094:
11071:
11069:
11068:
11063:
11061:
11060:
11040:
11038:
11037:
11032:
11030:
11029:
11005:
11003:
11002:
10997:
10995:
10994:
10974:
10972:
10971:
10966:
10961:
10957:
10956:
10955:
10943:
10942:
10933:
10932:
10914:
10913:
10901:
10900:
10891:
10890:
10860:
10858:
10857:
10852:
10850:
10849:
10831:
10830:
10814:
10812:
10811:
10806:
10804:
10803:
10785:
10784:
10768:
10766:
10765:
10760:
10755:
10754:
10736:
10735:
10711:
10709:
10708:
10703:
10701:
10700:
10682:
10681:
10662:
10660:
10659:
10654:
10652:
10645:
10643:
10642:
10633:
10632:
10623:
10614:
10613:
10592:
10588:
10586:
10585:
10576:
10575:
10566:
10557:
10556:
10524:rational variety
10469:
10467:
10466:
10461:
10459:
10458:
10442:
10440:
10439:
10434:
10429:
10428:
10419:
10418:
10394:
10393:
10384:
10383:
10354:
10352:
10351:
10346:
10341:
10340:
10324:
10322:
10321:
10316:
10311:
10310:
10291:
10289:
10288:
10283:
10281:
10280:
10268:
10267:
10241:
10239:
10238:
10233:
10228:
10227:
10215:
10214:
10192:
10190:
10189:
10184:
10182:
10181:
10163:
10162:
10124:
10122:
10121:
10116:
10079:
10077:
10076:
10071:
10012:
10010:
10009:
10004:
10002:
10001:
9964:
9949:
9947:
9946:
9941:
9924:
9923:
9893:
9889:
9882:
9870:
9868:
9867:
9862:
9848:
9847:
9835:
9800:
9799:
9778:It is the ideal
9777:
9775:
9774:
9769:
9764:
9763:
9743:
9741:
9740:
9735:
9730:
9729:
9709:
9702:with respect to
9700:
9699:
9686:
9684:
9683:
9678:
9676:
9675:
9655:
9653:
9652:
9647:
9642:
9641:
9625:
9623:
9622:
9617:
9612:
9611:
9591:
9575:
9573:
9572:
9567:
9544:
9524:
9523:
9479:This is done by
9451:
9449:
9448:
9443:
9438:
9437:
9425:
9424:
9395:
9393:
9392:
9387:
9379:
9378:
9366:
9365:
9356:
9355:
9337:
9336:
9324:
9323:
9314:
9313:
9283:
9281:
9280:
9275:
9267:
9266:
9254:
9253:
9244:
9243:
9231:
9230:
9218:
9217:
9208:
9207:
9195:
9194:
9189:
9186:
9151:
9149:
9148:
9143:
9132:
9131:
9115:
9113:
9112:
9107:
9096:
9095:
9071:
9069:
9068:
9063:
9052:
9051:
9035:
9033:
9032:
9027:
9016:
9015:
8996:
8994:
8993:
8988:
8986:
8979:
8977:
8967:
8966:
8956:
8946:
8945:
8935:
8926:
8925:
8904:
8900:
8898:
8888:
8887:
8877:
8867:
8866:
8856:
8847:
8846:
8804:
8802:
8801:
8796:
8769:
8755:
8753:
8752:
8747:
8729:
8727:
8726:
8721:
8703:
8701:
8700:
8695:
8647:
8638:In other words,
8634:
8632:
8631:
8626:
8618:
8617:
8584:
8583:
8556:
8555:
8534:
8533:
8498:
8472:
8430:
8426:
8413:
8411:
8410:
8405:
8397:
8396:
8378:
8377:
8353:
8352:
8334:
8333:
8314:
8312:
8311:
8306:
8304:
8303:
8285:
8284:
8266:
8264:
8263:
8258:
8190:
8188:
8187:
8182:
8155:
8149:
8143:
8141:
8140:
8135:
8104:
8102:
8101:
8096:
8069:
8067:
8066:
8061:
7998:
7996:
7995:
7990:
7961:
7960:
7942:
7941:
7929:
7928:
7910:
7909:
7861:
7859:
7858:
7853:
7829:
7827:
7826:
7821:
7800:
7798:
7797:
7792:
7764:
7762:
7761:
7756:
7751:
7749:
7748:
7747:
7725:
7711:
7706:
7705:
7696:
7695:
7685:
7680:
7651:
7649:
7648:
7643:
7641:
7640:
7624:
7622:
7621:
7616:
7614:
7613:
7604:
7603:
7593:
7588:
7528:general position
7518:of the zeros of
7392:
7369:
7365:
7361:
7350:
7346:
7328:
7317:
7306:
7295:
7284:of a polynomial
7238:
7222:
7218:
7211:
7207:
7194:
7192:
7191:
7186:
7184:
7119:
7118:
7103:
7102:
7074:
7073:
6963:
6962:
6911:
6907:
6888:
6884:
6882:
6881:
6876:
6852:
6850:
6849:
6844:
6818:
6816:
6815:
6810:
6783:
6779:
6775:
6762:
6750:
6738:
6724:
6722:
6721:
6716:
6695:
6693:
6692:
6687:
6676:
6675:
6620:
6616:
6613:is reducible by
6612:
6608:
6604:
6597:
6595:
6594:
6589:
6542:
6540:
6539:
6534:
6501:
6497:
6493:
6485:
6483:
6482:
6477:
6469:
6468:
6444:
6442:
6441:
6436:
6428:
6427:
6402:
6400:
6399:
6394:
6364:
6362:
6361:
6356:
6339:
6317:
6315:
6314:
6309:
6297:
6295:
6294:
6289:
6254:
6222:rational numbers
6219:
6217:
6216:
6211:
6209:
6183:
6179:
6172:
6167:
6166:
6152:
6148:
6143:
6142:
6101:
6097:
6089:
6085:
6074:
6070:
6054:
6052:
6051:
6046:
6041:
6021:ideal membership
6013:
6011:
6010:
6005:
5997:
5982:
5978:
5971:
5964:
5960:
5956:
5952:
5948:
5941:
5937:
5933:
5929:
5914:
5910:
5897:
5893:
5885:
5881:
5877:
5866:
5862:
5855:
5847:
5845:
5844:
5839:
5834:
5833:
5815:
5814:
5771:
5769:
5768:
5763:
5754:
5752:
5741:
5724:
5699:
5697:
5686:
5669:
5609:
5605:
5601:
5594:
5590:
5586:
5579:
5577:
5576:
5571:
5562:
5560:
5549:
5532:
5529:
5527:
5507:
5498:
5496:
5485:
5468:
5465:
5463:
5443:
5406:
5402:
5395:
5393:
5392:
5387:
5382:
5374:
5372:
5355:
5344:
5341:
5339:
5319:
5310:
5308:
5291:
5280:
5277:
5275:
5255:
5247:
5243:
5239:
5234:
5232:
5215:
5204:
5201:
5199:
5179:
5174:
5155:
5151:
5146:
5144:
5127:
5116:
5113:
5111:
5091:
5086:
5031:
5029:
5028:
5023:
5021:
5020:
5004:
5000:
4996:
4988:
4986:
4985:
4980:
4969:
4952:
4951:
4930:
4913:
4912:
4872:
4868:
4846:
4844:
4843:
4838:
4833:
4832:
4816:
4814:
4813:
4808:
4806:
4805:
4789:
4785:
4783:
4782:
4777:
4772:
4771:
4759:
4758:
4746:
4745:
4725:
4723:
4722:
4717:
4715:
4714:
4705:
4703:
4699:
4698:
4679:
4675:
4674:
4664:
4659:
4658:
4649:
4647:
4643:
4642:
4623:
4619:
4618:
4608:
4603:
4599:
4598:
4597:
4588:
4586:
4582:
4581:
4562:
4558:
4557:
4547:
4539:
4538:
4521:
4517:
4516:
4515:
4506:
4504:
4500:
4499:
4480:
4476:
4475:
4465:
4457:
4456:
4439:
4438:
4419:
4417:
4416:
4411:
4409:
4408:
4392:
4390:
4389:
4384:
4382:
4381:
4365:
4363:
4362:
4357:
4355:
4354:
4338:
4336:
4335:
4330:
4328:
4327:
4311:
4309:
4308:
4303:
4298:
4297:
4281:
4274:
4272:
4271:
4266:
4255:
4254:
4233:
4232:
4217:
4216:
4201:
4200:
4181:
4174:
4172:
4171:
4166:
4164:
4163:
4151:
4150:
4134:
4132:
4131:
4126:
4124:
4123:
4107:
4105:
4104:
4099:
4097:
4096:
4080:
4076:
4074:
4073:
4068:
4066:
4065:
4053:
4052:
4036:
4032:
4030:
4029:
4024:
4022:
4021:
4009:
4008:
3992:
3990:
3989:
3984:
3979:
3978:
3966:
3965:
3949:
3938:
3928:
3926:
3925:
3920:
3906:
3905:
3881:
3880:
3864:
3862:
3861:
3856:
3848:
3847:
3835:
3834:
3818:
3811:
3809:
3808:
3803:
3801:
3800:
3781:
3779:
3778:
3773:
3759:
3758:
3734:
3733:
3721:
3720:
3701:
3690:
3689:
3665:
3664:
3643:
3642:
3626:
3624:
3623:
3611:
3610:
3588:
3570:
3566:
3564:
3563:
3558:
3556:
3555:
3536:
3534:
3533:
3528:
3517:
3516:
3504:
3503:
3491:
3490:
3471:
3460:
3459:
3447:
3446:
3425:
3424:
3408:
3406:
3405:
3393:
3392:
3370:
3356:
3354:
3353:
3348:
3343:
3342:
3326:
3324:
3323:
3318:
3313:
3312:
3296:
3294:
3293:
3288:
3283:
3282:
3266:
3264:
3263:
3258:
3256:
3255:
3239:
3237:
3236:
3231:
3229:
3228:
3212:
3210:
3209:
3204:
3199:
3198:
3176:
3174:
3173:
3168:
3154:
3153:
3132:
3131:
3110:
3109:
3094:
3093:
3069:
3068:
3056:
3055:
3033:
3018:
3014:
3007:
3005:
3004:
2999:
2997:
2996:
2980:
2978:
2977:
2972:
2967:
2966:
2950:
2948:
2947:
2942:
2940:
2939:
2924:is reducible by
2923:
2919:
2917:
2916:
2911:
2909:
2908:
2885:
2879:
2877:
2876:
2871:
2869:
2846:
2845:
2823:
2822:
2810:
2809:
2793:
2792:
2772:
2770:
2769:
2764:
2756:
2755:
2743:
2742:
2717:
2715:
2714:
2709:
2698:
2697:
2682:
2681:
2669:
2668:
2643:
2641:
2640:
2635:
2594:
2583:
2579:
2571:
2567:
2563:
2559:
2557:
2556:
2551:
2521:
2520:
2495:
2493:
2492:
2487:
2485:
2484:
2468:
2464:
2457:
2455:
2454:
2449:
2440:
2439:
2429:
2392:
2388:
2384:
2377:
2373:
2369:
2365:
2357:
2353:
2349:
2342:
2338:
2334:
2330:
2326:
2322:
2318:
2306:
2302:
2295:
2287:
2279:
2275:
2271:
2264:
2262:
2261:
2256:
2247:
2245:
2225:
2222:
2220:
2200:
2171:
2170:
2151:
2147:
2143:
2135:
2127:
2123:
2119:
2115:
2107:
2100:
2096:
2092:
2088:
2080:
2076:
2068:
2064:
2056:
2048:
2040:
1995:
1993:
1992:
1987:
1982:
1980:
1960:
1952:
1913:
1909:
1905:
1890:
1886:
1882:
1880:
1879:
1874:
1871:
1867:
1866:
1854:
1853:
1837:
1824:
1820:
1819:
1807:
1806:
1790:
1775:is the monomial
1774:
1770:
1766:
1748:
1744:
1740:
1738:
1737:
1732:
1730:
1722:
1713:
1711:
1710:
1705:
1699:
1698:
1697:
1685:
1684:
1674:
1661:
1660:
1659:
1647:
1646:
1636:
1624:
1616:
1607:
1605:
1604:
1599:
1597:
1589:
1580:
1576:
1572:
1568:
1566:
1565:
1560:
1558:
1557:
1545:
1544:
1528:
1520:
1516:
1510:
1504:
1502:
1501:
1496:
1488:
1487:
1469:
1468:
1443:
1441:
1440:
1435:
1430:
1429:
1411:
1410:
1385:
1383:
1382:
1377:
1374:
1373:
1372:
1362:
1349:
1348:
1347:
1337:
1315:
1313:
1312:
1307:
1304:
1303:
1302:
1292:
1279:
1278:
1277:
1267:
1236:
1232:
1202:
1194:
1177:leading monomial
1170:
1150:Basic operations
1083:
1081:
1080:
1075:
1053:
1051:
1050:
1045:
1006:
1002:
998:
977:
975:
974:
969:
958:
957:
939:
938:
922:
920:
919:
914:
912:
911:
902:
901:
891:
886:
864:
860:
856:
849:
845:
843:
842:
837:
832:
831:
813:
812:
761:
759:
758:
753:
748:
747:
734:
733:
732:
722:
709:
708:
707:
697:
681:
679:
678:
673:
668:
667:
649:
648:
619:
617:
616:
611:
606:
605:
587:
586:
561:
559:
558:
553:
551:
550:
534:
532:
531:
526:
520:
519:
518:
508:
495:
494:
493:
483:
459:
457:
456:
451:
449:
448:
428:
424:
422:
421:
416:
414:
413:
397:
395:
394:
389:
384:
383:
374:
373:
355:
354:
345:
344:
328:
326:
325:
320:
315:
314:
296:
295:
256:
252:
245:
243:
242:
237:
232:
231:
213:
212:
153:polynomial rings
138:Heisuke Hironaka
113:Wolfgang Gröbner
105:Bruno Buchberger
60:
51:
27:, computational
25:computer algebra
13567:
13566:
13562:
13561:
13560:
13558:
13557:
13556:
13522:
13521:
13520:
13515:
13464:
13403:
13346:Linear equation
13316:
13307:
13256:"Gröbner Basis"
13224:
13207:
13186:"Gröbner basis"
13184:
13176:
13139:(11): 5171–92.
13085:
13065:
13064:
13049:
13035:
12990:"Gröbner Bases"
12981:
12932:
12890:
12874:
12855:
12832:
12820:. Vol. 3.
12808:
12806:Further reading
12803:
12802:
12795:
12777:
12773:
12742:
12738:
12709:
12705:
12676:
12672:
12659:cite conference
12656:
12655:
12627:
12620:
12570:
12566:
12537:
12533:
12526:
12505:
12501:
12469:
12463:
12459:
12452:
12427:
12418:
12413:
12384:
12375:
12370:
12350:
12316:
12313:
12312:
12295:
12291:
12285:
12281:
12279:
12276:
12275:
12258:
12254:
12239:
12235:
12233:
12230:
12229:
12209:
12205:
12190:
12186:
12178:
12175:
12174:
12157:
12153:
12138:
12134:
12132:
12129:
12128:
12124:
12103:
12099:
12084:
12080:
12078:
12075:
12074:
12029:
12023:
12019:
12013:
12009:
12005:
12001:
11996:
11987:
11983:
11968:
11964:
11956:
11953:
11952:
11948:
11944:
11922:
11919:
11918:
11911:
11907:
11896:
11894:Generalizations
11856:
11852:
11851:
11847:
11845:
11842:
11841:
11810:
11806:
11805:
11801:
11799:
11796:
11795:
11758:
11754:
11753:
11749:
11747:
11744:
11743:
11717:
11713:
11712:
11708:
11706:
11703:
11702:
11698:
11667:
11663:
11645:
11641:
11635:
11631:
11623:
11620:
11619:
11596:
11592:
11584:
11581:
11580:
11573:
11561:
11528:
11524:
11512:
11508:
11500:
11497:
11496:
11476:
11472:
11464:
11461:
11460:
11456:
11448:
11444:
11437:
11206:
11184:
11181:
11180:
11163:
11159:
11157:
11154:
11153:
11135:
11132:
11131:
11114:
11110:
11108:
11105:
11104:
11088:
11085:
11084:
11056:
11052:
11050:
11047:
11046:
11025:
11021:
11019:
11016:
11015:
10990:
10986:
10984:
10981:
10980:
10951:
10947:
10938:
10934:
10928:
10924:
10909:
10905:
10896:
10892:
10886:
10882:
10881:
10877:
10869:
10866:
10865:
10845:
10841:
10826:
10822:
10820:
10817:
10816:
10799:
10795:
10780:
10776:
10774:
10771:
10770:
10750:
10746:
10731:
10727:
10725:
10722:
10721:
10696:
10692:
10677:
10673:
10671:
10668:
10667:
10650:
10649:
10638:
10634:
10628:
10624:
10622:
10615:
10609:
10605:
10602:
10601:
10590:
10589:
10581:
10577:
10571:
10567:
10565:
10558:
10552:
10548:
10544:
10542:
10539:
10538:
10520:
10477:
10454:
10450:
10448:
10445:
10444:
10424:
10420:
10414:
10410:
10389:
10385:
10379:
10375:
10367:
10364:
10363:
10336:
10332:
10330:
10327:
10326:
10306:
10302:
10300:
10297:
10296:
10276:
10272:
10263:
10259:
10251:
10248:
10247:
10223:
10219:
10210:
10206:
10198:
10195:
10194:
10177:
10173:
10158:
10154:
10152:
10149:
10148:
10089:
10086:
10085:
10041:
10038:
10037:
10022:
9997:
9993:
9985:
9982:
9981:
9962:
9919:
9915:
9907:
9904:
9903:
9891:
9887:
9880:
9843:
9839:
9831:
9795:
9791:
9783:
9780:
9779:
9759:
9755:
9753:
9750:
9749:
9725:
9721:
9719:
9716:
9715:
9707:
9697:
9696:
9671:
9667:
9665:
9662:
9661:
9637:
9633:
9631:
9628:
9627:
9607:
9603:
9601:
9598:
9597:
9589:
9540:
9519:
9515:
9513:
9510:
9509:
9489:
9462:
9433:
9429:
9420:
9416:
9405:
9402:
9401:
9374:
9370:
9361:
9357:
9351:
9347:
9332:
9328:
9319:
9315:
9309:
9305:
9300:
9297:
9296:
9262:
9258:
9249:
9245:
9239:
9235:
9226:
9222:
9213:
9209:
9203:
9199:
9190:
9185:
9184:
9182:
9179:
9178:
9160:Implicitization
9127:
9123:
9121:
9118:
9117:
9091:
9087:
9085:
9082:
9081:
9047:
9043:
9041:
9038:
9037:
9011:
9007:
9005:
9002:
9001:
8984:
8983:
8962:
8958:
8957:
8941:
8937:
8936:
8934:
8927:
8921:
8917:
8914:
8913:
8902:
8901:
8883:
8879:
8878:
8862:
8858:
8857:
8855:
8848:
8842:
8838:
8834:
8832:
8829:
8828:
8818:algebraic curve
8810:
8775:
8772:
8771:
8761:
8760:if and only if
8735:
8732:
8731:
8709:
8706:
8705:
8668:
8665:
8664:
8648:is obtained by
8639:
8613:
8609:
8579:
8575:
8551:
8547:
8529:
8525:
8511:
8508:
8507:
8490:
8464:
8462:
8453:
8446:
8437:
8428:
8424:
8421:
8392:
8388:
8367:
8363:
8348:
8344:
8329:
8325:
8320:
8317:
8316:
8299:
8295:
8280:
8276:
8274:
8271:
8270:
8234:
8231:
8230:
8219:Zariski closure
8161:
8158:
8157:
8151:
8145:
8114:
8111:
8110:
8075:
8072:
8071:
8040:
8037:
8036:
7956:
7952:
7937:
7933:
7924:
7920:
7905:
7901:
7893:
7890:
7889:
7875:
7838:
7835:
7834:
7806:
7803:
7802:
7777:
7774:
7773:
7743:
7739:
7726:
7712:
7710:
7701:
7697:
7691:
7687:
7681:
7670:
7664:
7661:
7660:
7636:
7632:
7630:
7627:
7626:
7609:
7605:
7599:
7595:
7589:
7578:
7572:
7569:
7568:
7504:Krull dimension
7488:
7405:
7399:
7390:
7381:
7374:
7371:
7367:
7363:
7360:
7359:
7355:
7352:
7348:
7345:
7336:
7330:
7326:
7308:
7304:
7293:
7278:
7265:
7249:
7224:
7220:
7216:
7209:
7199:
7182:
7181:
7123:
7114:
7110:
7098:
7094:
7091:
7090:
7069:
7065:
7028:
7010:
7009:
6958:
6954:
6941:
6922:
6920:
6917:
6916:
6909:
6893:
6886:
6858:
6855:
6854:
6823:
6820:
6819:
6789:
6786:
6785:
6781:
6777:
6767:
6753:
6741:
6729:
6704:
6701:
6700:
6671:
6667:
6629:
6626:
6625:
6618:
6614:
6610:
6606:
6602:
6550:
6547:
6546:
6507:
6504:
6503:
6499:
6495:
6491:
6464:
6460:
6452:
6449:
6448:
6423:
6419:
6411:
6408:
6407:
6370:
6367:
6366:
6335:
6327:
6324:
6323:
6303:
6300:
6299:
6283:
6280:
6279:
6272:
6252:
6234:
6205:
6203:
6200:
6199:
6181:
6177:
6170:
6164:
6163:
6150:
6146:
6140:
6139:
6135:
6107:Dickson's lemma
6099:
6095:
6087:
6083:
6072:
6068:
6065:
6037:
6032:
6029:
6028:
6016:
5993:
5988:
5985:
5984:
5980:
5976:
5969:
5962:
5958:
5954:
5953:of elements of
5950:
5946:
5939:
5935:
5931:
5927:
5918:
5912:
5908:
5895:
5891:
5883:
5879:
5875:
5864:
5860:
5853:
5850:polynomial ring
5829:
5825:
5810:
5806:
5792:
5789:
5788:
5785:
5742:
5725:
5723:
5687:
5670:
5668:
5629:
5626:
5625:
5607:
5603:
5599:
5592:
5588:
5584:
5550:
5533:
5531:
5511:
5506:
5486:
5469:
5467:
5447:
5442:
5419:
5416:
5415:
5404:
5400:
5380:
5379:
5356:
5345:
5343:
5323:
5318:
5292:
5281:
5279:
5259:
5254:
5245:
5244:
5216:
5205:
5203:
5183:
5178:
5164:
5163:
5159:
5128:
5117:
5115:
5095:
5090:
5076:
5075:
5071:
5064:
5042:
5040:
5037:
5036:
5016:
5012:
5010:
5007:
5006:
5002:
4998:
4994:
4959:
4947:
4943:
4920:
4908:
4904:
4881:
4878:
4877:
4870:
4866:
4855:
4828:
4824:
4822:
4819:
4818:
4801:
4797:
4795:
4792:
4791:
4787:
4767:
4763:
4754:
4750:
4741:
4737:
4735:
4732:
4731:
4710:
4706:
4694:
4690:
4680:
4670:
4666:
4665:
4663:
4654:
4650:
4638:
4634:
4624:
4614:
4610:
4609:
4607:
4593:
4589:
4577:
4573:
4563:
4553:
4549:
4548:
4546:
4534:
4530:
4529:
4525:
4511:
4507:
4495:
4491:
4481:
4471:
4467:
4466:
4464:
4452:
4448:
4447:
4443:
4434:
4430:
4428:
4425:
4424:
4404:
4400:
4398:
4395:
4394:
4377:
4373:
4371:
4368:
4367:
4350:
4346:
4344:
4341:
4340:
4323:
4319:
4317:
4314:
4313:
4293:
4289:
4287:
4284:
4283:
4279:
4250:
4246:
4228:
4224:
4212:
4208:
4196:
4192:
4190:
4187:
4186:
4182:the polynomial
4179:
4159:
4155:
4146:
4142:
4140:
4137:
4136:
4119:
4115:
4113:
4110:
4109:
4092:
4088:
4086:
4083:
4082:
4078:
4061:
4057:
4048:
4044:
4042:
4039:
4038:
4034:
4017:
4013:
4004:
4000:
3998:
3995:
3994:
3974:
3970:
3961:
3957:
3955:
3952:
3951:
3940:
3936:
3901:
3897:
3876:
3872:
3870:
3867:
3866:
3843:
3839:
3830:
3826:
3824:
3821:
3820:
3816:
3796:
3792:
3790:
3787:
3786:
3754:
3750:
3729:
3725:
3716:
3712:
3685:
3681:
3660:
3656:
3638:
3634:
3619:
3615:
3606:
3602:
3582:
3579:
3578:
3568:
3551:
3547:
3545:
3542:
3541:
3512:
3508:
3499:
3495:
3486:
3482:
3455:
3451:
3442:
3438:
3420:
3416:
3401:
3397:
3388:
3384:
3364:
3361:
3360:
3338:
3334:
3332:
3329:
3328:
3308:
3304:
3302:
3299:
3298:
3278:
3274:
3272:
3269:
3268:
3251:
3247:
3245:
3242:
3241:
3224:
3220:
3218:
3215:
3214:
3194:
3190:
3185:
3182:
3181:
3149:
3145:
3127:
3123:
3105:
3101:
3089:
3085:
3064:
3060:
3051:
3047:
3027:
3024:
3023:
3016:
3009:
2992:
2988:
2986:
2983:
2982:
2962:
2958:
2956:
2953:
2952:
2935:
2931:
2929:
2926:
2925:
2921:
2904:
2900:
2895:
2892:
2891:
2883:
2867:
2866:
2847:
2841:
2837:
2834:
2833:
2818:
2814:
2805:
2801:
2794:
2788:
2784:
2780:
2778:
2775:
2774:
2751:
2747:
2738:
2734:
2723:
2720:
2719:
2693:
2689:
2677:
2673:
2664:
2660:
2649:
2646:
2645:
2620:
2617:
2616:
2605:
2593:
2585:
2581:
2577:
2569:
2565:
2561:
2516:
2512:
2501:
2498:
2497:
2480:
2476:
2474:
2471:
2470:
2466:
2462:
2435:
2431:
2419:
2401:
2398:
2397:
2390:
2386:
2382:
2375:
2371:
2367:
2363:
2355:
2351:
2347:
2340:
2336:
2332:
2328:
2324:
2320:
2316:
2304:
2300:
2289:
2285:
2277:
2273:
2269:
2229:
2224:
2204:
2199:
2166:
2162:
2160:
2157:
2156:
2149:
2145:
2141:
2129:
2125:
2121:
2117:
2113:
2105:
2098:
2094:
2090:
2082:
2078:
2074:
2066:
2058:
2050:
2046:
2038:
2002:
1961:
1953:
1951:
1925:
1922:
1921:
1911:
1907:
1895:
1888:
1884:
1862:
1858:
1849:
1845:
1838:
1833:
1815:
1811:
1802:
1798:
1791:
1786:
1780:
1777:
1776:
1772:
1768:
1756:
1746:
1742:
1721:
1719:
1716:
1715:
1693:
1689:
1680:
1676:
1675:
1670:
1655:
1651:
1642:
1638:
1637:
1632:
1615:
1613:
1610:
1609:
1588:
1586:
1583:
1582:
1578:
1574:
1570:
1553:
1549:
1540:
1536:
1534:
1531:
1530:
1526:
1518:
1514:
1508:
1483:
1479:
1464:
1460:
1449:
1446:
1445:
1425:
1421:
1406:
1402:
1391:
1388:
1387:
1368:
1364:
1363:
1358:
1343:
1339:
1338:
1333:
1321:
1318:
1317:
1298:
1294:
1293:
1288:
1273:
1269:
1268:
1263:
1251:
1248:
1247:
1244:
1234:
1230:
1213:
1196:
1184:
1168:
1157:
1152:
1060:
1057:
1056:
1015:
1012:
1011:
1004:
1000:
996:
989:
983:
953:
949:
934:
930:
928:
925:
924:
907:
903:
897:
893:
887:
876:
870:
867:
866:
862:
858:
854:
847:
827:
823:
808:
804:
793:
790:
789:
743:
739:
728:
724:
723:
718:
703:
699:
698:
693:
687:
684:
683:
663:
659:
644:
640:
629:
626:
625:
622:exponent vector
601:
597:
582:
578:
567:
564:
563:
546:
542:
540:
537:
536:
514:
510:
509:
504:
489:
485:
484:
479:
473:
470:
469:
444:
440:
438:
435:
434:
426:
409:
405:
403:
400:
399:
379:
375:
369:
365:
350:
346:
340:
336:
334:
331:
330:
310:
306:
291:
287:
273:
270:
269:
254:
250:
227:
223:
208:
204:
190:
187:
186:
184:polynomial ring
176:
174:Polynomial ring
170:
168:Polynomial ring
165:
125:Nikolai Günther
56:
47:
45:polynomial ring
17:
12:
11:
5:
13565:
13555:
13554:
13549:
13544:
13539:
13534:
13517:
13516:
13514:
13513:
13508:
13503:
13498:
13493:
13488:
13483:
13478:
13472:
13470:
13466:
13465:
13463:
13462:
13457:
13452:
13447:
13442:
13437:
13432:
13427:
13422:
13417:
13411:
13409:
13405:
13404:
13402:
13401:
13396:
13391:
13386:
13385:
13384:
13374:
13373:
13372:
13370:Cubic equation
13362:
13361:
13360:
13350:
13349:
13348:
13338:
13333:
13327:
13325:
13318:
13317:
13306:
13305:
13298:
13291:
13283:
13277:
13276:
13267:
13248:
13242:
13237:
13228:
13222:
13200:
13182:
13175:
13174:External links
13172:
13171:
13170:
13119:
13078:
13039:
13033:
13020:
12985:
12979:
12922:
12878:
12872:
12859:
12853:
12836:
12830:
12807:
12804:
12801:
12800:
12793:
12771:
12752:(2): 147–158.
12736:
12725:(5): 1654–61.
12703:
12691:(3): 303–325,
12679:Mayr, Ernst W.
12670:
12618:
12564:
12531:
12524:
12499:
12457:
12450:
12430:Lazard, Daniel
12415:
12414:
12412:
12409:
12408:
12407:
12404:Regular chains
12401:
12396:
12391:
12383:
12380:
12374:
12371:
12369:
12366:
12338:
12335:
12332:
12329:
12326:
12323:
12320:
12298:
12294:
12288:
12284:
12261:
12257:
12253:
12250:
12247:
12242:
12238:
12217:
12212:
12208:
12204:
12201:
12198:
12193:
12189:
12185:
12182:
12160:
12156:
12152:
12149:
12146:
12141:
12137:
12119:is a basis of
12106:
12102:
12098:
12095:
12092:
12087:
12083:
12061:
12057:
12054:
12051:
12048:
12045:
12042:
12039:
12036:
12032:
12026:
12022:
12016:
12012:
12008:
12004:
11999:
11995:
11990:
11986:
11982:
11979:
11976:
11971:
11967:
11963:
11960:
11932:
11929:
11926:
11895:
11892:
11868:
11865:
11862:
11859:
11855:
11850:
11840:or containing
11829:
11822:
11819:
11816:
11813:
11809:
11804:
11783:
11776:
11773:
11770:
11767:
11764:
11761:
11757:
11752:
11727:
11720:
11716:
11711:
11684:
11679:
11676:
11673:
11670:
11666:
11662:
11657:
11654:
11651:
11648:
11644:
11638:
11634:
11630:
11627:
11607:
11604:
11599:
11595:
11591:
11588:
11566:linear algebra
11545:
11540:
11537:
11534:
11531:
11527:
11523:
11520:
11515:
11511:
11507:
11504:
11484:
11479:
11475:
11471:
11468:
11436:
11433:
11339:FGLM algorithm
11304:linear algebra
11277:
11276:
11265:
11262:
11255:
11252:Hensel lifting
11224:
11205:
11202:
11201:
11200:
11188:
11166:
11162:
11152:Eliminate the
11150:
11139:
11117:
11113:
11092:
11075:Therefore, if
11059:
11055:
11028:
11024:
10993:
10989:
10976:
10975:
10964:
10960:
10954:
10950:
10946:
10941:
10937:
10931:
10927:
10923:
10920:
10917:
10912:
10908:
10904:
10899:
10895:
10889:
10885:
10880:
10876:
10873:
10848:
10844:
10840:
10837:
10834:
10829:
10825:
10802:
10798:
10794:
10791:
10788:
10783:
10779:
10758:
10753:
10749:
10745:
10742:
10739:
10734:
10730:
10699:
10695:
10691:
10688:
10685:
10680:
10676:
10664:
10663:
10648:
10641:
10637:
10631:
10627:
10621:
10618:
10616:
10612:
10608:
10604:
10603:
10600:
10595:
10593:
10591:
10584:
10580:
10574:
10570:
10564:
10561:
10559:
10555:
10551:
10547:
10546:
10519:
10516:
10476:
10473:
10472:
10471:
10457:
10453:
10432:
10427:
10423:
10417:
10413:
10409:
10406:
10403:
10400:
10397:
10392:
10388:
10382:
10378:
10374:
10371:
10356:
10344:
10339:
10335:
10314:
10309:
10305:
10295:Saturating by
10293:
10279:
10275:
10271:
10266:
10262:
10258:
10255:
10246:Saturating by
10231:
10226:
10222:
10218:
10213:
10209:
10205:
10202:
10180:
10176:
10172:
10169:
10166:
10161:
10157:
10114:
10111:
10108:
10105:
10102:
10099:
10096:
10093:
10069:
10066:
10063:
10060:
10057:
10054:
10051:
10048:
10045:
10021:
10018:
10000:
9996:
9992:
9989:
9939:
9936:
9933:
9930:
9927:
9922:
9918:
9914:
9911:
9860:
9857:
9854:
9851:
9846:
9842:
9838:
9834:
9830:
9827:
9824:
9821:
9818:
9815:
9812:
9809:
9806:
9803:
9798:
9794:
9790:
9787:
9767:
9762:
9758:
9733:
9728:
9724:
9674:
9670:
9645:
9640:
9636:
9615:
9610:
9606:
9565:
9562:
9559:
9556:
9553:
9550:
9547:
9543:
9539:
9536:
9533:
9530:
9527:
9522:
9518:
9504:by an element
9488:
9485:
9461:
9458:
9441:
9436:
9432:
9428:
9423:
9419:
9415:
9412:
9409:
9385:
9382:
9377:
9373:
9369:
9364:
9360:
9354:
9350:
9346:
9343:
9340:
9335:
9331:
9327:
9322:
9318:
9312:
9308:
9304:
9285:
9284:
9273:
9270:
9265:
9261:
9257:
9252:
9248:
9242:
9238:
9234:
9229:
9225:
9221:
9216:
9212:
9206:
9202:
9198:
9193:
9141:
9138:
9135:
9130:
9126:
9105:
9102:
9099:
9094:
9090:
9061:
9058:
9055:
9050:
9046:
9025:
9022:
9019:
9014:
9010:
8998:
8997:
8982:
8976:
8973:
8970:
8965:
8961:
8955:
8952:
8949:
8944:
8940:
8933:
8930:
8928:
8924:
8920:
8916:
8915:
8912:
8907:
8905:
8903:
8897:
8894:
8891:
8886:
8882:
8876:
8873:
8870:
8865:
8861:
8854:
8851:
8849:
8845:
8841:
8837:
8836:
8814:rational curve
8809:
8806:
8794:
8791:
8788:
8785:
8782:
8779:
8745:
8742:
8739:
8719:
8716:
8713:
8693:
8690:
8687:
8684:
8681:
8678:
8675:
8672:
8636:
8635:
8624:
8621:
8616:
8612:
8608:
8605:
8602:
8599:
8596:
8593:
8590:
8587:
8582:
8578:
8574:
8571:
8568:
8565:
8562:
8559:
8554:
8550:
8546:
8543:
8540:
8537:
8532:
8528:
8524:
8521:
8518:
8515:
8458:
8451:
8442:
8435:
8420:
8417:
8403:
8400:
8395:
8391:
8387:
8384:
8381:
8376:
8373:
8370:
8366:
8362:
8359:
8356:
8351:
8347:
8343:
8340:
8337:
8332:
8328:
8324:
8302:
8298:
8294:
8291:
8288:
8283:
8279:
8256:
8253:
8250:
8247:
8244:
8241:
8238:
8180:
8177:
8174:
8171:
8168:
8165:
8133:
8130:
8127:
8124:
8121:
8118:
8094:
8091:
8088:
8085:
8082:
8079:
8059:
8056:
8053:
8050:
8047:
8044:
7988:
7985:
7982:
7979:
7976:
7973:
7970:
7967:
7964:
7959:
7955:
7951:
7948:
7945:
7940:
7936:
7932:
7927:
7923:
7919:
7916:
7913:
7908:
7904:
7900:
7897:
7874:
7871:
7851:
7848:
7845:
7842:
7819:
7816:
7813:
7810:
7790:
7787:
7784:
7781:
7766:
7765:
7754:
7746:
7742:
7738:
7735:
7732:
7729:
7724:
7721:
7718:
7715:
7709:
7704:
7700:
7694:
7690:
7684:
7679:
7676:
7673:
7669:
7639:
7635:
7612:
7608:
7602:
7598:
7592:
7587:
7584:
7581:
7577:
7565:Hilbert series
7487:
7484:
7470:of the ideal.
7401:Main article:
7398:
7395:
7386:
7379:
7372:
7357:
7356:
7353:
7341:
7334:
7298:if and only if
7277:
7274:
7264:
7261:
7248:
7245:
7196:
7195:
7180:
7177:
7174:
7171:
7168:
7165:
7162:
7159:
7156:
7153:
7150:
7147:
7144:
7141:
7138:
7135:
7132:
7129:
7126:
7124:
7122:
7117:
7113:
7109:
7106:
7101:
7097:
7093:
7092:
7089:
7086:
7083:
7080:
7077:
7072:
7068:
7064:
7061:
7058:
7055:
7052:
7049:
7046:
7043:
7040:
7037:
7034:
7031:
7029:
7027:
7024:
7021:
7018:
7015:
7012:
7011:
7008:
7005:
7002:
6999:
6996:
6993:
6990:
6987:
6984:
6981:
6978:
6975:
6972:
6969:
6966:
6961:
6957:
6953:
6950:
6947:
6944:
6942:
6940:
6937:
6934:
6931:
6928:
6925:
6924:
6874:
6871:
6868:
6865:
6862:
6842:
6839:
6836:
6833:
6830:
6827:
6808:
6805:
6802:
6799:
6796:
6793:
6764:
6763:
6751:
6739:
6714:
6711:
6708:
6697:
6696:
6685:
6682:
6679:
6674:
6670:
6666:
6663:
6660:
6657:
6654:
6651:
6648:
6645:
6642:
6639:
6636:
6633:
6599:
6598:
6587:
6584:
6581:
6578:
6575:
6572:
6569:
6566:
6563:
6560:
6557:
6554:
6532:
6529:
6526:
6523:
6520:
6517:
6514:
6511:
6488:
6487:
6475:
6472:
6467:
6463:
6459:
6456:
6446:
6434:
6431:
6426:
6422:
6418:
6415:
6392:
6389:
6386:
6383:
6380:
6377:
6374:
6354:
6351:
6348:
6345:
6342:
6338:
6334:
6331:
6307:
6287:
6278:The zeroes of
6271:
6268:
6233:
6230:
6208:
6134:
6131:
6064:
6061:
6044:
6040:
6036:
6015:
6014:
6003:
6000:
5996:
5992:
5983:-vector space
5973:
5966:
5965:produces zero;
5943:
5923:
5917:
5916:
5904:
5900:
5899:
5837:
5832:
5828:
5824:
5821:
5818:
5813:
5809:
5805:
5802:
5799:
5796:
5784:
5781:
5773:
5772:
5761:
5758:
5751:
5748:
5745:
5740:
5737:
5734:
5731:
5728:
5721:
5718:
5715:
5712:
5709:
5706:
5703:
5696:
5693:
5690:
5685:
5682:
5679:
5676:
5673:
5666:
5663:
5660:
5657:
5654:
5651:
5648:
5645:
5642:
5639:
5636:
5633:
5581:
5580:
5569:
5566:
5559:
5556:
5553:
5548:
5545:
5542:
5539:
5536:
5526:
5523:
5520:
5517:
5514:
5510:
5505:
5502:
5495:
5492:
5489:
5484:
5481:
5478:
5475:
5472:
5462:
5459:
5456:
5453:
5450:
5446:
5441:
5438:
5435:
5432:
5429:
5426:
5423:
5397:
5396:
5385:
5378:
5371:
5368:
5365:
5362:
5359:
5354:
5351:
5348:
5338:
5335:
5332:
5329:
5326:
5322:
5317:
5314:
5307:
5304:
5301:
5298:
5295:
5290:
5287:
5284:
5274:
5271:
5268:
5265:
5262:
5258:
5253:
5250:
5248:
5246:
5242:
5238:
5231:
5228:
5225:
5222:
5219:
5214:
5211:
5208:
5198:
5195:
5192:
5189:
5186:
5182:
5177:
5173:
5170:
5167:
5162:
5158:
5154:
5150:
5143:
5140:
5137:
5134:
5131:
5126:
5123:
5120:
5110:
5107:
5104:
5101:
5098:
5094:
5089:
5085:
5082:
5079:
5074:
5070:
5067:
5065:
5063:
5060:
5057:
5054:
5051:
5048:
5045:
5044:
5019:
5015:
4991:
4990:
4978:
4975:
4972:
4968:
4965:
4962:
4958:
4955:
4950:
4946:
4942:
4939:
4936:
4933:
4929:
4926:
4923:
4919:
4916:
4911:
4907:
4903:
4900:
4897:
4894:
4891:
4888:
4885:
4854:
4848:
4836:
4831:
4827:
4804:
4800:
4775:
4770:
4766:
4762:
4757:
4753:
4749:
4744:
4740:
4728:
4727:
4713:
4709:
4702:
4697:
4693:
4689:
4686:
4683:
4678:
4673:
4669:
4662:
4657:
4653:
4646:
4641:
4637:
4633:
4630:
4627:
4622:
4617:
4613:
4606:
4602:
4596:
4592:
4585:
4580:
4576:
4572:
4569:
4566:
4561:
4556:
4552:
4545:
4542:
4537:
4533:
4528:
4524:
4520:
4514:
4510:
4503:
4498:
4494:
4490:
4487:
4484:
4479:
4474:
4470:
4463:
4460:
4455:
4451:
4446:
4442:
4437:
4433:
4420:respectively:
4407:
4403:
4380:
4376:
4353:
4349:
4326:
4322:
4301:
4296:
4292:
4276:
4275:
4264:
4261:
4258:
4253:
4249:
4245:
4242:
4239:
4236:
4231:
4227:
4223:
4220:
4215:
4211:
4207:
4204:
4199:
4195:
4162:
4158:
4154:
4149:
4145:
4122:
4118:
4095:
4091:
4064:
4060:
4056:
4051:
4047:
4020:
4016:
4012:
4007:
4003:
3982:
3977:
3973:
3969:
3964:
3960:
3918:
3915:
3912:
3909:
3904:
3900:
3896:
3893:
3890:
3887:
3884:
3879:
3875:
3854:
3851:
3846:
3842:
3838:
3833:
3829:
3799:
3795:
3783:
3782:
3771:
3768:
3765:
3762:
3757:
3753:
3749:
3746:
3743:
3740:
3737:
3732:
3728:
3719:
3715:
3711:
3708:
3704:
3699:
3696:
3693:
3688:
3684:
3680:
3677:
3674:
3671:
3668:
3663:
3659:
3655:
3652:
3649:
3641:
3637:
3633:
3629:
3622:
3618:
3609:
3605:
3601:
3598:
3595:
3591:
3586:
3554:
3550:
3538:
3537:
3526:
3523:
3520:
3515:
3511:
3507:
3502:
3498:
3489:
3485:
3481:
3478:
3474:
3469:
3466:
3463:
3458:
3454:
3450:
3445:
3441:
3437:
3434:
3431:
3423:
3419:
3415:
3411:
3404:
3400:
3391:
3387:
3383:
3380:
3377:
3373:
3368:
3346:
3341:
3337:
3316:
3311:
3307:
3286:
3281:
3277:
3254:
3250:
3227:
3223:
3202:
3197:
3193:
3189:
3178:
3177:
3166:
3163:
3160:
3157:
3152:
3148:
3144:
3141:
3138:
3135:
3130:
3126:
3122:
3119:
3116:
3113:
3108:
3104:
3100:
3097:
3092:
3088:
3084:
3081:
3078:
3075:
3072:
3067:
3063:
3054:
3050:
3046:
3043:
3040:
3036:
3031:
2995:
2991:
2970:
2965:
2961:
2938:
2934:
2907:
2903:
2899:
2865:
2862:
2859:
2856:
2853:
2850:
2848:
2844:
2840:
2836:
2835:
2832:
2829:
2826:
2821:
2817:
2813:
2808:
2804:
2800:
2797:
2795:
2791:
2787:
2783:
2782:
2762:
2759:
2754:
2750:
2746:
2741:
2737:
2733:
2730:
2727:
2707:
2704:
2701:
2696:
2692:
2688:
2685:
2680:
2676:
2672:
2667:
2663:
2659:
2656:
2653:
2633:
2630:
2627:
2624:
2604:
2601:
2589:
2549:
2546:
2543:
2540:
2537:
2534:
2531:
2528:
2525:
2519:
2515:
2511:
2508:
2505:
2483:
2479:
2459:
2458:
2447:
2444:
2438:
2434:
2428:
2425:
2422:
2418:
2414:
2411:
2408:
2405:
2313:lead-reducible
2266:
2265:
2254:
2251:
2244:
2241:
2238:
2235:
2232:
2228:
2219:
2216:
2213:
2210:
2207:
2203:
2198:
2195:
2192:
2189:
2186:
2183:
2180:
2177:
2174:
2169:
2165:
2081:is a multiple
2043:lead-reducible
2028:Lead-reduction
2001:
1998:
1997:
1996:
1985:
1979:
1976:
1973:
1970:
1967:
1964:
1959:
1956:
1950:
1947:
1944:
1941:
1938:
1935:
1932:
1929:
1870:
1865:
1861:
1857:
1852:
1848:
1844:
1841:
1836:
1832:
1828:
1823:
1818:
1814:
1810:
1805:
1801:
1797:
1794:
1789:
1785:
1728:
1725:
1703:
1696:
1692:
1688:
1683:
1679:
1673:
1669:
1665:
1658:
1654:
1650:
1645:
1641:
1635:
1631:
1627:
1622:
1619:
1608:is defined as
1595:
1592:
1573:; that is, if
1556:
1552:
1548:
1543:
1539:
1507:One says that
1494:
1491:
1486:
1482:
1478:
1475:
1472:
1467:
1463:
1459:
1456:
1453:
1433:
1428:
1424:
1420:
1417:
1414:
1409:
1405:
1401:
1398:
1395:
1371:
1367:
1361:
1357:
1353:
1346:
1342:
1336:
1332:
1328:
1325:
1301:
1297:
1291:
1287:
1283:
1276:
1272:
1266:
1262:
1258:
1255:
1243:
1240:
1239:
1238:
1227:
1224:
1212:
1209:
1156:
1153:
1151:
1148:
1139:
1138:
1128:
1118:
1086:
1085:
1073:
1070:
1067:
1064:
1054:
1043:
1040:
1037:
1034:
1031:
1028:
1025:
1022:
1019:
987:Monomial order
985:Main article:
982:
979:
967:
964:
961:
956:
952:
948:
945:
942:
937:
933:
910:
906:
900:
896:
890:
885:
882:
879:
875:
835:
830:
826:
822:
819:
816:
811:
807:
803:
800:
797:
751:
746:
742:
738:
731:
727:
721:
717:
713:
706:
702:
696:
692:
671:
666:
662:
658:
655:
652:
647:
643:
639:
636:
633:
620:is called the
609:
604:
600:
596:
593:
590:
585:
581:
577:
574:
571:
549:
545:
524:
517:
513:
507:
503:
499:
492:
488:
482:
478:
466:power products
447:
443:
412:
408:
387:
382:
378:
372:
368:
364:
361:
358:
353:
349:
343:
339:
318:
313:
309:
305:
302:
299:
294:
290:
286:
283:
280:
277:
235:
230:
226:
222:
219:
216:
211:
207:
203:
200:
197:
194:
172:Main article:
169:
166:
164:
161:
142:standard bases
92:for computing
15:
9:
6:
4:
3:
2:
13564:
13553:
13550:
13548:
13545:
13543:
13540:
13538:
13535:
13533:
13530:
13529:
13527:
13512:
13511:Gröbner basis
13509:
13507:
13504:
13502:
13499:
13497:
13494:
13492:
13489:
13487:
13484:
13482:
13479:
13477:
13476:Factorization
13474:
13473:
13471:
13467:
13461:
13458:
13456:
13453:
13451:
13448:
13446:
13443:
13441:
13438:
13436:
13433:
13431:
13428:
13426:
13423:
13421:
13418:
13416:
13413:
13412:
13410:
13408:By properties
13406:
13400:
13397:
13395:
13392:
13390:
13387:
13383:
13380:
13379:
13378:
13375:
13371:
13368:
13367:
13366:
13363:
13359:
13356:
13355:
13354:
13351:
13347:
13344:
13343:
13342:
13339:
13337:
13334:
13332:
13329:
13328:
13326:
13324:
13319:
13315:
13311:
13304:
13299:
13297:
13292:
13290:
13285:
13284:
13281:
13275:
13271:
13268:
13263:
13262:
13257:
13254:
13249:
13246:
13243:
13241:
13238:
13234:
13229:
13225:
13219:
13215:
13214:
13206:
13201:
13197:
13193:
13192:
13187:
13183:
13181:
13178:
13177:
13166:
13162:
13157:
13152:
13147:
13142:
13138:
13134:
13133:
13128:
13124:
13120:
13117:
13111:
13107:
13103:
13099:
13095:
13091:
13084:
13079:
13075:
13069:
13061:
13057:
13056:
13048:
13044:
13040:
13036:
13034:0-471-97442-0
13030:
13026:
13021:
13016:
13011:
13007:
13003:
12999:
12995:
12991:
12986:
12982:
12976:
12972:
12968:
12960:
12956:
12952:
12948:
12944:
12940:
12939:
12931:
12927:
12923:
12918:
12913:
12909:
12905:
12904:
12899:
12889:
12888:
12883:
12879:
12875:
12873:0-387-97971-9
12869:
12865:
12860:
12856:
12850:
12846:
12842:
12837:
12833:
12831:0-8218-3804-0
12827:
12823:
12819:
12815:
12810:
12809:
12796:
12790:
12786:
12782:
12775:
12767:
12763:
12759:
12755:
12751:
12747:
12740:
12732:
12728:
12724:
12720:
12719:
12714:
12707:
12699:
12694:
12690:
12686:
12685:
12680:
12674:
12666:
12660:
12652:
12648:
12643:
12638:
12634:
12633:
12625:
12623:
12614:
12610:
12605:
12600:
12595:
12590:
12586:
12582:
12578:
12574:
12568:
12559:
12554:
12550:
12546:
12542:
12535:
12527:
12525:0-387-94680-2
12521:
12517:
12513:
12512:O'Shea, Donal
12509:
12508:Cox, David A.
12503:
12495:
12491:
12487:
12483:
12479:
12475:
12468:
12461:
12453:
12447:
12443:
12439:
12435:
12431:
12425:
12423:
12421:
12416:
12405:
12402:
12400:
12397:
12395:
12392:
12389:
12386:
12385:
12379:
12365:
12363:
12362:Weyl algebras
12359:
12354:
12336:
12333:
12330:
12327:
12324:
12321:
12318:
12296:
12292:
12286:
12282:
12259:
12255:
12251:
12248:
12245:
12240:
12236:
12228:generated by
12210:
12206:
12202:
12199:
12196:
12191:
12187:
12180:
12158:
12154:
12150:
12147:
12144:
12139:
12135:
12127:generated by
12122:
12104:
12100:
12096:
12093:
12090:
12085:
12081:
12059:
12052:
12049:
12046:
12043:
12040:
12037:
12034:
12024:
12020:
12014:
12010:
12002:
11997:
11988:
11984:
11980:
11977:
11974:
11969:
11965:
11958:
11930:
11927:
11924:
11917:
11905:
11901:
11891:
11889:
11884:
11863:
11853:
11848:
11827:
11817:
11807:
11802:
11781:
11771:
11765:
11762:
11759:
11755:
11750:
11741:
11725:
11718:
11714:
11709:
11695:
11682:
11674:
11668:
11664:
11660:
11652:
11646:
11636:
11632:
11625:
11605:
11597:
11593:
11586:
11577:
11571:
11567:
11559:
11543:
11535:
11525:
11521:
11513:
11509:
11477:
11473:
11452:
11442:
11432:
11430:
11426:
11422:
11418:
11414:
11411:
11406:
11404:
11400:
11396:
11392:
11388:
11384:
11380:
11376:
11372:
11367:
11362:
11360:
11356:
11352:
11348:
11344:
11340:
11335:
11333:
11332:HFE challenge
11329:
11325:
11321:
11316:
11314:
11310:
11305:
11301:
11300:row reduction
11296:
11294:
11290:
11286:
11282:
11274:
11270:
11266:
11263:
11260:
11256:
11253:
11249:
11245:
11241:
11237:
11233:
11229:
11225:
11222:
11218:
11217:
11216:
11214:
11210:
11186:
11164:
11160:
11151:
11137:
11115:
11111:
11090:
11082:
11081:
11080:
11078:
11073:
11057:
11053:
11044:
11026:
11022:
11013:
11009:
10991:
10987:
10962:
10958:
10952:
10948:
10944:
10939:
10935:
10929:
10925:
10921:
10918:
10915:
10910:
10906:
10902:
10897:
10893:
10887:
10883:
10878:
10874:
10871:
10864:
10863:
10862:
10846:
10842:
10838:
10835:
10832:
10827:
10823:
10800:
10796:
10792:
10789:
10786:
10781:
10777:
10756:
10751:
10747:
10743:
10740:
10737:
10732:
10728:
10719:
10715:
10697:
10693:
10689:
10686:
10683:
10678:
10674:
10646:
10639:
10635:
10629:
10625:
10619:
10617:
10610:
10606:
10598:
10594:
10582:
10578:
10572:
10568:
10562:
10560:
10553:
10549:
10537:
10536:
10535:
10533:
10529:
10526:of dimension
10525:
10515:
10513:
10509:
10505:
10501:
10497:
10493:
10488:
10485:
10481:
10455:
10451:
10430:
10425:
10421:
10415:
10411:
10407:
10404:
10401:
10398:
10395:
10390:
10386:
10380:
10376:
10372:
10369:
10361:
10357:
10342:
10337:
10333:
10312:
10307:
10303:
10294:
10277:
10273:
10269:
10264:
10260:
10256:
10253:
10245:
10244:
10243:
10229:
10224:
10220:
10216:
10211:
10207:
10203:
10200:
10178:
10174:
10170:
10167:
10164:
10159:
10155:
10145:
10143:
10139:
10135:
10130:
10128:
10109:
10106:
10103:
10100:
10094:
10091:
10083:
10067:
10061:
10058:
10055:
10052:
10046:
10043:
10035:
10031:
10027:
10017:
10015:
9994:
9990:
9987:
9980:
9978:
9972:
9968:
9959:
9957:
9953:
9937:
9934:
9931:
9928:
9925:
9916:
9912:
9909:
9901:
9897:
9884:
9878:
9874:
9855:
9852:
9849:
9844:
9840:
9828:
9825:
9816:
9813:
9810:
9807:
9801:
9792:
9788:
9785:
9765:
9760:
9756:
9747:
9731:
9726:
9722:
9713:
9705:
9701:
9692:
9690:
9672:
9668:
9659:
9643:
9638:
9634:
9613:
9608:
9604:
9596:is the ideal
9595:
9587:
9583:
9579:
9563:
9557:
9554:
9551:
9548:
9541:
9534:
9528:
9525:
9520:
9516:
9507:
9503:
9499:
9494:
9484:
9482:
9477:
9475:
9471:
9470:algebraic set
9467:
9457:
9455:
9434:
9430:
9426:
9421:
9417:
9407:
9399:
9383:
9375:
9371:
9367:
9362:
9358:
9352:
9348:
9344:
9341:
9338:
9333:
9329:
9325:
9320:
9316:
9310:
9306:
9295:in the ideal
9294:
9290:
9271:
9263:
9259:
9255:
9250:
9246:
9240:
9236:
9232:
9227:
9223:
9219:
9214:
9210:
9204:
9200:
9191:
9177:
9176:
9175:
9173:
9169:
9165:
9161:
9157:
9155:
9136:
9128:
9124:
9100:
9092:
9088:
9079:
9075:
9056:
9048:
9044:
9020:
9012:
9008:
8980:
8971:
8963:
8959:
8950:
8942:
8938:
8931:
8929:
8922:
8918:
8910:
8906:
8892:
8884:
8880:
8871:
8863:
8859:
8852:
8850:
8843:
8839:
8827:
8826:
8825:
8823:
8819:
8815:
8805:
8792:
8789:
8786:
8783:
8780:
8777:
8768:
8764:
8759:
8743:
8740:
8737:
8717:
8714:
8711:
8691:
8688:
8685:
8679:
8676:
8673:
8662:
8658:
8654:
8651:
8646:
8642:
8622:
8614:
8610:
8603:
8600:
8597:
8591:
8588:
8585:
8580:
8576:
8569:
8566:
8563:
8557:
8552:
8548:
8544:
8541:
8538:
8535:
8530:
8526:
8522:
8516:
8513:
8506:
8505:
8504:
8502:
8497:
8493:
8488:
8484:
8480:
8476:
8471:
8467:
8461:
8457:
8450:
8445:
8441:
8434:
8416:
8401:
8393:
8389:
8385:
8382:
8379:
8374:
8371:
8368:
8364:
8357:
8349:
8345:
8341:
8338:
8335:
8330:
8326:
8300:
8296:
8292:
8289:
8286:
8281:
8277:
8267:
8254:
8248:
8242:
8239:
8236:
8228:
8224:
8220:
8216:
8212:
8208:
8204:
8199:
8197:
8192:
8175:
8169:
8166:
8163:
8154:
8148:
8128:
8122:
8119:
8116:
8109:). Moreover,
8108:
8089:
8083:
8080:
8077:
8054:
8048:
8045:
8042:
8034:
8030:
8026:
8022:
8018:
8014:
8010:
8006:
8002:
7986:
7980:
7977:
7974:
7968:
7965:
7957:
7953:
7949:
7946:
7943:
7938:
7934:
7930:
7925:
7921:
7917:
7914:
7911:
7906:
7902:
7895:
7886:
7884:
7880:
7870:
7868:
7863:
7846:
7840:
7831:
7814:
7808:
7785:
7779:
7771:
7752:
7744:
7736:
7733:
7730:
7719:
7713:
7707:
7702:
7698:
7692:
7688:
7677:
7674:
7671:
7667:
7659:
7658:
7657:
7655:
7637:
7633:
7610:
7606:
7600:
7596:
7585:
7582:
7579:
7575:
7566:
7561:
7559:
7555:
7551:
7547:
7542:
7539:
7537:
7533:
7529:
7525:
7521:
7517:
7513:
7509:
7505:
7501:
7497:
7493:
7483:
7481:
7476:
7471:
7469:
7465:
7461:
7457:
7453:
7449:
7445:
7440:
7438:
7434:
7429:
7427:
7426:complex field
7423:
7419:
7415:
7410:
7404:
7394:
7389:
7385:
7378:
7344:
7340:
7333:
7329:generated by
7323:
7321:
7315:
7311:
7302:
7299:
7291:
7287:
7283:
7273:
7270:
7260:
7256:
7254:
7244:
7242:
7236:
7232:
7228:
7213:
7206:
7202:
7175:
7172:
7169:
7166:
7163:
7157:
7154:
7148:
7145:
7142:
7139:
7136:
7130:
7127:
7125:
7120:
7115:
7111:
7107:
7104:
7099:
7095:
7087:
7084:
7078:
7075:
7070:
7066:
7059:
7056:
7050:
7047:
7044:
7041:
7035:
7032:
7030:
7025:
7022:
7019:
7016:
7013:
7006:
7003:
7000:
6997:
6991:
6988:
6985:
6982:
6976:
6973:
6967:
6964:
6959:
6955:
6948:
6945:
6943:
6938:
6935:
6932:
6929:
6926:
6915:
6914:
6913:
6905:
6901:
6897:
6890:
6869:
6866:
6863:
6840:
6834:
6831:
6828:
6806:
6800:
6797:
6794:
6774:
6770:
6761:
6757:
6752:
6749:
6745:
6740:
6737:
6733:
6728:
6727:
6726:
6712:
6709:
6706:
6683:
6680:
6677:
6672:
6668:
6664:
6661:
6655:
6652:
6649:
6643:
6640:
6637:
6634:
6631:
6624:
6623:
6622:
6585:
6582:
6579:
6576:
6573:
6570:
6567:
6564:
6561:
6558:
6555:
6552:
6545:
6544:
6543:
6530:
6524:
6521:
6518:
6512:
6509:
6473:
6470:
6465:
6461:
6457:
6454:
6447:
6432:
6429:
6424:
6420:
6416:
6413:
6406:
6405:
6404:
6387:
6384:
6381:
6375:
6372:
6349:
6346:
6343:
6332:
6329:
6305:
6285:
6276:
6267:
6265:
6261:
6256:
6250:
6245:
6243:
6239:
6232:Special cases
6229:
6227:
6223:
6196:
6194:
6189:
6186:
6174:
6168:
6159:
6155:
6144:
6130:
6128:
6122:
6120:
6116:
6112:
6108:
6103:
6093:
6080:
6078:
6060:
6058:
6042:
6038:
6034:
6026:
6022:
6001:
5998:
5994:
5990:
5974:
5967:
5944:
5926:a polynomial
5925:
5922:
5906:
5903:
5889:
5888:
5887:
5874:
5870:
5857:
5852:over a field
5851:
5830:
5826:
5822:
5819:
5816:
5811:
5807:
5800:
5797:
5794:
5780:
5778:
5759:
5756:
5735:
5729:
5726:
5716:
5710:
5707:
5704:
5701:
5680:
5674:
5671:
5661:
5655:
5652:
5649:
5643:
5640:
5637:
5631:
5624:
5623:
5622:
5620:
5615:
5613:
5596:
5567:
5564:
5543:
5537:
5534:
5521:
5515:
5512:
5508:
5503:
5500:
5479:
5473:
5470:
5457:
5451:
5448:
5444:
5439:
5433:
5430:
5427:
5421:
5414:
5413:
5412:
5410:
5383:
5376:
5366:
5360:
5357:
5333:
5327:
5324:
5320:
5315:
5312:
5302:
5296:
5293:
5269:
5263:
5260:
5256:
5251:
5249:
5240:
5236:
5226:
5220:
5217:
5193:
5187:
5184:
5180:
5175:
5160:
5156:
5152:
5148:
5138:
5132:
5129:
5105:
5099:
5096:
5092:
5087:
5072:
5068:
5066:
5058:
5055:
5052:
5046:
5035:
5034:
5033:
5017:
5013:
4973:
4970:
4953:
4948:
4944:
4940:
4934:
4931:
4914:
4909:
4905:
4901:
4895:
4892:
4889:
4883:
4876:
4875:
4874:
4864:
4863:critical pair
4860:
4852:
4847:
4834:
4829:
4825:
4802:
4798:
4773:
4768:
4764:
4760:
4755:
4751:
4747:
4742:
4738:
4711:
4707:
4695:
4691:
4684:
4681:
4676:
4671:
4667:
4660:
4655:
4651:
4639:
4635:
4628:
4625:
4620:
4615:
4611:
4604:
4600:
4594:
4590:
4578:
4574:
4567:
4564:
4559:
4554:
4550:
4543:
4540:
4535:
4531:
4526:
4522:
4518:
4512:
4508:
4496:
4492:
4485:
4482:
4477:
4472:
4468:
4461:
4458:
4453:
4449:
4444:
4440:
4435:
4431:
4423:
4422:
4421:
4405:
4401:
4378:
4374:
4351:
4347:
4324:
4320:
4299:
4294:
4290:
4262:
4259:
4256:
4251:
4247:
4243:
4240:
4237:
4234:
4229:
4225:
4221:
4218:
4213:
4209:
4205:
4202:
4197:
4193:
4185:
4184:
4183:
4176:
4160:
4156:
4152:
4147:
4143:
4120:
4116:
4093:
4089:
4062:
4058:
4054:
4049:
4045:
4018:
4014:
4010:
4005:
4001:
3980:
3975:
3971:
3967:
3962:
3958:
3948:
3944:
3934:
3929:
3916:
3913:
3910:
3907:
3902:
3898:
3894:
3891:
3888:
3885:
3882:
3877:
3873:
3852:
3849:
3844:
3840:
3836:
3831:
3827:
3813:
3797:
3793:
3769:
3766:
3763:
3760:
3755:
3751:
3747:
3744:
3741:
3738:
3735:
3730:
3726:
3717:
3713:
3709:
3706:
3702:
3697:
3694:
3691:
3686:
3682:
3678:
3675:
3672:
3669:
3666:
3661:
3657:
3653:
3650:
3647:
3639:
3635:
3631:
3627:
3620:
3616:
3607:
3603:
3599:
3596:
3593:
3589:
3584:
3577:
3576:
3575:
3572:
3552:
3548:
3524:
3521:
3518:
3513:
3509:
3505:
3500:
3496:
3487:
3483:
3479:
3476:
3472:
3467:
3464:
3461:
3456:
3452:
3448:
3443:
3439:
3435:
3432:
3429:
3421:
3417:
3413:
3409:
3402:
3398:
3389:
3385:
3381:
3378:
3375:
3371:
3366:
3359:
3358:
3357:
3344:
3339:
3335:
3314:
3309:
3305:
3284:
3279:
3275:
3252:
3248:
3225:
3221:
3200:
3195:
3191:
3187:
3164:
3161:
3158:
3155:
3150:
3146:
3142:
3139:
3136:
3133:
3128:
3124:
3120:
3117:
3114:
3111:
3106:
3102:
3098:
3095:
3090:
3086:
3082:
3079:
3076:
3073:
3070:
3065:
3061:
3052:
3048:
3044:
3041:
3038:
3034:
3029:
3022:
3021:
3020:
3013:
2993:
2989:
2968:
2963:
2959:
2936:
2932:
2905:
2901:
2897:
2888:
2880:
2863:
2860:
2857:
2854:
2851:
2849:
2842:
2838:
2830:
2827:
2824:
2819:
2815:
2811:
2806:
2802:
2798:
2796:
2789:
2785:
2760:
2752:
2748:
2744:
2739:
2735:
2728:
2725:
2705:
2702:
2699:
2694:
2690:
2686:
2683:
2678:
2674:
2670:
2665:
2661:
2657:
2654:
2651:
2631:
2628:
2625:
2622:
2614:
2609:
2600:
2598:
2592:
2588:
2575:
2547:
2541:
2535:
2532:
2529:
2523:
2517:
2513:
2506:
2503:
2481:
2477:
2445:
2442:
2436:
2432:
2426:
2423:
2420:
2416:
2412:
2409:
2406:
2403:
2396:
2395:
2394:
2379:
2361:
2344:
2314:
2310:
2297:
2293:
2283:
2252:
2249:
2239:
2233:
2230:
2226:
2214:
2208:
2205:
2201:
2196:
2193:
2190:
2184:
2181:
2178:
2172:
2167:
2163:
2155:
2154:
2153:
2139:
2133:
2120:be a term of
2111:
2104:Suppose that
2102:
2086:
2072:
2062:
2054:
2044:
2037:A polynomial
2035:
2032:
2029:
2025:
2023:
2019:
2015:
2012:occurring in
2011:
2010:row reduction
2007:
1983:
1974:
1971:
1968:
1957:
1954:
1948:
1942:
1939:
1936:
1930:
1927:
1920:
1919:
1918:
1915:
1903:
1899:
1894:
1863:
1859:
1855:
1850:
1846:
1834:
1830:
1826:
1816:
1812:
1808:
1803:
1799:
1787:
1783:
1764:
1760:
1755:
1750:
1726:
1723:
1701:
1694:
1690:
1686:
1681:
1677:
1671:
1667:
1663:
1656:
1652:
1648:
1643:
1639:
1633:
1629:
1625:
1620:
1617:
1593:
1590:
1554:
1550:
1546:
1541:
1537:
1524:
1513:
1505:
1492:
1484:
1480:
1476:
1473:
1470:
1465:
1461:
1454:
1451:
1426:
1422:
1418:
1415:
1412:
1407:
1403:
1396:
1393:
1369:
1365:
1359:
1355:
1351:
1344:
1340:
1334:
1330:
1326:
1323:
1299:
1295:
1289:
1285:
1281:
1274:
1270:
1264:
1260:
1256:
1253:
1228:
1225:
1222:
1218:
1217:
1216:
1208:
1204:
1200:
1192:
1188:
1182:
1178:
1174:
1165:
1163:
1147:
1145:
1136:
1132:
1129:
1126:
1122:
1119:
1116:
1112:
1108:
1105:
1104:
1103:
1100:
1098:
1093:
1091:
1071:
1068:
1065:
1062:
1055:
1041:
1038:
1035:
1032:
1029:
1023:
1020:
1017:
1010:
1009:
1008:
994:
988:
978:
965:
962:
959:
954:
950:
946:
943:
940:
935:
931:
908:
904:
898:
894:
888:
883:
880:
877:
873:
853:
828:
824:
820:
817:
814:
809:
805:
798:
795:
786:
784:
780:
775:
774:ordered pairs
771:
767:
762:
749:
744:
740:
736:
729:
725:
719:
715:
711:
704:
700:
694:
690:
664:
660:
656:
653:
650:
645:
641:
634:
631:
623:
602:
598:
594:
591:
588:
583:
579:
572:
569:
547:
543:
522:
515:
511:
505:
501:
497:
490:
486:
480:
476:
467:
463:
445:
441:
432:
410:
406:
385:
380:
376:
370:
366:
362:
359:
356:
351:
347:
341:
337:
311:
307:
303:
300:
297:
292:
288:
281:
278:
275:
267:
262:
260:
249:
228:
224:
220:
217:
214:
209:
205:
198:
195:
192:
185:
181:
175:
160:
158:
154:
150:
145:
143:
139:
135:
131:
126:
122:
118:
114:
110:
106:
101:
99:
95:
91:
86:
84:
83:rational maps
80:
76:
72:
68:
64:
59:
55:
50:
46:
42:
38:
37:Gröbner basis
34:
30:
26:
22:
13510:
13506:Discriminant
13425:Multivariate
13274:Scholarpedia
13259:
13212:
13189:
13146:math/0411514
13136:
13130:
13115:
13093:
13089:
13068:cite journal
13059:
13053:
13024:
13000:(10): 7763.
12997:
12994:Scholarpedia
12993:
12970:
12942:
12936:
12907:
12901:
12886:
12863:
12840:
12813:
12780:
12774:
12749:
12745:
12739:
12722:
12716:
12706:
12688:
12682:
12673:
12631:
12584:
12580:
12567:
12548:
12544:
12534:
12518:. Springer.
12515:
12502:
12480:(2): 35–48.
12477:
12473:
12460:
12433:
12394:Graver basis
12376:
12355:
12120:
11904:free modules
11897:
11885:
11696:
11575:
11570:vector space
11453:
11438:
11412:
11407:
11363:
11358:
11346:
11336:
11317:
11313:vector space
11297:
11278:
11272:
11207:
11076:
11074:
11042:
11007:
10977:
10717:
10713:
10665:
10534:of the form
10527:
10521:
10511:
10507:
10503:
10499:
10495:
10492:hypersurface
10489:
10478:
10359:
10146:
10141:
10137:
10133:
10131:
10126:
10081:
10033:
10029:
10025:
10023:
10013:
9974:
9970:
9960:
9955:
9951:
9899:
9895:
9894:and 1−
9885:
9876:
9872:
9745:
9711:
9706:of an ideal
9703:
9695:
9693:
9688:
9657:
9593:
9588:of an ideal
9586:localization
9585:
9581:
9577:
9508:is the ring
9505:
9501:
9498:localization
9497:
9490:
9480:
9478:
9463:
9453:
9397:
9292:
9288:
9286:
9167:
9158:
9077:
9073:
8999:
8824:of the form
8811:
8766:
8762:
8757:
8660:
8656:
8652:
8649:
8644:
8640:
8637:
8500:
8495:
8491:
8486:
8482:
8478:
8474:
8469:
8465:
8459:
8455:
8448:
8443:
8439:
8432:
8422:
8268:
8226:
8222:
8206:
8200:
8195:
8193:
8152:
8146:
8106:
8032:
8028:
8024:
8020:
8016:
8012:
8008:
8004:
8000:
7887:
7878:
7876:
7864:
7832:
7769:
7767:
7653:
7562:
7557:
7553:
7549:
7545:
7543:
7540:
7536:hypersurface
7531:
7519:
7511:
7507:
7506:of the ring
7499:
7495:
7494:of an ideal
7491:
7489:
7472:
7467:
7463:
7459:
7455:
7451:
7447:
7446:of an ideal
7443:
7441:
7433:inconsistent
7430:
7417:
7406:
7387:
7383:
7376:
7342:
7338:
7331:
7324:
7319:
7318:is equal to
7313:
7309:
7300:
7292:of an ideal
7289:
7285:
7279:
7268:
7266:
7257:
7250:
7234:
7230:
7226:
7214:
7204:
7200:
7197:
6903:
6899:
6895:
6891:
6772:
6768:
6765:
6759:
6755:
6747:
6743:
6735:
6731:
6698:
6600:
6490:By reducing
6489:
6321:
6257:
6246:
6235:
6197:
6190:
6187:
6175:
6162:
6160:
6156:
6138:
6136:
6123:
6118:
6104:
6091:
6081:
6066:
6017:
5949:-polynomial
5919:
5901:
5858:
5786:
5774:
5618:
5616:
5611:
5597:
5582:
5408:
5398:
4992:
4862:
4859:S-polynomial
4858:
4856:
4850:
4729:
4277:
4177:
3946:
3942:
3930:
3814:
3784:
3573:
3539:
3179:
3011:
2889:
2881:
2610:
2606:
2596:
2590:
2586:
2460:
2380:
2359:
2345:
2312:
2308:
2298:
2291:
2281:
2267:
2137:
2131:
2109:
2103:
2084:
2070:
2060:
2052:
2042:
2036:
2033:
2027:
2026:
2021:
2005:
2003:
1916:
1901:
1897:
1892:
1762:
1758:
1753:
1751:
1522:
1511:
1506:
1245:
1214:
1205:
1198:
1190:
1186:
1180:
1176:
1173:leading term
1172:
1166:
1158:
1140:
1134:
1130:
1124:
1120:
1114:
1110:
1106:
1101:
1094:
1089:
1087:
990:
787:
770:ordered list
763:
621:
465:
431:coefficients
430:
263:
177:
157:Ore algebras
146:
141:
134:power series
129:
102:
87:
57:
48:
36:
18:
13455:Homogeneous
13450:Square-free
13445:Irreducible
13310:Polynomials
12945:: 374–383.
12474:SIGSAM Bull
12399:Janet basis
11391:Mathematica
9879:belongs to
8650:eliminating
8207:elimination
7879:elimination
7873:Elimination
7524:hyperplanes
4853:-polynomial
2951:and not by
2360:normal form
1910:instead of
1144:elimination
993:total order
79:projections
21:mathematics
13526:Categories
13415:Univariate
13096:(2): 3–6.
12642:2104.03572
12411:References
11916:direct sum
11900:submodules
11441:complexity
11435:Complexity
11379:Macaulay 2
11281:heuristics
11259:heuristics
11240:GMPlibrary
11008:base point
10358:Adding to
10355:and so on.
9698:saturation
9689:saturation
9500:of a ring
9481:saturating
9460:Saturation
8704:such that
8211:projection
8205:, is that
6780:belong to
6502:such that
6260:singletons
6249:unit ideal
6242:zero ideal
6111:Noetherian
5945:for every
5878:. The set
5783:Definition
5777:associates
3933:Buchberger
2393:, one has
2116:, and let
2089:. (So, if
1569:for every
1517:, or that
1097:well-order
535:where the
433:, and the
398:where the
266:polynomial
13501:Resultant
13440:Trinomial
13420:Bivariate
13261:MathWorld
13196:EMS Press
13027:. Wiley.
12959:189834323
12594:1304.1238
12334:≤
12328:≤
12322:≤
12249:…
12200:…
12148:…
12094:…
12050:≤
12044:≤
12038:≤
11978:…
11928:⊕
11858:Ω
11812:Ω
11530:Ω
11503:Ω
11467:Ω
11320:full rank
11273:degrevlex
11083:Saturate
11010:), every
10945:−
10919:…
10903:−
10836:…
10790:…
10741:…
10687:…
10599:⋮
10408:−
10399:…
10373:−
10270:⋯
10217:⋯
10168:…
10104:−
10095:∪
10056:−
10047:∪
9999:∞
9932:∩
9921:∞
9853:∈
9829:∈
9823:∃
9817:∣
9811:∈
9797:∞
9552:−
9466:triangles
9411:↦
9381:⟩
9368:−
9342:…
9326:−
9303:⟨
9256:−
9220:−
9172:resultant
8911:⋮
8787:∩
8781:∈
8741:∈
8715:∈
8677:−
8620:⟩
8601:−
8589:…
8567:−
8539:…
8520:⟨
8383:…
8339:…
8290:⋯
8240:∩
8225:into the
8167:∩
8120:∩
8081:∩
8046:∩
7947:…
7915:…
7734:−
7683:∞
7668:∑
7591:∞
7576:∑
7492:dimension
7296:yields 0
7282:reduction
7170:−
7143:−
7108:−
7076:−
7057:−
7048:−
7020:−
7004:−
6989:−
6974:−
6965:−
6933:−
6678:−
6653:−
6644:−
6580:−
6562:−
6528:⟩
6516:⟨
6471:−
6430:−
6391:⟩
6379:⟨
6238:empty set
6129:, below.
6090:of their
6063:Existence
5869:generates
5820:…
5730:
5711:
5705:−
5675:
5656:
5538:
5516:
5504:−
5474:
5452:
5361:
5328:
5316:−
5297:
5264:
5221:
5188:
5176:−
5157:−
5133:
5100:
5088:−
4954:
4941:−
4915:
4761:−
4661:−
4544:−
4523:−
4462:−
4257:−
4219:−
4153:−
4055:−
4011:−
3968:−
3908:−
3850:−
3761:−
3648:−
3594:−
3519:−
3430:−
3376:−
3345::
3188:−
3115:−
3099:−
3077:−
3039:−
2861:−
2825:−
2671:−
2536:
2530:≤
2507:
2424:∈
2417:∑
2309:reducible
2234:
2209:
2197:−
2173:
2110:reducible
2071:reducible
2006:reduction
2000:Reduction
1931:
1827:⋯
1687:−
1664:⋯
1649:−
1547:≤
1474:…
1416:…
1352:⋯
1282:⋯
1125:degrevlex
1066:≤
1036:≤
1027:⟺
1021:≤
960:∈
944:…
874:∑
818:…
712:⋯
654:…
592:…
498:⋯
462:monomials
429:, called
360:⋯
301:…
218:…
67:dimension
13486:Division
13435:Binomial
13430:Monomial
13110:37070503
12928:(1970).
12884:(1965).
12785:Springer
12514:(1997).
12382:See also
12073:, where
12060:⟩
12003:⟨
11399:SageMath
11395:SINGULAR
11242:. Also,
11228:integers
10959:⟩
10879:⟨
7625:, where
7422:rational
6725:we have
6601:None of
5403:and the
3703:→
3628:→
3590:→
3473:→
3410:→
3372:→
3035:→
2469:and the
1917:One has
1523:multiple
464:(called
13198:, 2001
13165:5656701
13002:Bibcode
12766:2515114
12494:1819694
11568:on the
11410:library
11311:of the
9902:, then
9154:coprime
8454:, ...,
8447:} and {
8438:, ...,
7502:is the
7337:, ...,
6220:of the
6165:reduced
6141:minimal
2568:, then
1512:divides
772:of the
257:is the
246:over a
52:over a
13323:degree
13220:
13163:
13108:
13031:
12977:
12957:
12870:
12851:
12828:
12791:
12764:
12613:149627
12611:
12522:
12492:
12448:
11947:to be
11738:Using
11425:GitHub
11413:Msolve
10666:where
9584:. The
9576:where
9000:where
8816:is an
8213:of an
7768:where
7532:degree
7468:degree
7382:, ...,
7362:is in
7303:is in
7269:unique
5930:is in
5583:where
5407:, the
4993:where
2584:, and
2461:where
1891:. The
1189:), lm(
1135:lexdeg
850:, the
180:ideals
130:et al.
96:, and
77:under
13208:(PDF)
13161:S2CID
13141:arXiv
13106:S2CID
13086:(PDF)
13050:(PDF)
12955:S2CID
12933:(PDF)
12891:(PDF)
12762:S2CID
12637:arXiv
12609:S2CID
12589:arXiv
12490:S2CID
12470:(PDF)
12349:. If
11429:Julia
11403:SymPy
11387:Maple
11383:Magma
11371:CoCoA
11309:basis
11179:from
9656:When
8194:This
8027:. If
7865:Most
6193:up to
5873:ideal
5867:that
5848:be a
4366:, by
2773:with
2718:, by
2615:with
2272:from
1529:, if
1521:is a
1221:merge
923:with
248:field
182:in a
163:Tools
54:field
43:in a
13312:and
13218:ISBN
13074:link
13029:ISBN
12975:ISBN
12868:ISBN
12849:ISBN
12826:ISBN
12789:ISBN
12665:link
12520:ISBN
12446:ISBN
11522:>
11439:The
11401:and
11267:The
11250:and
11234:and
11226:The
10712:are
9975:The
9965:the
9694:The
9491:The
9152:are
9116:and
9036:and
8730:and
8427:and
8293:>
8287:>
8003:and
7490:The
7370:and
7280:The
7219:and
7208:and
6853:and
6776:and
6758:) =
6746:) =
6734:) =
6710:>
6605:and
6322:Let
5859:Let
5787:Let
5602:and
5591:and
5001:and
4869:and
4861:or
4393:and
4339:and
3941:0 =
3267:and
2626:>
2152:by
2136:. A
2004:The
1896:lcm(
1887:and
1771:and
1757:gcd(
1752:The
1745:and
1444:and
1316:and
1246:Let
1195:and
1179:and
1115:plex
781:and
460:are
35:, a
13321:By
13272:on
13151:doi
13137:359
13098:doi
13010:doi
12947:doi
12912:doi
12754:doi
12727:doi
12693:doi
12647:doi
12599:doi
12553:doi
12482:doi
12438:doi
12360:or
11902:of
11375:GAP
11287:by
11103:by
10036:in
9898:in
9886:If
9748:to
9710:in
9626:of
9592:of
9396:If
9187:Res
8655:in
8423:If
7526:in
7375:∪ {
6766:As
6754:lt(
6742:lt(
6730:lt(
6494:by
5961:by
5938:by
5886:if
5871:an
5608:lcm
5595:.
5585:gcd
5405:gcd
5401:lcm
5014:red
4995:lcm
4945:red
4906:red
4817:or
4135:by
4077:to
3865:or
3213:of
3008:by
2920:of
2580:by
2576:of
2389:by
2366:by
2362:of
2350:by
2331:by
2323:of
2315:by
2311:or
2307:is
2290:lm(
2284:of
2164:red
2144:by
2140:of
2130:lm(
2112:by
2108:is
2083:lm(
2073:by
2069:is
2059:lm(
2051:lm(
2041:is
1963:gcd
1928:lcm
1912:min
1908:max
1840:min
1793:min
1767:of
1525:of
1197:lc(
1185:lt(
1113:or
1111:lex
788:If
268:in
151:or
119:'s
81:or
19:In
13528::
13258:.
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13188:,
13159:.
13149:.
13135:.
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13104:.
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13092:.
13088:.
13070:}}
13066:{{
13060:52
13058:.
13052:.
13008:.
12996:.
12992:.
12969:.
12953:.
12941:.
12935:.
12908:41
12906:.
12900:.
12847:.
12843:.
12824:.
12816:.
12783:.
12760:.
12750:13
12748:.
12723:40
12721:.
12715:.
12689:13
12687:,
12661:}}
12657:{{
12645:.
12621:^
12607:.
12597:.
12585:80
12583:.
12579:.
12549:24
12547:.
12543:.
12488:.
12478:37
12476:.
12472:.
12444:.
12419:^
12364:.
12311:,
11890:.
11397:,
11393:,
11389:,
11385:,
11381:,
11377:,
11373:,
11361:.
11334:.
11246:,
10129:.
10016:.
9979:of
9896:ft
9883:.
9691:.
9076:≤
8812:A
8765:=
8643:∩
8494:∩
8468:∩
7560:.
7454:,
7428:.
7393:.
7322:.
7312:∪{
7233:,
7229:,
7212:.
7203:,
6902:,
6898:,
6889:.
6771:,
6748:xy
6621::
6611:xk
6266:.
6255:.
6244:.
6079:.
6059:.
5779:.
5727:lm
5708:lc
5672:lm
5653:lc
5535:lm
5513:lc
5471:lm
5449:lc
5358:lm
5325:lc
5294:lm
5261:lc
5218:lm
5185:lc
5130:lm
5097:lc
4788:xy
3945:−
3571:.
3019::
3010:–2
2864:2.
2533:lm
2504:lm
2343:.
2296:.
2231:lm
2206:lc
2118:cm
1914:.
1900:,
1761:,
1749:.
1203:.
1175:,
1133:,
1092:.
1007:,
1003:,
999:,
785:.
159:.
85:.
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13295:t
13288:v
13264:.
13235:.
13226:.
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13112:.
13100::
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13037:.
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12920:.
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12857:.
12834:.
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12768:.
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12733:.
12729::
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12639::
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12561:.
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12528:.
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12440::
12351:R
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11998:/
11994:]
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11962:[
11959:R
11949:0
11945:L
11931:L
11925:R
11912:R
11908:L
11867:)
11864:n
11861:(
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11849:d
11828:,
11821:)
11818:n
11815:(
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11775:)
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11766:o
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11760:n
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11726:.
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11710:d
11699:n
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11675:n
11672:(
11669:O
11665:D
11661:=
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11643:)
11637:n
11633:D
11629:(
11626:O
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11603:)
11598:n
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11590:(
11587:O
11576:D
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11562:D
11544:.
11539:)
11536:n
11533:(
11526:D
11519:)
11514:n
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11506:(
11483:)
11478:n
11474:D
11470:(
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11445:n
11275:.
11187:G
11165:i
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11112:p
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10963:.
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10930:0
10926:p
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10875:=
10872:I
10847:n
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10833:,
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10793:,
10787:,
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10757:.
10752:k
10748:t
10744:,
10738:,
10733:1
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10694:p
10690:,
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10675:p
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10640:0
10636:p
10630:n
10626:p
10620:=
10611:n
10607:x
10583:0
10579:p
10573:1
10569:p
10563:=
10554:1
10550:x
10528:k
10512:f
10508:f
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10452:t
10431:,
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10265:1
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10257:=
10254:f
10230:,
10225:k
10221:f
10212:1
10208:f
10204:=
10201:f
10179:k
10175:f
10171:,
10165:,
10160:1
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10142:F
10138:F
10134:F
10127:t
10113:}
10110:f
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10098:{
10092:F
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10068:,
10065:}
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10044:F
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9988:I
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9935:R
9929:J
9926:=
9917:f
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9856:I
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9833:N
9826:k
9820:(
9814:R
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9802:=
9793:f
9789::
9786:I
9766:.
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9746:R
9732:I
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9723:R
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9669:R
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9639:f
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9542:/
9538:]
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9529:R
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9418:x
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9363:n
9359:x
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9269:)
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9101:t
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8975:)
8972:t
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8964:n
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8954:)
8951:t
8948:(
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8932:=
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8893:t
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8885:1
8881:g
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8872:t
8869:(
8864:1
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8853:=
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8840:x
8793:.
8790:J
8784:I
8778:b
8767:b
8763:a
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8623:.
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8553:m
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8536:,
8531:1
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8517:=
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8492:I
8487:t
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8466:I
8460:k
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8399:}
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8361:{
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8350:k
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8336:,
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8323:{
8301:n
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8278:x
8255:.
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8249:Y
8246:[
8243:K
8237:I
8227:Y
8223:I
8179:]
8176:Y
8173:[
8170:K
8164:G
8153:K
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8132:]
8129:Y
8126:[
8123:K
8117:G
8093:]
8090:Y
8087:[
8084:K
8078:I
8058:]
8055:Y
8052:[
8049:K
8043:G
8033:I
8029:G
8025:X
8021:X
8017:Y
8013:X
8009:X
8005:Y
8001:X
7987:,
7984:]
7981:Y
7978:,
7975:X
7972:[
7969:K
7966:=
7963:]
7958:m
7954:y
7950:,
7944:,
7939:1
7935:y
7931:,
7926:n
7922:x
7918:,
7912:,
7907:1
7903:x
7899:[
7896:K
7850:)
7847:t
7844:(
7841:P
7818:)
7815:1
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7809:P
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7786:t
7783:(
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7770:d
7753:,
7745:d
7741:)
7737:t
7731:1
7728:(
7723:)
7720:t
7717:(
7714:P
7708:=
7703:i
7699:t
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7689:d
7678:0
7675:=
7672:i
7654:i
7638:i
7634:d
7611:i
7607:t
7601:i
7597:d
7586:0
7583:=
7580:i
7558:x
7554:x
7550:S
7546:S
7520:I
7512:I
7510:/
7508:R
7500:R
7496:I
7464:G
7460:x
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7452:x
7448:I
7444:G
7391:}
7388:k
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7380:1
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7373:J
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7364:J
7358:I
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7332:f
7327:I
7320:G
7316:}
7314:f
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7301:f
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7225:{
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7155:+
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7149:k
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7131:y
7128:=
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7116:2
7112:x
7105:f
7100:2
7096:y
7088:0
7085:=
7082:)
7079:y
7071:2
7067:y
7063:(
7060:x
7054:)
7051:x
7045:y
7042:x
7039:(
7036:y
7033:=
7026:h
7023:x
7017:k
7014:y
7007:h
7001:f
6998:=
6995:)
6992:x
6986:y
6983:x
6980:(
6977:x
6971:)
6968:y
6960:2
6956:x
6952:(
6949:y
6946:=
6939:k
6936:x
6930:f
6927:y
6910:I
6906:}
6904:h
6900:k
6896:f
6894:{
6887:I
6873:}
6870:k
6867:,
6864:h
6861:{
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6838:}
6835:h
6832:,
6829:f
6826:{
6807:,
6804:}
6801:k
6798:,
6795:f
6792:{
6782:I
6778:h
6773:k
6769:f
6760:y
6756:h
6744:k
6736:x
6732:f
6713:y
6707:x
6684:.
6681:y
6673:2
6669:y
6665:=
6662:f
6659:)
6656:1
6650:y
6647:(
6641:k
6638:x
6635:=
6632:h
6619:I
6615:f
6607:k
6603:f
6586:.
6583:x
6577:y
6574:x
6571:=
6568:f
6565:x
6559:g
6556:=
6553:k
6531::
6525:k
6522:,
6519:f
6513:=
6510:I
6500:k
6496:f
6492:g
6486:.
6474:x
6466:3
6462:x
6458:=
6455:g
6445:,
6433:y
6425:2
6421:x
6417:=
6414:f
6388:g
6385:,
6382:f
6376:=
6373:I
6353:]
6350:y
6347:,
6344:x
6341:[
6337:Q
6333:=
6330:R
6306:g
6286:f
6253:1
6207:Q
6182:I
6178:I
6171:1
6151:I
6147:I
6119:S
6113:(
6100:G
6096:G
6092:S
6088:G
6084:G
6073:G
6069:G
6043:I
6039:/
6035:R
6002:.
5999:I
5995:/
5991:R
5981:F
5977:G
5970:R
5963:G
5959:s
5955:G
5951:s
5947:S
5940:G
5936:f
5932:I
5928:f
5915:.
5913:G
5909:I
5898:,
5896:G
5892:I
5884:I
5880:G
5876:I
5865:R
5861:G
5854:F
5836:]
5831:n
5827:x
5823:,
5817:,
5812:1
5808:x
5804:[
5801:F
5798:=
5795:R
5760:;
5757:g
5750:d
5747:c
5744:g
5739:)
5736:f
5733:(
5720:)
5717:f
5714:(
5702:f
5695:d
5692:c
5689:g
5684:)
5681:g
5678:(
5665:)
5662:g
5659:(
5650:=
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5644:g
5641:,
5638:f
5635:(
5632:S
5619:S
5612:S
5604:g
5600:f
5593:g
5589:f
5568:;
5565:g
5558:d
5555:c
5552:g
5547:)
5544:f
5541:(
5525:)
5522:g
5519:(
5509:1
5501:f
5494:d
5491:c
5488:g
5483:)
5480:g
5477:(
5461:)
5458:f
5455:(
5445:1
5440:=
5437:)
5434:g
5431:,
5428:f
5425:(
5422:S
5409:S
5384:.
5377:g
5370:)
5367:g
5364:(
5353:m
5350:c
5347:l
5337:)
5334:g
5331:(
5321:1
5313:f
5306:)
5303:f
5300:(
5289:m
5286:c
5283:l
5273:)
5270:f
5267:(
5257:1
5252:=
5241:)
5237:f
5230:)
5227:f
5224:(
5213:m
5210:c
5207:l
5197:)
5194:f
5191:(
5181:1
5172:m
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5166:l
5161:(
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5149:g
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5136:(
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5119:l
5109:)
5106:g
5103:(
5093:1
5084:m
5081:c
5078:l
5073:(
5069:=
5062:)
5059:g
5056:,
5053:f
5050:(
5047:S
5018:1
5003:g
4999:f
4989:;
4977:)
4974:f
4971:,
4967:m
4964:c
4961:l
4957:(
4949:1
4938:)
4935:g
4932:,
4928:m
4925:c
4922:l
4918:(
4910:1
4902:=
4899:)
4896:g
4893:,
4890:f
4887:(
4884:S
4871:g
4867:f
4851:S
4835:.
4830:3
4826:g
4803:2
4799:g
4774:.
4769:2
4765:f
4756:3
4752:f
4748:=
4743:3
4739:g
4726:.
4712:2
4708:g
4701:)
4696:2
4692:g
4688:(
4685:t
4682:l
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4645:)
4640:1
4636:g
4632:(
4629:t
4626:l
4621:y
4616:2
4612:x
4605:=
4601:)
4595:1
4591:g
4584:)
4579:1
4575:g
4571:(
4568:t
4565:l
4560:y
4555:2
4551:x
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4532:x
4527:(
4519:)
4513:2
4509:g
4502:)
4497:2
4493:g
4489:(
4486:t
4483:l
4478:y
4473:2
4469:x
4459:y
4454:2
4450:x
4445:(
4441:=
4436:3
4432:g
4406:1
4402:g
4379:2
4375:g
4352:2
4348:g
4325:1
4321:g
4300:y
4295:2
4291:x
4280:S
4263:.
4260:y
4252:3
4248:y
4244:+
4241:x
4238:2
4235:=
4230:2
4226:g
4222:x
4214:1
4210:g
4206:y
4203:=
4198:3
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4180:G
4161:2
4157:f
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4121:2
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4094:3
4090:f
4079:G
4063:2
4059:f
4050:3
4046:f
4035:G
4019:3
4015:f
4006:2
4002:f
3981:.
3976:3
3972:f
3963:2
3959:f
3947:f
3943:f
3937:S
3917:.
3914:y
3911:2
3903:3
3899:y
3895:2
3892:+
3889:x
3886:2
3883:=
3878:3
3874:f
3853:y
3845:3
3841:y
3837:=
3832:2
3828:f
3817:f
3798:3
3794:f
3770:.
3767:y
3764:2
3756:3
3752:y
3748:2
3745:+
3742:x
3739:2
3736:=
3731:3
3727:f
3718:2
3714:g
3710:y
3707:2
3698:y
3695:2
3692:+
3687:3
3683:y
3679:2
3676:+
3673:x
3670:2
3667:+
3662:2
3658:y
3654:x
3651:2
3640:1
3636:g
3632:y
3621:1
3617:f
3608:1
3604:g
3600:x
3597:2
3585:f
3569:f
3553:2
3549:f
3525:.
3522:y
3514:3
3510:y
3506:=
3501:2
3497:f
3488:2
3484:g
3480:y
3477:2
3468:y
3465:3
3462:+
3457:3
3453:y
3449:+
3444:2
3440:y
3436:x
3433:2
3422:2
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3414:x
3403:1
3399:f
3390:1
3386:g
3382:x
3379:2
3367:f
3340:2
3336:g
3315:,
3310:2
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3285:,
3280:2
3276:g
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3226:1
3222:f
3201:y
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3192:x
3165:.
3162:y
3159:3
3156:+
3151:3
3147:y
3143:+
3140:x
3137:2
3134:+
3129:2
3125:y
3121:x
3118:2
3112:y
3107:2
3103:x
3096:=
3091:1
3087:g
3083:x
3080:2
3074:f
3071:=
3066:1
3062:f
3053:1
3049:g
3045:x
3042:2
3030:f
3017:f
3012:x
2994:1
2990:g
2969:.
2964:2
2960:g
2937:1
2933:g
2922:f
2906:3
2902:x
2898:2
2884:f
2858:y
2855:x
2852:=
2843:2
2839:g
2831:,
2828:1
2820:2
2816:y
2812:+
2807:2
2803:x
2799:=
2790:1
2786:g
2761:,
2758:}
2753:2
2749:g
2745:,
2740:1
2736:g
2732:{
2729:=
2726:G
2706:y
2703:3
2700:+
2695:3
2691:y
2687:+
2684:y
2679:2
2675:x
2666:3
2662:x
2658:2
2655:=
2652:f
2632:,
2629:y
2623:x
2591:g
2587:q
2582:g
2578:f
2570:h
2566:g
2562:G
2548:.
2545:)
2542:f
2539:(
2527:)
2524:g
2518:g
2514:q
2510:(
2482:g
2478:q
2467:G
2463:h
2446:,
2443:g
2437:g
2433:q
2427:G
2421:g
2413:+
2410:h
2407:=
2404:f
2391:G
2387:f
2383:h
2376:f
2372:G
2368:G
2364:f
2356:G
2352:G
2348:f
2341:G
2337:f
2333:G
2329:f
2325:G
2321:g
2317:G
2305:f
2301:G
2294:)
2292:f
2286:f
2278:m
2274:f
2270:m
2253:.
2250:g
2243:)
2240:g
2237:(
2227:m
2218:)
2215:g
2212:(
2202:c
2194:f
2191:=
2188:)
2185:g
2182:,
2179:f
2176:(
2168:1
2150:f
2146:g
2142:f
2134:)
2132:g
2126:m
2122:f
2114:g
2106:f
2099:f
2095:g
2091:f
2087:)
2085:g
2079:f
2075:g
2067:f
2063:)
2061:g
2055:)
2053:f
2047:g
2039:f
1984:.
1978:)
1975:N
1972:,
1969:M
1966:(
1958:N
1955:M
1949:=
1946:)
1943:N
1940:,
1937:M
1934:(
1904:)
1902:N
1898:M
1889:B
1885:A
1869:)
1864:n
1860:b
1856:,
1851:n
1847:a
1843:(
1835:n
1831:x
1822:)
1817:1
1813:b
1809:,
1804:1
1800:a
1796:(
1788:1
1784:x
1773:N
1769:M
1765:)
1763:N
1759:M
1747:M
1743:N
1727:M
1724:N
1702:.
1695:n
1691:a
1682:n
1678:b
1672:n
1668:x
1657:1
1653:a
1644:1
1640:b
1634:1
1630:x
1626:=
1621:M
1618:N
1594:M
1591:N
1579:B
1575:A
1571:i
1555:i
1551:b
1542:i
1538:a
1527:M
1519:N
1515:N
1509:M
1493:.
1490:]
1485:n
1481:b
1477:,
1471:,
1466:1
1462:b
1458:[
1455:=
1452:B
1432:]
1427:n
1423:a
1419:,
1413:,
1408:1
1404:a
1400:[
1397:=
1394:A
1370:n
1366:b
1360:n
1356:x
1345:1
1341:b
1335:1
1331:x
1327:=
1324:N
1300:n
1296:a
1290:n
1286:x
1275:1
1271:a
1265:1
1261:x
1257:=
1254:M
1235:m
1231:m
1201:)
1199:p
1193:)
1191:p
1187:p
1169:p
1137:.
1127:.
1084:.
1072:P
1069:M
1063:M
1042:P
1039:N
1033:P
1030:M
1024:N
1018:M
1005:P
1001:N
997:M
966:.
963:R
955:k
951:g
947:,
941:,
936:1
932:g
909:i
905:f
899:i
895:g
889:k
884:1
881:=
878:i
863:R
859:F
855:F
848:R
834:}
829:k
825:f
821:,
815:,
810:1
806:f
802:{
799:=
796:F
750:.
745:A
741:X
737:=
730:n
726:a
720:n
716:x
705:1
701:a
695:1
691:x
670:]
665:n
661:x
657:,
651:,
646:1
642:x
638:[
635:=
632:X
608:]
603:n
599:a
595:,
589:,
584:1
580:a
576:[
573:=
570:A
548:i
544:a
523:,
516:n
512:a
506:n
502:x
491:1
487:a
481:1
477:x
446:i
442:M
427:K
411:i
407:c
386:,
381:m
377:M
371:m
367:c
363:+
357:+
352:1
348:M
342:1
338:c
317:]
312:n
308:x
304:,
298:,
293:1
289:x
285:[
282:K
279:=
276:R
255:K
251:K
234:]
229:n
225:x
221:,
215:,
210:1
206:x
202:[
199:K
196:=
193:R
58:K
49:K
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