191:
942:, it has infinitely many solutions. It is thus not possible to enumerate them. It follows that, in this case, solving may only mean "finding a description of the solutions from which the relevant properties of the solutions are easy to extract". There is no commonly accepted such description. In fact there are many different "relevant properties", which involve almost every subfield of
1003:
of the coefficients of the system. There are several ways to represent the solution in an algebraic closure, which are discussed below. All of them allow one to compute a numerical approximation of the solutions by solving one or several univariate equations. For this computation, it is preferable to
3601:
There are at least four software packages which can solve zero-dimensional systems automatically (by automatically, one means that no human intervention is needed between input and output, and thus that no knowledge of the method by the user is needed). There are also several other software packages
2960:
may be used if the number of equations is equal to the number of variables. It does not allow one to find all the solutions nor to prove that there is no solution. But it is very fast when starting from a point which is close to a solution. Therefore, it is a basic tool for the homotopy continuation
2892:
The RUR is uniquely defined for a given separating variable, independently of any algorithm, and it preserves the multiplicities of the roots. This is a notable difference with triangular decompositions (even the equiprojectable decomposition), which, in general, do not preserve multiplicities. The
1501:
are usually represented as polynomials in a generator of the field which satisfies some univariate polynomial equation. To work with a polynomial system whose coefficients belong to a number field, it suffices to consider this generator as a new variable and to add the equation of the generator to
979:
For zero-dimensional systems, solving consists of computing all the solutions. There are two different ways of outputting the solutions. The most common way is possible only for real or complex solutions, and consists of outputting numeric approximations of the solutions. Such a solution is called
177:
Searching for solutions that belong to a specific set is a problem which is generally much more difficult, and is outside the scope of this article, except for the case of the solutions in a given finite field. For the case of solutions of which all components are integers or rational numbers, see
2967:
is rarely used for solving polynomial systems, but it succeeded, circa 1970, in showing that a system of 81 quadratic equations in 56 variables is not inconsistent. With the other known methods, this remains beyond the possibilities of modern technology, as of 2022. This method consists simply in
483:
3651:
The third solver is
Bertini, written by D. J. Bates, J. D. Hauenstein, A. J. Sommese, and C. W. Wampler. Bertini uses numerical homotopy continuation with adaptive precision. In addition to computing zero-dimensional solution sets, both PHCpack and Bertini are capable of working with positive
1407:
In the case of this simple example, it may be unclear whether the system is, or not, easier to solve than the equation. On more complicated examples, one lacks systematic methods for solving directly the equation, while software are available for automatically solving the corresponding system.
3558:
To deduce the numeric values of the solutions from a RUR seems easy: it suffices to compute the roots of the univariate polynomial and to substitute them in the other equations. This is not so easy because the evaluation of a polynomial at the roots of another polynomial is highly unstable.
2011:
The solutions of this system are obtained by solving the first univariate equation, substituting the solutions in the other equations, then solving the second equation which is now univariate, and so on. The definition of regular chains implies that the univariate equation obtained from
2223:
The first issue has been solved by Dahan and Schost: Among the sets of regular chains that represent a given set of solutions, there is a set for which the coefficients are explicitly bounded in terms of the size of the input system, with a nearly optimal bound. This set, called
2511:
902:
This exponential behavior makes solving polynomial systems difficult and explains why there are few solvers that are able to automatically solve systems with BĂ©zout's bound higher than, say, 25 (three equations of degree 3 or five equations of degree 2 are beyond this bound).
3617:
numbers, they are converted to rational numbers) and outputs the real solutions represented either (optionally) as intervals of rational numbers or as floating point approximations of arbitrary precision. If the system is not zero dimensional, this is signaled as an error.
2896:
For zero-dimensional systems, the RUR allows retrieval of the numeric values of the solutions by solving a single univariate polynomial and substituting them in rational functions. This allows production of certified approximations of the solutions to any given precision.
2006:
3621:
Internally, this solver, designed by F. Rouillier computes first a Gröbner basis and then a
Rational Univariate Representation from which the required approximation of the solutions are deduced. It works routinely for systems having up to a few hundred complex solutions.
2887:
2968:
minimizing the sum of the squares of the equations. If zero is found as a local minimum, then it is attained at a solution. This method works for overdetermined systems, but outputs an empty information if all local minimums which are found are positive.
2059:
Every zero-dimensional system of polynomial equations is equivalent (i.e. has the same solutions) to a finite number of regular chains. Several regular chains may be needed, as it is the case for the following system which has three solutions.
3647:
The second solver is PHCpack, written under the direction of J. Verschelde. PHCpack implements the homotopy continuation method. This solver computes the isolated complex solutions of polynomial systems having as many equations as variables.
3852:. Moreover, recent algorithms for decomposing polynomial systems into triangular decompositions produce regular chains with coefficients matching the results of Dahan and Schost. In proc. ISSAC'04, pages 103--110, ACM Press, 2004
1366:
2986:
This method divides into three steps. First an upper bound on the number of solutions is computed. This bound has to be as sharp as possible. Therefore, it is computed by, at least, four different methods and the best value, say
326:
289:
2181:
682:
3736:
3321:
2300:
3197:
between the two systems is considered. It consists, for example, of the straight line between the two systems, but other paths may be considered, in particular to avoid some singularities, in the system
2983:
This is a semi-numeric method which supposes that the number of equations is equal to the number of variables. This method is relatively old but it has been dramatically improved in the last decades.
1822:
769:(with polynomials as coefficients) of the first members of the equations. Most but not all overdetermined systems, when constructed with random coefficients, are inconsistent. For example, the system
538:, are less often used, as their elements cannot be represented in a computer (only approximations of real numbers can be used in computations, and these approximations are always rational numbers).
607:
331:
209:
160:
This article is about the methods for solving, that is, finding all solutions or describing them. As these methods are designed for being implemented in a computer, emphasis is given on fields
2711:
1172:
3188:
3068:
1244:
3562:
The roots of the univariate polynomial have thus to be computed at a high precision which may not be defined once for all. There are two algorithms which fulfill this requirement.
794:
if the number of equations is lower than the number of the variables. An underdetermined system is either inconsistent or has infinitely many complex solutions (or solutions in an
911:
The first thing to do for solving a polynomial system is to decide whether it is inconsistent, zero-dimensional or positive dimensional. This may be done by the computation of a
3644:
several times, doubling the precision each time, until solutions remain stable, as the substitution of the roots in the equations of the input variables can be highly unstable.
3439:
737:. However, it has been shown that, for the case of the singular points of a surface of degree 6, the maximum number of solutions is 65, and is reached by the Barth surface.
721:, shown in the figure is the geometric representation of the solutions of a polynomial system reduced to a single equation of degree 6 in 3 variables. Some of its numerous
3548:
2201:
There is also an algorithm which is specific to the zero-dimensional case and is competitive, in this case, with the direct algorithms. It consists in computing first the
1561:
1527:
2219:
To deduce the numeric values of the solutions from the output, one has to solve univariate polynomials with approximate coefficients, which is a highly unstable problem.
2212:
This representation of the solutions are fully convenient for coefficients in a finite field. However, for rational coefficients, two aspects have to be taken care of:
1004:
use a representation that involves solving only one univariate polynomial per solution, because computing the roots of a polynomial which has approximate coefficients
3505:
3472:
3395:
3369:
3345:
3128:
3108:
3088:
3005:
309:
The subject of this article is the study of generalizations of such an examples, and the description of the methods that are used for computing the solutions.
2266:, described below, allows computing such a special regular chain, satisfying DahanâSchost bound, by starting from either a regular chain or a Gröbner basis.
3550:
Too large, Newton's convergence may be slow and may even jump from a solution path to another one. Too small, and the number of steps slows down the method.
1255:
478:{\displaystyle {\begin{aligned}f_{1}\left(x_{1},\ldots ,x_{m}\right)&=0\\&\;\;\vdots \\f_{n}\left(x_{1},\ldots ,x_{m}\right)&=0,\end{aligned}}}
2278:
or RUR is a representation of the solutions of a zero-dimensional polynomial system over the rational numbers which has been introduced by F. Rouillier.
3885:
2946:
work also for polynomial systems. However the specific methods will generally be preferred, as the general methods generally do not allow one to find
204:
2066:
1502:
the equations of the system. Thus solving a polynomial system over a number field is reduced to solving another system over the rational numbers.
2926:
of the ideal). In practice, this provides an output with much smaller coefficients, especially in the case of systems with high multiplicities.
602:
817:
if it has a finite number of complex solutions (or solutions in an algebraically closed field). This terminology comes from the fact that the
1600:
The usual way of representing the solutions is through zero-dimensional regular chains. Such a chain consists of a sequence of polynomials
729:
has no solution in general (that is if the coefficients are not specific). If it has a finite number of solutions, this number is at most
3204:
2950:
solutions. In particular, when a general method does not find any solution, this is usually not an indication that there is no solution.
2506:{\displaystyle {\begin{cases}h(x_{0})=0\\x_{1}=g_{1}(x_{0})/g_{0}(x_{0})\\\quad \vdots \\x_{n}=g_{n}(x_{0})/g_{0}(x_{0}),\end{cases}}}
2001:{\displaystyle {\begin{cases}f_{1}(x_{1})=0\\f_{2}(x_{1},x_{2})=0\\\quad \vdots \\f_{n}(x_{1},x_{2},\ldots ,x_{n})=0.\end{cases}}}
306:. These solutions can easily be checked by substitution, but more work is needed for proving that there are no other solutions.
722:
2893:
RUR shares with equiprojectable decomposition the property of producing an output with coefficients of relatively small size.
4164:(SIAM ed.). Society for Industrial and Applied Mathematics (SIAM, 3600 Market Street, Floor 6, Philadelphia, PA 19104).
4131:
1039:
3691:
3663:, written by Marc Moreno-Maza and collaborators. It contains various functions for solving polynomial systems by means of
2929:
Contrarily to triangular decompositions and equiprojectable decompositions, the RUR is not defined in positive dimension.
2882:{\displaystyle {\begin{cases}t^{3}-t=0\\x={\frac {t^{2}+2t-1}{3t^{2}-1}}\\y={\frac {t^{2}-2t-1}{3t^{2}-1}}.\\\end{cases}}}
966:
988:
if it is provided with a bound on the error of the approximations, and if this bound separates the different solutions.
4150:
3681:
2655:
For example, for the system in the previous section, every linear combination of the variable, except the multiples of
807:
2209:, then deducing the lexicographical Gröbner basis by FGLM algorithm and finally applying the Lextriangular algorithm.
1090:
4169:
822:
4143:
Ideals, varieties, and algorithms : an introduction to computational algebraic geometry and commutative algebra
780:
is overdetermined (having two equations but only one unknown), but it is not inconsistent since it has the solution
3941:
932:
3133:
3013:
3580:. This algorithms computes the real roots, isolated in intervals of arbitrary small width. It is implemented in
725:
are visible on the image. They are the solutions of a system of 4 equations of degree 5 in 3 variables. Such an
4188:
3602:
which may be useful for solving zero-dimensional systems. Some of them are listed after the automatic solvers.
2053:
960:
889:
3899:
Rouillier, Fabrice (1999). "Solving Zero-Dimensional
Systems Through the Rational Univariate Representation".
2943:
2195:
1183:
1038:. Such an equation may be converted into a polynomial system by expanding the sines and cosines in it (using
2571:
Given a zero-dimensional polynomial system over the rational numbers, the RUR has the following properties.
803:
762:
2216:
The output may involve huge integers which may make the computation and the use of the result problematic.
4228:
3577:
970:
3935:
3640:, which computes the complex roots of univariate polynomials to any precision. It is recommended to run
2964:
2605:
795:
758:
574:
2235:
The second issue is generally solved by outputting regular chains of a special form, sometimes called
4218:
1005:
927:
of some element of the Gröbner basis which is a pure power of this variable. For this test, the best
3404:
2720:
2309:
2075:
1831:
1264:
3863:
3686:
2914:
of the RUR may be factorized, and this gives a RUR for every irreducible factor. This provides the
2187:
1035:
973:
711:
502:
3993:"Algorithm 795: PHCpack: A general-purpose solver for polynomial systems by homotopy continuation"
3889:.In proc. ISSAC'2011, pages 83-90, ACM Press, 2011 and Journal of Symbolic Computation (to appear)
3510:
3992:
1542:
1508:
3576:
Uspensky's algorithm of
Collins and Akritas, improved by Rouillier and Zimmermann and based on
1498:
949:
A natural example of such a question concerning positive-dimensional systems is the following:
814:
791:
750:
707:
28:
3752:"Triangular Sets for Solving Polynomial Systems: a Comparative Implementation of Four Methods"
1575:
In the case of a finite field, the same transformation allows always supposing that the field
837:
734:
166:
in which computation (including equality testing) is easy and efficient, that is the field of
3477:
3444:
2978:
2919:
746:
726:
3636:
To extract all the complex solutions from a rational univariate representation, one may use
4223:
4122:
Bates, Daniel J.; Sommese, Andrew J.; Hauenstein, Jonathan D.; Wampler, Charles W. (2013).
4061:
1492:
179:
110:
8:
3613:
takes as input any polynomial system over the rational numbers (if some coefficients are
3374:
2923:
799:
578:
523:
194:
The numerous singular points of the Barth sextic are the solutions of a polynomial system
83:
4065:
4208:
4015:
3916:
3676:
3354:
3330:
3113:
3093:
3073:
2990:
2957:
2229:
1361:{\displaystyle {\begin{cases}s^{3}+4c^{3}-3c&=0\\s^{2}+c^{2}-1&=0.\end{cases}}}
943:
766:
569:
that satisfies all equations of the polynomial system. The solutions are sought in the
3848:
3833:
2575:
All but a finite number linear combinations of the variables are separating variables.
832:
A zero-dimensional system with as many equations as variables is sometimes said to be
4184:
4165:
4162:
Solving polynomial systems using continuation for engineering and scientific problems
4146:
4127:
2642:
996:
818:
4019:
3920:
2578:
When the separating variable is chosen, the RUR exists and is unique. In particular
995:. It uses the fact that, for a zero-dimensional system, the solutions belong to the
596:
The set of solutions is not always finite; for example, the solutions of the system
593:
solutions are much more difficult problems that are not considered in this article.
4213:
4069:
4007:
3968:
3908:
3829:
3800:
3763:
3656:
3626:
3606:
3581:
3553:
2254:. For getting such regular chains, one may have to add a further variable, called
2202:
931:(that is the one which leads generally to the fastest computation) is usually the
924:
912:
798:
that contains the coefficients of the equations). This is a non-trivial result of
3862:
Dahan, Xavier; Moreno Maza, Marc; Schost, Eric; Wu, Wenyuan; Xie, Yuzhen (2005).
3784:
952:
590:
527:
167:
148:
134:
113:
4034:
284:{\displaystyle {\begin{aligned}x^{2}+y^{2}-5&=0\\xy-2&=0.\end{aligned}}}
3789:"Efficient Computation of Zero-Dimensional Gröbner Basis by Change of Ordering"
3614:
2598:
The solutions of the system are in one-to-one correspondence with the roots of
2206:
2052:
solutions, provided that there is no multiple root in this resolution process (
928:
754:
749:
if the number of equations is higher than the number of variables. A system is
582:
570:
144:
4074:
4049:
3973:
3956:
3933:
4202:
4088:
4038:. Proceedings of the 1976 ACM Symposium on Symbolic and Algebraic Computation
3664:
3566:
2191:
2176:{\displaystyle {\begin{cases}x^{2}-1=0\\(x-1)(y-1)=0\\y^{2}-1=0.\end{cases}}}
1810:
1595:
757:
solution (or, if the coefficients are not complex numbers, no solution in an
718:
3805:
3788:
3768:
3751:
531:
171:
4011:
3912:
2190:
of an arbitrary polynomial system (not necessarily zero-dimensional) into
976:
and therefore cannot be used in practice, except for very small examples.
677:{\displaystyle {\begin{aligned}x(x-1)&=0\\x(y-1)&=0\end{aligned}}}
586:
535:
3371:
solutions during this deformation. This gives the desired solutions for
3090:
solutions that are easy to compute. This new system has the same number
2228:, depends only on the choice of the coordinates. This allows the use of
1529:, a system over the rational numbers is obtained by adding the equation
1493:
Coefficients in a number field or in a finite field with non-prime order
3886:
Algorithms for
Computing Triangular Decomposition of Polynomial Systems
3507:
by Newton's method. The difficulty here is to well choose the value of
498:
61:
2617:
The solutions of the system are obtained by substituting the roots of
3130:
of equations and the same general structure as the system to solve,
2281:
A RUR of a zero-dimensional system consists in a linear combination
3316:{\displaystyle (1-t)g_{1}+tf_{1}=0,\,\ldots ,\,(1-t)g_{n}+tf_{n}=0}
3194:
3820:
Lazard, D. (1992). "Solving zero-dimensional algebraic systems".
3641:
3637:
3570:
706:. Even when the solution set is finite, there is, in general, no
4126:. Philadelphia: Society for Industrial and Applied Mathematics.
876:
solutions. This bound is sharp. If all the degrees are equal to
840:
asserts that a well-behaved system whose equations have degrees
3554:
Numerically solving from the rational univariate representation
3327:
The homotopy continuation consists in deforming the parameter
190:
4121:
4099:
4035:
Polynomial Real Root
Isolation Using Descartes' Rule of Signs
3722:
3710:
2232:
for computing efficiently the equiprojectable decomposition.
546:
3625:
The rational univariate representation may be computed with
2942:
The general numerical algorithms which are designed for any
1584:
825:
zero. A system with infinitely many solutions is said to be
710:
of the solutions (in the case of a single equation, this is
157:, which is isomorphic to a subfield of the complex numbers.
3934:
Saugata Basu; Richard
Pollack; Marie-Françoise Roy (2006).
2875:
2595:
are defined independently of any algorithm to compute them.
2499:
2169:
1994:
1354:
522:, with integer coefficients, or coefficients in some fixed
1428:
elements, one is primarily interested in the solutions in
991:
The other way of representing the solutions is said to be
4110:
965:. The classical algorithm for solving these question is
3861:
2953:
Nevertheless, two methods deserve to be mentioned here.
198:
A simple example of a system of polynomial equations is
3782:
3070:
of polynomial equations is generated which has exactly
915:
of the left-hand sides of the equations. The system is
2614:
equals the multiplicity of the corresponding solution.
955:
has a finite number of real solutions and compute them
3957:"Thirty years of Polynomial System Solving, and now?"
3513:
3480:
3447:
3407:
3377:
3357:
3333:
3207:
3136:
3116:
3096:
3076:
3016:
2993:
2714:
2303:
2269:
2069:
1825:
1545:
1511:
1258:
1186:
1093:
919:
if this Gröbner basis is reduced to 1. The system is
605:
329:
207:
888:
and is exponential in the number of variables. (The
4140:
4124:
Numerically solving polynomial systems with
Bertini
3937:
Algorithms in real algebraic geometry, chapter 12.4
3737:
A Package for
Solving Parametric Polynomial Systems
3734:Songxin Liang, J. Gerhard, D.J. Jeffrey, G. Moroz,
963:
of the set of real solutions of a polynomial system
740:
4111:Bertini: Software for Numerical Algebraic Geometry
4047:
3864:"Lifting techniques for triangular decompositions"
3542:
3499:
3466:
3433:
3389:
3363:
3339:
3315:
3182:
3122:
3102:
3082:
3062:
2999:
2881:
2505:
2175:
2000:
1555:
1521:
1360:
1238:
1166:
676:
477:
283:
98:of a polynomial system is a set of values for the
4200:
4141:Cox, David; Little, John; O'Shea, Donal (1997).
4054:Journal of Computational and Applied Mathematics
4050:"Efficient isolation of polynomial's real roots"
3573:computes all the complex roots to any precision.
1451:, it suffices, for restricting the solutions to
1167:{\displaystyle \cos(3x)=4\cos ^{3}(x)-3\cos(x),}
2972:
1249:is equivalent to solving the polynomial system
577:containing the coefficients. In particular, in
1411:
4183:. Providence, RI: American Mathematical Soc.
4032:George E. Collins and Alkiviadis G. Akritas,
2937:
2186:There are several algorithms for computing a
2207:graded reverse lexicographic order (grevlex)
534:. Other fields of coefficients, such as the
3740:. Communications in Computer Algebra (2009)
3183:{\displaystyle f_{1}=0,\,\ldots ,\,f_{n}=0}
3063:{\displaystyle g_{1}=0,\,\ldots ,\,g_{n}=0}
2677:, is a separating variable. If one chooses
1389:of this system, there is a unique solution
3990:
2705:as a separating variable, then the RUR is
1440:are exactly the solutions of the equation
1416:When solving a system over a finite field
1016:
401:
400:
16:Roots of multiple multivariate polynomials
4178:
4073:
4000:ACM Transactions on Mathematical Software
3986:
3984:
3972:
3898:
3804:
3767:
3749:
3265:
3258:
3163:
3156:
3043:
3036:
1784:which does not have any common zero with
1585:Algebraic representation of the solutions
1021:A trigonometric equation is an equation
585:solutions are sought. Searching for the
189:
143:is generally assumed to be the field of
4181:Solving systems of polynomial equations
3704:
1239:{\displaystyle \sin ^{3}(x)+\cos(3x)=0}
957:. A generalization of this question is
951:decide if a polynomial system over the
4201:
4159:
3981:
3954:
3819:
2932:
304:) = (1, 2), (2, 1), (-1, -2), (-2, -1)
4048:Rouillier, F.; Zimmerman, P. (2004).
2632:does not have any multiple root then
1084:For example, because of the identity
147:, because each solution belongs to a
4145:(2nd ed.). New York: Springer.
3596:
2900:Moreover, the univariate polynomial
127:, and make all equations true. When
3883:Changbo Chen and Marc Moreno-Maza.
3849:Sharp Estimates for Triangular Sets
3750:Aubry, P.; Maza, M. Moreno (1999).
3474:are deduced from the solutions for
967:cylindrical algebraic decomposition
959:find at least one solution in each
906:
13:
3682:Systems of polynomial inequalities
2276:rational univariate representation
2270:Rational univariate representation
2264:rational univariate representation
1505:For example, if a system contains
923:if, for every variable there is a
14:
4240:
3901:Appl. Algebra Eng. Commun. Comput
3692:Wu's method of characteristic set
3110:of variables and the same number
1589:
761:containing the coefficients). By
294:Its solutions are the four pairs
2918:of the given ideal (that is the
741:Basic properties and definitions
4104:
4093:
4082:
4041:
4026:
3948:
3927:
3892:
3822:Journal of Symbolic Computation
3793:Journal of Symbolic Computation
3787:; Lazard, D.; Mora, T. (1993).
2415:
2250:but the first one are equal to
1919:
808:Krull's principal ideal theorem
314:system of polynomial equations,
3877:
3873:. ACM Press. pp. 108â105.
3855:
3846:Xavier Dahan and Eric Schost.
3840:
3813:
3776:
3743:
3728:
3716:
3434:{\displaystyle t_{1}<t_{2}}
3278:
3266:
3220:
3208:
2553:are univariate polynomials in
2522:is a univariate polynomial in
2490:
2477:
2459:
2446:
2408:
2395:
2377:
2364:
2328:
2315:
2196:regular semi-algebraic systems
2131:
2119:
2116:
2104:
2054:fundamental theorem of algebra
1982:
1937:
1906:
1880:
1857:
1844:
1815:triangular system of equations
1227:
1218:
1206:
1200:
1158:
1152:
1137:
1131:
1109:
1100:
890:fundamental theorem of algebra
802:that involves, in particular,
657:
645:
625:
613:
21:system of polynomial equations
1:
3834:10.1016/S0747-7171(08)80086-7
3697:
3010:In the second step, a system
2944:system of nonlinear equations
2226:equiprojectable decomposition
2034:and thus that the system has
1011:
320:is a collection of equations
185:
3543:{\displaystyle t_{2}-t_{1}:}
2973:Homotopy continuation method
2294:, and a system of equations
1070:and adding the new equation
1006:is a highly unstable problem
933:graded reverse lexicographic
545:of a polynomial system is a
7:
3670:
3652:dimensional solution sets.
2258:, which is given the index
1556:{\displaystyle {\sqrt {2}}}
1522:{\displaystyle {\sqrt {2}}}
1412:Solutions in a finite field
1040:sum and difference formulas
10:
4245:
4160:Morgan, Alexander (1987).
2976:
2938:General solving algorithms
1593:
1393:of the equation such that
796:algebraically closed field
759:algebraically closed field
575:algebraically closed field
573:, or more generally in an
64:in several variables, say
4179:Sturmfels, Bernd (2002).
4089:Release 2.3.86 of PHCpack
4075:10.1016/j.cam.2003.08.015
3974:10.1016/j.jsc.2008.03.004
3871:Proceedings of ISAAC 2005
3655:The fourth solver is the
2290:of the variables, called
1717:only, which has a degree
804:Hilbert's Nullstellensatz
763:Hilbert's Nullstellensatz
3991:Verschelde, Jan (1999).
3687:Triangular decomposition
3578:Descartes' rule of signs
2188:triangular decomposition
1572:in the other equations.
1036:trigonometric polynomial
974:computational complexity
3955:Lazard, Daniel (2009).
3500:{\displaystyle t=t_{1}}
3467:{\displaystyle t=t_{2}}
2961:method described below.
2623:in the other equations.
1478:for each variable
1017:Trigonometric equations
765:this means that 1 is a
109:s which belong to some
3806:10.1006/jsco.1993.1051
3769:10.1006/jsco.1999.0270
3544:
3501:
3468:
3435:
3391:
3365:
3341:
3317:
3184:
3124:
3104:
3084:
3064:
3001:
2883:
2507:
2177:
2002:
1557:
1523:
1499:algebraic number field
1457:, to add the equation
1362:
1240:
1168:
708:closed-form expression
678:
479:
285:
195:
29:simultaneous equations
4012:10.1145/317275.317286
3913:10.1007/s002000050114
3545:
3502:
3469:
3436:
3392:
3366:
3342:
3318:
3185:
3125:
3105:
3085:
3065:
3002:
2979:Homotopy continuation
2920:primary decomposition
2884:
2508:
2178:
2003:
1669:such that, for every
1558:
1524:
1434:. As the elements of
1363:
1241:
1177:solving the equation
1169:
1058:by two new variables
882:, this bound becomes
821:of the solutions has
727:overdetermined system
679:
526:, often the field of
480:
286:
193:
3511:
3478:
3445:
3441:, the solutions for
3405:
3375:
3355:
3331:
3205:
3134:
3114:
3094:
3074:
3014:
2991:
2712:
2562:of degree less than
2301:
2067:
1823:
1543:
1509:
1256:
1184:
1091:
940:positive-dimensional
892:is the special case
827:positive-dimensional
712:AbelâRuffini theorem
603:
327:
205:
180:Diophantine equation
111:algebraically closed
23:(sometimes simply a
4066:2004JCoAM.162...33R
3390:{\displaystyle t=1}
2933:Solving numerically
2916:prime decomposition
2292:separating variable
2256:separating variable
1765:is a polynomial in
1743:the coefficient of
1699:is a polynomial in
1581:has a prime order.
1497:The elements of an
961:connected component
800:commutative algebra
579:characteristic zero
4229:Algebraic geometry
3677:Elimination theory
3540:
3497:
3464:
3431:
3387:
3361:
3337:
3313:
3180:
3120:
3100:
3080:
3060:
2997:
2879:
2874:
2503:
2498:
2173:
2168:
1998:
1993:
1553:
1519:
1371:For each solution
1358:
1353:
1236:
1164:
971:doubly exponential
944:algebraic geometry
767:linear combination
674:
672:
475:
473:
281:
279:
196:
4133:978-1-61197-269-6
4100:Bates et al. 2013
3723:Bates et al. 2013
3711:Bates et al. 2013
3597:Software packages
3569:, implemented in
3364:{\displaystyle N}
3340:{\displaystyle t}
3123:{\displaystyle n}
3103:{\displaystyle n}
3083:{\displaystyle N}
3000:{\displaystyle N}
2867:
2805:
1551:
1517:
997:algebraic closure
938:If the system is
819:algebraic variety
318:polynomial system
25:polynomial system
4236:
4219:Computer algebra
4194:
4175:
4156:
4137:
4113:
4108:
4102:
4097:
4091:
4086:
4080:
4079:
4077:
4045:
4039:
4030:
4024:
4023:
3997:
3988:
3979:
3978:
3976:
3952:
3946:
3945:
3931:
3925:
3924:
3896:
3890:
3881:
3875:
3874:
3868:
3859:
3853:
3844:
3838:
3837:
3817:
3811:
3810:
3808:
3780:
3774:
3773:
3771:
3762:(1â2): 125â154.
3747:
3741:
3732:
3726:
3720:
3714:
3708:
3549:
3547:
3546:
3541:
3536:
3535:
3523:
3522:
3506:
3504:
3503:
3498:
3496:
3495:
3473:
3471:
3470:
3465:
3463:
3462:
3440:
3438:
3437:
3432:
3430:
3429:
3417:
3416:
3396:
3394:
3393:
3388:
3370:
3368:
3367:
3362:
3347:from 0 to 1 and
3346:
3344:
3343:
3338:
3322:
3320:
3319:
3314:
3306:
3305:
3290:
3289:
3248:
3247:
3232:
3231:
3189:
3187:
3186:
3181:
3173:
3172:
3146:
3145:
3129:
3127:
3126:
3121:
3109:
3107:
3106:
3101:
3089:
3087:
3086:
3081:
3069:
3067:
3066:
3061:
3053:
3052:
3026:
3025:
3006:
3004:
3003:
2998:
2913:
2888:
2886:
2885:
2880:
2878:
2877:
2868:
2866:
2859:
2858:
2845:
2829:
2828:
2818:
2806:
2804:
2797:
2796:
2783:
2767:
2766:
2756:
2732:
2731:
2704:
2703:
2701:
2700:
2697:
2694:
2676:
2666:
2660:
2650:
2640:
2631:
2622:
2613:
2608:of each root of
2603:
2594:
2583:
2567:
2561:
2552:
2536:
2530:
2521:
2512:
2510:
2509:
2504:
2502:
2501:
2489:
2488:
2476:
2475:
2466:
2458:
2457:
2445:
2444:
2432:
2431:
2407:
2406:
2394:
2393:
2384:
2376:
2375:
2363:
2362:
2350:
2349:
2327:
2326:
2289:
2261:
2253:
2249:
2239:, for which all
2182:
2180:
2179:
2174:
2172:
2171:
2153:
2152:
2087:
2086:
2051:
2033:
2022:
2007:
2005:
2004:
1999:
1997:
1996:
1981:
1980:
1962:
1961:
1949:
1948:
1936:
1935:
1905:
1904:
1892:
1891:
1879:
1878:
1856:
1855:
1843:
1842:
1813:is associated a
1804:
1792:
1783:
1764:
1753:
1739:
1728:
1716:
1698:
1685:
1674:
1668:
1640:
1616:
1580:
1571:
1562:
1560:
1559:
1554:
1552:
1547:
1538:
1528:
1526:
1525:
1520:
1518:
1513:
1488:
1477:
1456:
1450:
1439:
1433:
1427:
1421:
1403:
1402:
1392:
1388:
1367:
1365:
1364:
1359:
1357:
1356:
1336:
1335:
1323:
1322:
1292:
1291:
1276:
1275:
1245:
1243:
1242:
1237:
1196:
1195:
1173:
1171:
1170:
1165:
1127:
1126:
1080:
1069:
1063:
1057:
1049:
1033:
1027:
984:. A solution is
953:rational numbers
925:leading monomial
921:zero-dimensional
907:What is solving?
898:
887:
881:
875:
857:
838:BĂ©zout's theorem
815:zero-dimensional
786:
779:
735:BĂ©zout's theorem
732:
705:
698:
683:
681:
680:
675:
673:
568:
528:rational numbers
521:
496:
484:
482:
481:
476:
474:
457:
453:
452:
451:
433:
432:
418:
417:
396:
382:
378:
377:
376:
358:
357:
343:
342:
305:
290:
288:
287:
282:
280:
234:
233:
221:
220:
168:rational numbers
165:
156:
142:
135:rational numbers
133:is the field of
132:
126:
120:
108:
90:
81:
59:
48:
4244:
4243:
4239:
4238:
4237:
4235:
4234:
4233:
4199:
4198:
4197:
4191:
4172:
4153:
4134:
4117:
4116:
4109:
4105:
4098:
4094:
4087:
4083:
4046:
4042:
4031:
4027:
3995:
3989:
3982:
3961:J. Symb. Comput
3953:
3949:
3942:Springer-Verlag
3932:
3928:
3897:
3893:
3882:
3878:
3866:
3860:
3856:
3845:
3841:
3818:
3814:
3783:FaugĂšre, J.C.;
3781:
3777:
3756:J. Symb. Comput
3748:
3744:
3733:
3729:
3721:
3717:
3709:
3705:
3700:
3673:
3599:
3556:
3531:
3527:
3518:
3514:
3512:
3509:
3508:
3491:
3487:
3479:
3476:
3475:
3458:
3454:
3446:
3443:
3442:
3425:
3421:
3412:
3408:
3406:
3403:
3402:
3401:means that, if
3376:
3373:
3372:
3356:
3353:
3352:
3332:
3329:
3328:
3301:
3297:
3285:
3281:
3243:
3239:
3227:
3223:
3206:
3203:
3202:
3168:
3164:
3141:
3137:
3135:
3132:
3131:
3115:
3112:
3111:
3095:
3092:
3091:
3075:
3072:
3071:
3048:
3044:
3021:
3017:
3015:
3012:
3011:
2992:
2989:
2988:
2981:
2975:
2958:Newton's method
2940:
2935:
2911:
2901:
2873:
2872:
2854:
2850:
2846:
2824:
2820:
2819:
2817:
2808:
2807:
2792:
2788:
2784:
2762:
2758:
2757:
2755:
2746:
2745:
2727:
2723:
2716:
2715:
2713:
2710:
2709:
2698:
2695:
2686:
2685:
2683:
2678:
2668:
2662:
2656:
2646:
2639:
2633:
2627:
2618:
2609:
2599:
2593:
2585:
2579:
2563:
2560:
2554:
2551:
2544:
2538:
2532:
2529:
2523:
2517:
2497:
2496:
2484:
2480:
2471:
2467:
2462:
2453:
2449:
2440:
2436:
2427:
2423:
2420:
2419:
2412:
2411:
2402:
2398:
2389:
2385:
2380:
2371:
2367:
2358:
2354:
2345:
2341:
2338:
2337:
2322:
2318:
2305:
2304:
2302:
2299:
2298:
2288:
2282:
2272:
2259:
2251:
2248:
2240:
2230:modular methods
2167:
2166:
2148:
2144:
2141:
2140:
2101:
2100:
2082:
2078:
2071:
2070:
2068:
2065:
2064:
2050:
2041:
2035:
2032:
2024:
2021:
2013:
1992:
1991:
1976:
1972:
1957:
1953:
1944:
1940:
1931:
1927:
1924:
1923:
1916:
1915:
1900:
1896:
1887:
1883:
1874:
1870:
1867:
1866:
1851:
1847:
1838:
1834:
1827:
1826:
1824:
1821:
1820:
1803:
1794:
1791:
1785:
1782:
1772:
1766:
1763:
1755:
1752:
1744:
1738:
1730:
1726:
1718:
1715:
1706:
1700:
1697:
1689:
1676:
1670:
1666:
1657:
1650:
1642:
1638:
1631:
1624:
1618:
1614:
1607:
1601:
1598:
1592:
1587:
1576:
1570:
1564:
1546:
1544:
1541:
1540:
1536:
1530:
1512:
1510:
1507:
1506:
1495:
1487:
1479:
1475:
1466:
1458:
1452:
1441:
1435:
1429:
1423:
1417:
1414:
1400:
1394:
1390:
1386:
1379:
1372:
1352:
1351:
1343:
1331:
1327:
1318:
1314:
1311:
1310:
1302:
1287:
1283:
1271:
1267:
1260:
1259:
1257:
1254:
1253:
1191:
1187:
1185:
1182:
1181:
1122:
1118:
1092:
1089:
1088:
1071:
1065:
1059:
1051:
1043:
1029:
1022:
1019:
1014:
935:one (grevlex).
909:
893:
883:
877:
874:
865:
859:
856:
847:
841:
792:underdetermined
781:
770:
743:
730:
723:singular points
700:
688:
671:
670:
660:
639:
638:
628:
606:
604:
601:
600:
571:complex numbers
566:
557:
550:
520:
511:
505:
494:
489:
472:
471:
458:
447:
443:
428:
424:
423:
419:
413:
409:
406:
405:
394:
393:
383:
372:
368:
353:
349:
348:
344:
338:
334:
330:
328:
325:
324:
295:
278:
277:
267:
252:
251:
241:
229:
225:
216:
212:
208:
206:
203:
202:
188:
161:
152:
149:field extension
145:complex numbers
138:
128:
122:
116:
114:field extension
107:
99:
86:
80:
71:
65:
58:
50:
46:
37:
31:
17:
12:
11:
5:
4242:
4232:
4231:
4226:
4221:
4216:
4211:
4196:
4195:
4189:
4176:
4170:
4157:
4152:978-0387946801
4151:
4138:
4132:
4118:
4115:
4114:
4103:
4092:
4081:
4040:
4025:
4006:(2): 251â276.
3980:
3947:
3926:
3907:(9): 433â461.
3891:
3876:
3854:
3839:
3828:(2): 117â131.
3812:
3799:(4): 329â344.
3775:
3742:
3727:
3715:
3702:
3701:
3699:
3696:
3695:
3694:
3689:
3684:
3679:
3672:
3669:
3665:regular chains
3615:floating point
3598:
3595:
3594:
3593:
3574:
3555:
3552:
3539:
3534:
3530:
3526:
3521:
3517:
3494:
3490:
3486:
3483:
3461:
3457:
3453:
3450:
3428:
3424:
3420:
3415:
3411:
3386:
3383:
3380:
3360:
3336:
3325:
3324:
3312:
3309:
3304:
3300:
3296:
3293:
3288:
3284:
3280:
3277:
3274:
3271:
3268:
3264:
3261:
3257:
3254:
3251:
3246:
3242:
3238:
3235:
3230:
3226:
3222:
3219:
3216:
3213:
3210:
3179:
3176:
3171:
3167:
3162:
3159:
3155:
3152:
3149:
3144:
3140:
3119:
3099:
3079:
3059:
3056:
3051:
3047:
3042:
3039:
3035:
3032:
3029:
3024:
3020:
2996:
2977:Main article:
2974:
2971:
2970:
2969:
2962:
2939:
2936:
2934:
2931:
2909:
2890:
2889:
2876:
2871:
2865:
2862:
2857:
2853:
2849:
2844:
2841:
2838:
2835:
2832:
2827:
2823:
2816:
2813:
2810:
2809:
2803:
2800:
2795:
2791:
2787:
2782:
2779:
2776:
2773:
2770:
2765:
2761:
2754:
2751:
2748:
2747:
2744:
2741:
2738:
2735:
2730:
2726:
2722:
2721:
2719:
2653:
2652:
2637:
2624:
2615:
2596:
2589:
2576:
2558:
2549:
2542:
2527:
2514:
2513:
2500:
2495:
2492:
2487:
2483:
2479:
2474:
2470:
2465:
2461:
2456:
2452:
2448:
2443:
2439:
2435:
2430:
2426:
2422:
2421:
2418:
2414:
2413:
2410:
2405:
2401:
2397:
2392:
2388:
2383:
2379:
2374:
2370:
2366:
2361:
2357:
2353:
2348:
2344:
2340:
2339:
2336:
2333:
2330:
2325:
2321:
2317:
2314:
2311:
2310:
2308:
2286:
2271:
2268:
2244:
2221:
2220:
2217:
2192:regular chains
2184:
2183:
2170:
2165:
2162:
2159:
2156:
2151:
2147:
2143:
2142:
2139:
2136:
2133:
2130:
2127:
2124:
2121:
2118:
2115:
2112:
2109:
2106:
2103:
2102:
2099:
2096:
2093:
2090:
2085:
2081:
2077:
2076:
2074:
2046:
2039:
2028:
2017:
2009:
2008:
1995:
1990:
1987:
1984:
1979:
1975:
1971:
1968:
1965:
1960:
1956:
1952:
1947:
1943:
1939:
1934:
1930:
1926:
1925:
1922:
1918:
1917:
1914:
1911:
1908:
1903:
1899:
1895:
1890:
1886:
1882:
1877:
1873:
1869:
1868:
1865:
1862:
1859:
1854:
1850:
1846:
1841:
1837:
1833:
1832:
1830:
1807:
1806:
1798:
1789:
1777:
1770:
1759:
1748:
1741:
1734:
1722:
1711:
1704:
1693:
1662:
1655:
1646:
1636:
1629:
1622:
1612:
1605:
1594:Main article:
1591:
1590:Regular chains
1588:
1586:
1583:
1568:
1550:
1539:and replacing
1534:
1516:
1494:
1491:
1483:
1471:
1462:
1413:
1410:
1384:
1377:
1369:
1368:
1355:
1350:
1347:
1344:
1342:
1339:
1334:
1330:
1326:
1321:
1317:
1313:
1312:
1309:
1306:
1303:
1301:
1298:
1295:
1290:
1286:
1282:
1279:
1274:
1270:
1266:
1265:
1263:
1247:
1246:
1235:
1232:
1229:
1226:
1223:
1220:
1217:
1214:
1211:
1208:
1205:
1202:
1199:
1194:
1190:
1175:
1174:
1163:
1160:
1157:
1154:
1151:
1148:
1145:
1142:
1139:
1136:
1133:
1130:
1125:
1121:
1117:
1114:
1111:
1108:
1105:
1102:
1099:
1096:
1018:
1015:
1013:
1010:
969:, which has a
929:monomial order
908:
905:
870:
863:
852:
845:
747:overdetermined
742:
739:
685:
684:
669:
666:
663:
661:
659:
656:
653:
650:
647:
644:
641:
640:
637:
634:
631:
629:
627:
624:
621:
618:
615:
612:
609:
608:
562:
555:
516:
509:
503:indeterminates
492:
486:
485:
470:
467:
464:
461:
459:
456:
450:
446:
442:
439:
436:
431:
427:
422:
416:
412:
408:
407:
404:
399:
397:
395:
392:
389:
386:
384:
381:
375:
371:
367:
364:
361:
356:
352:
347:
341:
337:
333:
332:
292:
291:
276:
273:
270:
268:
266:
263:
260:
257:
254:
253:
250:
247:
244:
242:
240:
237:
232:
228:
224:
219:
215:
211:
210:
187:
184:
103:
76:
69:
54:
42:
35:
27:) is a set of
15:
9:
6:
4:
3:
2:
4241:
4230:
4227:
4225:
4222:
4220:
4217:
4215:
4212:
4210:
4207:
4206:
4204:
4192:
4186:
4182:
4177:
4173:
4171:9780898719031
4167:
4163:
4158:
4154:
4148:
4144:
4139:
4135:
4129:
4125:
4120:
4119:
4112:
4107:
4101:
4096:
4090:
4085:
4076:
4071:
4067:
4063:
4059:
4055:
4051:
4044:
4037:
4036:
4029:
4021:
4017:
4013:
4009:
4005:
4001:
3994:
3987:
3985:
3975:
3970:
3966:
3962:
3958:
3951:
3943:
3939:
3938:
3930:
3922:
3918:
3914:
3910:
3906:
3902:
3895:
3888:
3887:
3880:
3872:
3865:
3858:
3851:
3850:
3843:
3835:
3831:
3827:
3823:
3816:
3807:
3802:
3798:
3794:
3790:
3786:
3779:
3770:
3765:
3761:
3757:
3753:
3746:
3739:
3738:
3731:
3724:
3719:
3712:
3707:
3703:
3693:
3690:
3688:
3685:
3683:
3680:
3678:
3675:
3674:
3668:
3666:
3662:
3661:RegularChains
3658:
3653:
3649:
3645:
3643:
3639:
3634:
3632:
3628:
3623:
3619:
3616:
3612:
3608:
3603:
3591:
3587:
3583:
3579:
3575:
3572:
3568:
3567:Aberth method
3565:
3564:
3563:
3560:
3551:
3537:
3532:
3528:
3524:
3519:
3515:
3492:
3488:
3484:
3481:
3459:
3455:
3451:
3448:
3426:
3422:
3418:
3413:
3409:
3400:
3384:
3381:
3378:
3358:
3350:
3334:
3310:
3307:
3302:
3298:
3294:
3291:
3286:
3282:
3275:
3272:
3269:
3262:
3259:
3255:
3252:
3249:
3244:
3240:
3236:
3233:
3228:
3224:
3217:
3214:
3211:
3201:
3200:
3199:
3196:
3191:
3177:
3174:
3169:
3165:
3160:
3157:
3153:
3150:
3147:
3142:
3138:
3117:
3097:
3077:
3057:
3054:
3049:
3045:
3040:
3037:
3033:
3030:
3027:
3022:
3018:
3008:
2994:
2984:
2980:
2966:
2963:
2959:
2956:
2955:
2954:
2951:
2949:
2945:
2930:
2927:
2925:
2921:
2917:
2908:
2904:
2898:
2894:
2869:
2863:
2860:
2855:
2851:
2847:
2842:
2839:
2836:
2833:
2830:
2825:
2821:
2814:
2811:
2801:
2798:
2793:
2789:
2785:
2780:
2777:
2774:
2771:
2768:
2763:
2759:
2752:
2749:
2742:
2739:
2736:
2733:
2728:
2724:
2717:
2708:
2707:
2706:
2693:
2689:
2681:
2675:
2671:
2665:
2659:
2649:
2644:
2636:
2630:
2625:
2621:
2616:
2612:
2607:
2602:
2597:
2592:
2588:
2582:
2577:
2574:
2573:
2572:
2569:
2566:
2557:
2548:
2541:
2535:
2526:
2520:
2493:
2485:
2481:
2472:
2468:
2463:
2454:
2450:
2441:
2437:
2433:
2428:
2424:
2416:
2403:
2399:
2390:
2386:
2381:
2372:
2368:
2359:
2355:
2351:
2346:
2342:
2334:
2331:
2323:
2319:
2312:
2306:
2297:
2296:
2295:
2293:
2285:
2279:
2277:
2267:
2265:
2257:
2247:
2243:
2238:
2233:
2231:
2227:
2218:
2215:
2214:
2213:
2210:
2208:
2204:
2203:Gröbner basis
2199:
2197:
2193:
2189:
2163:
2160:
2157:
2154:
2149:
2145:
2137:
2134:
2128:
2125:
2122:
2113:
2110:
2107:
2097:
2094:
2091:
2088:
2083:
2079:
2072:
2063:
2062:
2061:
2057:
2055:
2049:
2045:
2038:
2031:
2027:
2020:
2016:
1988:
1985:
1977:
1973:
1969:
1966:
1963:
1958:
1954:
1950:
1945:
1941:
1932:
1928:
1920:
1912:
1909:
1901:
1897:
1893:
1888:
1884:
1875:
1871:
1863:
1860:
1852:
1848:
1839:
1835:
1828:
1819:
1818:
1817:
1816:
1812:
1811:regular chain
1801:
1797:
1788:
1780:
1776:
1769:
1762:
1758:
1751:
1747:
1742:
1737:
1733:
1725:
1721:
1714:
1710:
1703:
1696:
1692:
1688:
1687:
1686:
1684:
1680:
1673:
1665:
1661:
1654:
1649:
1645:
1635:
1628:
1621:
1611:
1604:
1597:
1596:Regular chain
1582:
1579:
1573:
1567:
1548:
1533:
1514:
1503:
1500:
1490:
1486:
1482:
1474:
1470:
1465:
1461:
1455:
1448:
1444:
1438:
1432:
1426:
1420:
1409:
1405:
1398:
1383:
1376:
1348:
1345:
1340:
1337:
1332:
1328:
1324:
1319:
1315:
1307:
1304:
1299:
1296:
1293:
1288:
1284:
1280:
1277:
1272:
1268:
1261:
1252:
1251:
1250:
1233:
1230:
1224:
1221:
1215:
1212:
1209:
1203:
1197:
1192:
1188:
1180:
1179:
1178:
1161:
1155:
1149:
1146:
1143:
1140:
1134:
1128:
1123:
1119:
1115:
1112:
1106:
1103:
1097:
1094:
1087:
1086:
1085:
1082:
1078:
1074:
1068:
1062:
1055:
1047:
1042:), replacing
1041:
1037:
1032:
1025:
1009:
1007:
1002:
999:of the field
998:
994:
989:
987:
983:
977:
975:
972:
968:
964:
962:
956:
954:
947:
945:
941:
936:
934:
930:
926:
922:
918:
914:
913:Gröbner basis
904:
900:
896:
891:
886:
880:
873:
869:
862:
855:
851:
844:
839:
835:
830:
828:
824:
820:
816:
811:
809:
805:
801:
797:
793:
788:
784:
777:
773:
768:
764:
760:
756:
753:if it has no
752:
748:
738:
736:
728:
724:
720:
719:Barth surface
715:
713:
709:
703:
696:
692:
667:
664:
662:
654:
651:
648:
642:
635:
632:
630:
622:
619:
616:
610:
599:
598:
597:
594:
592:
588:
584:
580:
576:
572:
565:
561:
554:
549:of values of
548:
544:
539:
537:
533:
529:
525:
519:
515:
508:
504:
500:
495:
468:
465:
462:
460:
454:
448:
444:
440:
437:
434:
429:
425:
420:
414:
410:
402:
398:
390:
387:
385:
379:
373:
369:
365:
362:
359:
354:
350:
345:
339:
335:
323:
322:
321:
319:
315:
310:
307:
303:
299:
274:
271:
269:
264:
261:
258:
255:
248:
245:
243:
238:
235:
230:
226:
222:
217:
213:
201:
200:
199:
192:
183:
181:
175:
173:
172:finite fields
169:
164:
158:
155:
150:
146:
141:
136:
131:
125:
119:
115:
112:
106:
102:
97:
92:
89:
85:
79:
75:
68:
63:
57:
53:
45:
41:
34:
30:
26:
22:
4180:
4161:
4142:
4123:
4106:
4095:
4084:
4060:(1): 33â50.
4057:
4053:
4043:
4033:
4028:
4003:
3999:
3964:
3960:
3950:
3936:
3929:
3904:
3900:
3894:
3884:
3879:
3870:
3857:
3847:
3842:
3825:
3821:
3815:
3796:
3792:
3778:
3759:
3755:
3745:
3735:
3730:
3718:
3706:
3660:
3654:
3650:
3646:
3635:
3630:
3624:
3620:
3610:
3604:
3600:
3589:
3585:
3561:
3557:
3398:
3348:
3326:
3192:
3009:
2985:
2982:
2965:Optimization
2952:
2947:
2941:
2928:
2915:
2906:
2902:
2899:
2895:
2891:
2691:
2687:
2679:
2673:
2669:
2663:
2657:
2654:
2647:
2634:
2628:
2619:
2610:
2606:multiplicity
2600:
2590:
2586:
2580:
2570:
2564:
2555:
2546:
2539:
2533:
2524:
2518:
2515:
2291:
2283:
2280:
2275:
2273:
2263:
2255:
2245:
2241:
2236:
2234:
2225:
2222:
2211:
2200:
2185:
2058:
2047:
2043:
2036:
2029:
2025:
2018:
2014:
2010:
1814:
1808:
1799:
1795:
1786:
1778:
1774:
1767:
1760:
1756:
1749:
1745:
1735:
1731:
1723:
1719:
1712:
1708:
1701:
1694:
1690:
1682:
1678:
1671:
1663:
1659:
1652:
1647:
1643:
1633:
1626:
1619:
1609:
1602:
1599:
1577:
1574:
1565:
1531:
1504:
1496:
1484:
1480:
1472:
1468:
1463:
1459:
1453:
1446:
1442:
1436:
1430:
1424:
1418:
1415:
1406:
1396:
1381:
1374:
1370:
1248:
1176:
1083:
1076:
1072:
1066:
1060:
1053:
1045:
1030:
1023:
1020:
1000:
992:
990:
985:
981:
978:
958:
950:
948:
939:
937:
920:
917:inconsistent
916:
910:
901:
894:
884:
878:
871:
867:
860:
858:has at most
853:
849:
842:
834:well-behaved
833:
831:
826:
813:A system is
812:
790:A system is
789:
782:
775:
771:
751:inconsistent
745:A system is
744:
716:
701:
694:
690:
687:are a point
686:
595:
563:
559:
552:
542:
540:
536:real numbers
532:finite field
517:
513:
506:
490:
487:
317:
313:
311:
308:
301:
297:
293:
197:
176:
162:
159:
153:
139:
129:
123:
117:
104:
100:
95:
93:
87:
82:, over some
77:
73:
66:
55:
51:
43:
39:
32:
24:
20:
18:
4224:Polynomials
3967:(3): 2009.
3725:, p. 8
3713:, p. 4
3611:RootFinding
3590:RootFinding
3584:(functions
3007:, is kept.
2237:shape lemma
2023:has degree
699:and a line
488:where each
62:polynomials
4203:Categories
4190:0821832514
3785:Gianni, P.
3698:References
2643:derivative
2531:of degree
1809:To such a
1675:such that
1012:Extensions
499:polynomial
186:Definition
49:where the
38:= 0, ...,
4209:Equations
3629:function
3609:function
3525:−
3399:Following
3349:following
3273:−
3260:…
3215:−
3158:…
3038:…
2861:−
2840:−
2831:−
2799:−
2778:−
2734:−
2417:⋮
2155:−
2126:−
2111:−
2089:−
1967:…
1921:⋮
1338:−
1294:−
1216:
1198:
1150:
1141:−
1129:
1098:
993:algebraic
986:certified
823:dimension
774:â 1 = 0,
697:) = (1,1)
652:−
620:−
438:…
403:⋮
363:…
262:−
236:−
4020:15485257
3921:25579305
3671:See also
3659:library
3631:Groebner
3195:homotopy
2604:and the
2584:and the
2205:for the
591:rational
543:solution
96:solution
4214:Algebra
4062:Bibcode
3642:MPSolve
3638:MPSolve
3571:MPSolve
3193:Then a
2924:radical
2922:of the
2702:
2684:
2641:is the
2545:, ...,
1793:, ...,
1773:, ...,
1707:, ...,
1658:, ...,
1641:, ...,
1537:â 2 = 0
1079:â 1 = 0
982:numeric
848:, ...,
778:â 1 = 0
755:complex
731:5 = 125
583:complex
558:, ...,
512:, ...,
501:in the
72:, ...,
4187:
4168:
4149:
4130:
4018:
3919:
3586:fsolve
2516:where
2262:. The
1727:> 0
1399:< 2
1028:where
581:, all
4016:S2CID
3996:(PDF)
3917:S2CID
3867:(PDF)
3657:Maple
3627:Maple
3607:Maple
3582:Maple
1422:with
1034:is a
733:, by
547:tuple
530:or a
524:field
497:is a
84:field
4185:ISBN
4166:ISBN
4147:ISBN
4128:ISBN
3605:The
3588:and
3419:<
3351:the
2667:and
2537:and
2274:The
2194:(or
2042:...
1677:1 â€
1395:0 â€
1064:and
1052:cos(
1050:and
1044:sin(
806:and
717:The
587:real
170:and
60:are
4070:doi
4058:162
4008:doi
3969:doi
3909:doi
3830:doi
3801:doi
3764:doi
2948:all
2645:of
2626:If
2198:).
2056:).
1802:â 1
1781:â1
1754:in
1729:in
1563:by
1476:= 0
1449:= 0
1213:cos
1189:sin
1147:cos
1120:cos
1095:cos
1026:= 0
899:.)
897:= 1
866:â
â
â
785:= 1
714:).
704:= 0
589:or
316:or
151:of
121:of
47:= 0
4205::
4068:.
4056:.
4052:.
4014:.
4004:25
4002:.
3998:.
3983:^
3965:44
3963:.
3959:.
3940:.
3915:.
3903:.
3869:.
3826:13
3824:.
3797:16
3795:.
3791:.
3760:28
3758:.
3754:.
3667:.
3633:.
3592:).
3397:.
3190:.
2690:â
2682:=
2672:+
2661:,
2568:.
2164:0.
1989:0.
1681:â€
1632:,
1617:,
1489:.
1467:â
1445:â
1404:.
1380:,
1349:0.
1081:.
1075:+
1008:.
946:.
836:.
829:.
810:.
787:.
541:A
312:A
300:,
275:0.
182:.
174:.
137:,
94:A
91:.
19:A
4193:.
4174:.
4155:.
4136:.
4078:.
4072::
4064::
4022:.
4010::
3977:.
3971::
3944:.
3923:.
3911::
3905:9
3836:.
3832::
3809:.
3803::
3772:.
3766::
3538::
3533:1
3529:t
3520:2
3516:t
3493:1
3489:t
3485:=
3482:t
3460:2
3456:t
3452:=
3449:t
3427:2
3423:t
3414:1
3410:t
3385:1
3382:=
3379:t
3359:N
3335:t
3323:.
3311:0
3308:=
3303:n
3299:f
3295:t
3292:+
3287:n
3283:g
3279:)
3276:t
3270:1
3267:(
3263:,
3256:,
3253:0
3250:=
3245:1
3241:f
3237:t
3234:+
3229:1
3225:g
3221:)
3218:t
3212:1
3209:(
3178:0
3175:=
3170:n
3166:f
3161:,
3154:,
3151:0
3148:=
3143:1
3139:f
3118:n
3098:n
3078:N
3058:0
3055:=
3050:n
3046:g
3041:,
3034:,
3031:0
3028:=
3023:1
3019:g
2995:N
2912:)
2910:0
2907:x
2905:(
2903:h
2870:.
2864:1
2856:2
2852:t
2848:3
2843:1
2837:t
2834:2
2826:2
2822:t
2815:=
2812:y
2802:1
2794:2
2790:t
2786:3
2781:1
2775:t
2772:2
2769:+
2764:2
2760:t
2753:=
2750:x
2743:0
2740:=
2737:t
2729:3
2725:t
2718:{
2699:2
2696:/
2692:y
2688:x
2680:t
2674:y
2670:x
2664:y
2658:x
2651:.
2648:h
2638:0
2635:g
2629:h
2620:h
2611:h
2601:h
2591:i
2587:g
2581:h
2565:D
2559:0
2556:x
2550:n
2547:g
2543:0
2540:g
2534:D
2528:0
2525:x
2519:h
2494:,
2491:)
2486:0
2482:x
2478:(
2473:0
2469:g
2464:/
2460:)
2455:0
2451:x
2447:(
2442:n
2438:g
2434:=
2429:n
2425:x
2409:)
2404:0
2400:x
2396:(
2391:0
2387:g
2382:/
2378:)
2373:0
2369:x
2365:(
2360:1
2356:g
2352:=
2347:1
2343:x
2335:0
2332:=
2329:)
2324:0
2320:x
2316:(
2313:h
2307:{
2287:0
2284:x
2260:0
2252:1
2246:i
2242:d
2161:=
2158:1
2150:2
2146:y
2138:0
2135:=
2132:)
2129:1
2123:y
2120:(
2117:)
2114:1
2108:x
2105:(
2098:0
2095:=
2092:1
2084:2
2080:x
2073:{
2048:n
2044:d
2040:1
2037:d
2030:i
2026:d
2019:i
2015:f
1986:=
1983:)
1978:n
1974:x
1970:,
1964:,
1959:2
1955:x
1951:,
1946:1
1942:x
1938:(
1933:n
1929:f
1913:0
1910:=
1907:)
1902:2
1898:x
1894:,
1889:1
1885:x
1881:(
1876:2
1872:f
1864:0
1861:=
1858:)
1853:1
1849:x
1845:(
1840:1
1836:f
1829:{
1805:.
1800:i
1796:f
1790:1
1787:f
1779:i
1775:x
1771:1
1768:x
1761:i
1757:f
1750:i
1746:x
1740:;
1736:i
1732:x
1724:i
1720:d
1713:i
1709:x
1705:1
1702:x
1695:i
1691:f
1683:n
1679:i
1672:i
1667:)
1664:n
1660:x
1656:1
1653:x
1651:(
1648:n
1644:f
1639:)
1637:2
1634:x
1630:1
1627:x
1625:(
1623:2
1620:f
1615:)
1613:1
1610:x
1608:(
1606:1
1603:f
1578:k
1569:2
1566:r
1549:2
1535:2
1532:r
1515:2
1485:i
1481:x
1473:i
1469:x
1464:i
1460:x
1454:k
1447:x
1443:x
1437:k
1431:k
1425:q
1419:k
1401:Ï
1397:x
1391:x
1387:)
1385:0
1382:s
1378:0
1375:c
1373:(
1346:=
1341:1
1333:2
1329:c
1325:+
1320:2
1316:s
1308:0
1305:=
1300:c
1297:3
1289:3
1285:c
1281:4
1278:+
1273:3
1269:s
1262:{
1234:0
1231:=
1228:)
1225:x
1222:3
1219:(
1210:+
1207:)
1204:x
1201:(
1193:3
1162:,
1159:)
1156:x
1153:(
1144:3
1138:)
1135:x
1132:(
1124:3
1116:4
1113:=
1110:)
1107:x
1104:3
1101:(
1077:c
1073:s
1067:c
1061:s
1056:)
1054:x
1048:)
1046:x
1031:g
1024:g
1001:k
895:n
885:d
879:d
872:n
868:d
864:1
861:d
854:n
850:d
846:1
843:d
783:x
776:x
772:x
702:x
695:y
693:,
691:x
689:(
668:0
665:=
658:)
655:1
649:y
646:(
643:x
636:0
633:=
626:)
623:1
617:x
614:(
611:x
567:)
564:m
560:x
556:1
553:x
551:(
518:m
514:x
510:1
507:x
493:h
491:f
469:,
466:0
463:=
455:)
449:m
445:x
441:,
435:,
430:1
426:x
421:(
415:n
411:f
391:0
388:=
380:)
374:m
370:x
366:,
360:,
355:1
351:x
346:(
340:1
336:f
302:y
298:x
296:(
272:=
265:2
259:y
256:x
249:0
246:=
239:5
231:2
227:y
223:+
218:2
214:x
163:k
154:k
140:K
130:k
124:k
118:K
105:i
101:x
88:k
78:n
74:x
70:1
67:x
56:i
52:f
44:h
40:f
36:1
33:f
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