1962:
2360:
10225:
5329:
2997:
39:
9957:
5063:
2486:
10220:{\displaystyle \Delta _{K}=\det \left({\begin{array}{cccc}\sigma _{1}(b_{1})&\sigma _{1}(b_{2})&\cdots &\sigma _{1}(b_{n})\\\sigma _{2}(b_{1})&\ddots &&\vdots \\\vdots &&\ddots &\vdots \\\sigma _{n}(b_{1})&\cdots &\cdots &\sigma _{n}(b_{n})\end{array}}\right)^{2}.}
5324:{\displaystyle {\begin{aligned}\operatorname {disc} _{x}(R)&=\operatorname {disc} _{x}(P)\operatorname {Res} _{x}(P,Q)^{2}\operatorname {disc} _{x}(Q)\\{}&=(-1)^{pq}\operatorname {disc} _{x}(P)\operatorname {Res} _{x}(P,Q)\operatorname {Res} _{x}(Q,P)\operatorname {disc} _{x}(Q),\end{aligned}}}
7376:
Discriminants of the second class arise for problems depending on coefficients, when degenerate instances or singularities of the problem are characterized by the vanishing of a single polynomial in the coefficients. This is the case for the discriminant of a polynomial, which is zero when two roots
3222:
The discriminant is zero if and only if at least two roots are equal. If the coefficients are real numbers and the discriminant is negative, then there are two real roots and two complex conjugate roots. Conversely, if the discriminant is positive, then the roots are either all real or all non-real.
2992:{\displaystyle {\begin{aligned}{}&256a^{3}e^{3}-192a^{2}bde^{2}-128a^{2}c^{2}e^{2}+144a^{2}cd^{2}e\\&{}-27a^{2}d^{4}+144ab^{2}ce^{2}-6ab^{2}d^{2}e-80abc^{2}de\\&{}+18abcd^{3}+16ac^{4}e-4ac^{3}d^{2}-27b^{4}e^{2}+18b^{3}cde\\&{}-4b^{3}d^{3}-4b^{2}c^{3}e+b^{2}c^{2}d^{2}\,.\end{aligned}}}
6205:
is a homogeneous polynomial of degre 2 which has only two there are only two terms, while the general homogeneous polynomial of degree two in three variables has 6 terms. The discriminant of the general cubic polynomial is a homogeneous polynomial of degree 4 in four variables; it has five terms,
1559:
2312:
The discriminant is zero if and only if at least two roots are equal. If the coefficients are real numbers, and the discriminant is not zero, the discriminant is positive if the roots are three distinct real numbers, and negative if there is one real root and two complex conjugate roots.
8957:
of dimension three is a surface that may be defined as the zeros of a polynomial of degree two in three variables. As for the conic sections there are two discriminants that may be naturally defined. Both are useful for getting information on the nature of a quadric surface.
6338:
that the sign of the discriminant provides useful information on the nature of the roots for polynomials of degree 2 and 3. For higher degrees, the information provided by the discriminant is less complete, but still useful. More precisely, for a polynomial of degree
3217:
7719:
4760:
1114:
3593:
3705:
6610:
4336:
7265:. This polynomial may be considered as a univariate polynomial in one of the indeterminates, with polynomials in the other indeterminates as coefficients. The discriminant with respect to the selected indeterminate defines a hypersurface
4499:
8872:
3465:
7018:
6920:
6780:
4017:
771:
1329:
2491:
517:
7457:
of the resultant, computed with generic coefficients. The restriction on the characteristic is needed because otherwise a common zero of the partial derivative is not necessarily a zero of the polynomial (see
2202:
1894:
331:. In the case of a cubic with real coefficients, the discriminant is positive if the polynomial has three distinct real roots, and negative if it has one real root and two distinct complex conjugate roots.
2395:) where the polynomial has a repeated root. The cuspidal edge corresponds to the polynomials with a triple root, and the self-intersection corresponds to the polynomials with two different repeated roots.
5068:
5791:
6152:
4876:
4167:
3322:
of the variable. As a projective transformation may be decomposed into a product of translations, homotheties and inversions, this results in the following formulas for simpler transformations, where
4996:
4824:
11209:
8780:
8293:
4598:
9160:
6039:
5575:
3072:
5867:
4939:
8625:
5944:
3860:
8362:
6825:
3067:
9325:
7544:
7060:
8880:
The second discriminant, which is the only one that is considered in many elementary textbooks, is the discriminant of the homogeneous part of degree two of the equation. It is equal to
4650:
2478:
6284:
10391:
7778:
5359:
1306:
7935:
4413:
3776:
8129:
4059:
8442:
2076:
10760:
3059:
1229:
820:
608:
9561:
4643:
4550:
2307:
986:
10620:
9667:
9451:
9396:
7990:
9703:
9506:
3488:
1750:
9737:
9634:
9591:
9358:
9193:
5023:
3896:
1796:
1607:
10534:
3744:
3616:
562:
338:
is zero if and only if the polynomial has a multiple root. For real coefficients and no multiple roots, the discriminant is positive if the number of non-real roots is a
220:
7851:
6436:
2251:
1620:, then the discriminant is positive. Unlike the previous definition, this expression is not obviously a polynomial in the coefficients, but this follows either from the
265:
10861:
8920:
7961:
7877:
7807:
6203:
4210:
11089:
10993:
9000:
4368:
4196:
305:
7502:
6317:
905:
9225:
9030:
4421:
939:
8058:
7219:
7192:
7131:
7104:
6680:
6653:
5671:
5628:
3364:
3304:
1179:
1144:
1119:
Historically, this sign has been chosen such that, over the reals, the discriminant will be positive when all the roots of the polynomial are real. The division by
978:
872:
11235:
10927:
10489:
11020:
10956:
10891:
11255:
10692:
8794:
5398:
This property follows immediately by substituting the expression for the resultant, and the discriminant, in terms of the roots of the respective polynomials.
11048:
10712:
10663:
10643:
10578:
10554:
10454:
10434:
10414:
3382:
6209:
For higher degrees, there may be monomials which satisfy above rules and do not appear in the discriminant. The first example is for the quartic polynomial
6931:
6833:
6688:
3903:
1669:
For small degrees, the discriminant is rather simple (see below), but for higher degrees, it may become unwieldy. For example, the discriminant of a
8070:
is equal to its
Hessian determinant. So, the discriminant of a quadratic form is a special case of the above general definition of a discriminant.
1554:{\displaystyle \operatorname {Disc} _{x}(A)=a_{n}^{2n-2}\prod _{i<j}(r_{i}-r_{j})^{2}=(-1)^{n(n-1)/2}a_{n}^{2n-2}\prod _{i\neq j}(r_{i}-r_{j}).}
7459:
8492:
if and only its discriminant is zero. In this case, either the surface may be decomposed in planes, or it has a unique singular point, and is a
11098:
10310:
A specific type of discriminant useful in the study of quadratic fields is the fundamental discriminant. It arises in the theory of integral
638:
1625:
7161:. In this case, above formulas and definition remain valid, if the discriminants are computed as if all polynomials would have the degree
400:
7447:. However, because of the integer coefficients resulting of the derivation, this multivariate resultant may be divisible by a power of
5431:
in the coefficients. This can be seen in two ways. In terms of the roots-and-leading-term formula, multiplying all the coefficients by
4504:
As the discriminant is defined in terms of a determinant, this property results immediately from the similar property of determinants.
2084:
1811:
10829:
10817:
9747:
7366:
347:
6206:
which is the maximum allowed by the above rules, while the general homogeneous polynomial of degree 4 in 4 variables has 35 terms.
11614:; Diaz y Diaz, Francisco; Olivier, Michel (2002), "A Survey of Discriminant Counting", in Fieker, Claus; Kohel, David R. (eds.),
8077:
of the vector space on which the quadratic form is defined) in the following sense: a linear change of variables is defined by a
5508:
in the roots. This follows from the expression of the discriminant in terms of the roots, which is the product of a constant and
5696:
9509:
8925:
and determines the shape of the conic section. If this discriminant is negative, the curve either has no real points, or is an
8489:
7508:. Several other classical types of discriminants, that are instances of the general definition are described in next sections.
7415:
6050:
4829:
4075:
10317:
9814:
states that there are only finitely many number fields of bounded discriminant, however determining this quantity is still an
6391:
If the discriminant is negative, the number of non-real roots is not a multiple of 4. That is, there is a nonnegative integer
4947:
4775:
11696:
11631:
8672:
8219:
8209:, a quadratic form over a field of characteristic different from 2 can be expressed, after a linear change of variables, in
4555:
9002:
be a polynomial of degree two in three variables that defines a real quadric surface. The first associated quadratic form,
1621:
103:
11369:
8500:. Over the reals, if the discriminant is positive, then the surface either has no real point or has everywhere a negative
6356:
If the discriminant is positive, the number of non-real roots is a multiple of 4. That is, there is a nonnegative integer
10863:
that has degree 2. The discriminant of a quadratic field plays a role analogous to the discriminant of a quadratic form.
9047:
5955:
5511:
1654:(degree 1) is rarely considered. If needed, it is commonly defined to be equal to 1 (using the usual conventions for the
75:
5802:
11876:
10766:
These conditions ensure that every fundamental discriminant corresponds uniquely to a specific type of quadratic form.
4881:
3212:{\displaystyle {\begin{aligned}{}&16c^{4}e-4c^{3}d^{2}-128c^{2}e^{2}+144cd^{2}e-27d^{4}+256e^{3}\,.\end{aligned}}}
11595:
11560:
11515:
11448:
11421:
11383:
11341:
8530:
7714:{\displaystyle Q(x_{1},\ldots ,x_{n})\ =\ \sum _{i=1}^{n}a_{ii}x_{i}^{2}+\sum _{1\leq i<j\leq n}a_{ij}x_{i}x_{j},}
122:
5875:
4755:{\displaystyle \varphi (\operatorname {Disc} _{x}(A))=\varphi (a_{n-1})^{2}\operatorname {Disc} _{x}(A^{\varphi }).}
3791:
82:
11947:
11413:
9787:
8304:
6788:
9239:
7026:
10494:
2405:
2339:), then the discriminant is a square of a rational number (or a number from the number field) if and only if the
1612:
This expression for the discriminant is often taken as a definition. It makes clear that if the polynomial has a
11301:
Sylvester, J. J. (1851). "On a remarkable discovery in the theory of canonical forms and of hyperdeterminants".
319:
coefficients, it is positive if the polynomial has two distinct real roots, and negative if it has two distinct
9033:
8478:
7377:
collapse. Most of the cases, where such a generalized discriminant is defined, are instances of the following.
6212:
1666:). There is no common convention for the discriminant of a constant polynomial (i.e., polynomial of degree 0).
1317:
1309:
60:
7730:
6166:
This restricts the possible terms in the discriminant. For the general quadratic polynomial, the discriminant
5337:
1947:, then the discriminant is the square of a rational number if and only if the two roots are rational numbers.
1252:
89:
11611:
9807:
9799:
7882:
5684:
56:
27:
9524:. If there are several singular points the surface consists of two planes, a double plane or a single line.
4373:
3749:
11871:
11855:
11616:
Algorithmic Number Theory, Proceedings, 5th
International Syposium, ANTS-V, University of Sydney, July 2002
8937:, or, if degenerated, a double line or two parallel lines. If the discriminant is positive, the curve is a
8095:
5411:
4032:
1616:, then its discriminant is zero, and that, in the case of real coefficients, if all the roots are real and
1109:{\displaystyle \operatorname {Disc} _{x}(A)={\frac {(-1)^{n(n-1)/2}}{a_{n}}}\operatorname {Res} _{x}(A,A')}
9776:
8411:
2019:
11891:
3005:
1188:
779:
567:
71:
10717:
9782:
The discriminant is one of the most basic invariants of a number field, and occurs in several important
9530:
4603:
4510:
3599:
This results from the expression in terms of the roots, or of the quasi-homogeneity of the discriminant.
2324:
times the discriminant, or its product with the square of a rational number; for example, the square of
2259:
11736:
9639:
9420:
9365:
8504:. If the discriminant is negative, the surface has real points, and has a negative Gaussian curvature.
10583:
7966:
3588:{\displaystyle \operatorname {Disc} _{x}(P(\alpha x))=\alpha ^{n(n-1)}\operatorname {Disc} _{x}(P(x))}
1913:
are real numbers, the polynomial has two distinct real roots if the discriminant is positive, and two
11689:
11484:
3319:
2317:
11022:
as its discriminant. This connects the theory of quadratic forms and the study of quadratic fields.
9672:
9475:
3700:{\displaystyle \operatorname {Disc} _{x}(P^{\mathrm {r} }\!\!\;(x))=\operatorname {Disc} _{x}(P(x))}
1701:
11937:
11881:
11030:
Fundamental discriminants can also be characterized by their prime factorization. Consider the set
8206:
7993:
7517:
7393:
6605:{\displaystyle A(x,y)=a_{0}x^{n}+a_{1}x^{n-1}y+\cdots +a_{n}y^{n}=\sum _{i=0}^{n}a_{i}x^{n-i}y^{i}}
4765:
When one is only interested in knowing whether a discriminant is zero (as is generally the case in
3259:
1709:
160:
9712:
9609:
9566:
9333:
9168:
8141:
the multiplication by a square. In other words, the discriminant of a quadratic form over a field
7441:
of these partial derivatives is zero, and this resultant may be considered as the discriminant of
5001:
4331:{\displaystyle A^{\varphi }=\varphi (a_{n})x^{n}+\varphi (a_{n-1})x^{n-1}+\cdots +\varphi (a_{0})}
3865:
1758:
1571:
11932:
11820:
11530:
In characteristic 2, the discriminant of a quadratic form is not defined, and is replaced by the
7411:
7262:
3714:
2208:
534:
371:
180:
49:
9983:
8933:, or, if degenerated, is reduced to a single point. If the discriminant is zero, the curve is a
7814:
2214:
228:
11850:
11845:
11840:
11718:
11333:
10536:
Not every integer can arise as a discriminant of an integral binary quadratic form. An integer
9791:
9783:
9753:
8886:
8073:
The discriminant of a quadratic form is invariant under linear changes of variables (that is a
8061:
7940:
7856:
7786:
7535:
7438:
6616:
6169:
5407:
4494:{\displaystyle \operatorname {Disc} _{x}(A^{\varphi })=\varphi (\operatorname {Disc} _{x}(A)).}
3276:
3263:
1565:
523:
383:
367:
339:
335:
11618:, Lecture Notes in Computer Science, vol. 2369, Berlin: Springer-Verlag, pp. 80–94,
11505:
11438:
11327:
11313:
11053:
10844:
8964:
4353:
4181:
281:
11942:
11830:
11810:
11585:
11407:
10965:
10311:
9458:
9403:
9399:
8877:
It is zero if the conic section degenerates into two lines, a double line or a single point.
8477:. The discriminant is zero if and only if the curve is decomposed in lines (possibly over an
7474:
6289:
3779:
877:
9200:
9005:
8473:
Geometrically, the discriminant of a quadratic form in three variables is the equation of a
1956:
1312:
of the field. (If the coefficients are real numbers, the roots may be taken in the field of
914:
11927:
11896:
11815:
11682:
11664:
11649:
9924:
9862:
8036:
7531:
7197:
7170:
7109:
7082:
6658:
6631:
5680:
5679:. This is a consequence of the general fact that every polynomial which is homogeneous and
5649:
5606:
5451:
3342:
3282:
2348:
2335:
If the polynomial is irreducible and its coefficients are rational numbers (or belong to a
1633:
1157:
1122:
956:
850:
175:
11570:
8867:{\displaystyle {\begin{vmatrix}a&b&d\\b&c&e\\d&e&f\end{vmatrix}}.}
7353:-discriminant allows one to compute all of the remarkable points of the curve, except the
96:
8:
11709:
11214:
10899:
10251:
9462:
8199:
8024:
3275:, the discriminant is zero if and only if the polynomial is not square-free or it has an
3237:
1629:
1240:
611:
156:
10459:
11886:
11753:
11748:
11674:
11403:
11240:
10998:
10934:
10869:
9764:
9466:
9411:
8501:
8485:
8078:
7434:
7424:
7284:
7238:
4766:
3460:{\displaystyle \operatorname {Disc} _{x}(P(x+\alpha ))=\operatorname {Disc} _{x}(P(x))}
2400:
1670:
1147:
363:
168:
152:
10668:
6157:
which is obtained by subtracting the second equation from the first one multiplied by
11731:
11637:
11627:
11591:
11556:
11511:
11444:
11417:
11389:
11379:
11365:
11337:
10556:
is a fundamental discriminant if and only if it meets one of the following criteria:
9890:
9513:
8521:
8400:
8183:
6375:
5030:
4062:
3252:
1961:
1914:
1802:
1651:
941:
Hence the discriminant—up to its sign—is defined as the quotient of the resultant of
629:
320:
275:
10622:) and is square-free, meaning it is not divisible by the square of any prime number.
7013:{\displaystyle \operatorname {Disc} _{y}(A)=x^{n(n-1)}\operatorname {Disc} ^{h}(A).}
6915:{\displaystyle \operatorname {Disc} _{x}(A)=y^{n(n-1)}\operatorname {Disc} ^{h}(A),}
6775:{\displaystyle \operatorname {Disc} _{x}(A(x,1))=\operatorname {Disc} _{y}(A(1,y)).}
11784:
11777:
11772:
11619:
11566:
11544:
11361:
11092:
11033:
10697:
10648:
10628:
10563:
10539:
10439:
10419:
10399:
9909:
9833:
9768:
9757:
8474:
7809:
7354:
5455:
4066:
1677:
1673:
1659:
831:
615:
324:
159:
of the coefficients of the original polynomial. The discriminant is widely used in
11906:
11277:
8186:
if one is the product of the other by a nonzero square). It follows that over the
7337:-coordinates of the singular points, of the points with a tangent parallel to the
2316:
The square root of a quantity strongly related to the discriminant appears in the
11794:
11789:
11741:
11726:
11645:
10291:
10240:
9811:
8954:
8950:
8445:
8195:
8159:
8074:
7370:
7314:
7242:
3248:
3241:
2329:
1944:
1681:
11510:. Springer-Verlag New York, Inc. ch. 10 ex. 10.14.4 & 10.17.4, pp. 154–156.
16:
Function of the coefficients of a polynomial that gives information on its roots
11765:
11760:
11552:
11357:
10258:, and like the absolute discriminant it indicates which primes are ramified in
9883:
9851:
8660:
8187:
7523:
7454:
5410:
in the coefficients; it is also a homogeneous polynomial in the roots and thus
4012:{\displaystyle P^{\mathrm {r} }\!\!\;(x)=x^{n}P(1/x)=a_{0}x^{n}+\cdots +a_{n}.}
1313:
354:
308:
11921:
11641:
11531:
9521:
9407:
8513:
7365:
There are two classes of the concept of discriminant. The first class is the
7288:
6350:
5026:
1655:
1613:
328:
164:
11623:
11464:
11375:
11237:
is a fundamental discriminant if and only if it is a product of elements of
1185:
computing the determinant. In any case, the discriminant is a polynomial in
10828:−3, −4, −7, −8, −11, −15, −19, −20, −23, −24, −31 (sequence A003657 in the
9815:
9772:
7527:
7397:
7246:
7221:
indeterminate, the substitution for them of their actual values being done
2344:
2340:
2336:
1663:
1151:
3471:
This results from the expression of the discriminant in terms of the roots
1899:
where the discriminant is zero if and only if the two roots are equal. If
10813:
10809:
10805:
10801:
10797:
10793:
10789:
10785:
9913:
8786:
8517:
8191:
8131:
and thus multiplies the discriminant by the square of the determinant of
8011:
6328:
1617:
827:
316:
312:
271:
148:
136:
20:
3279:
which is not separable (i.e., the irreducible factor is a polynomial in
382:
The term "discriminant" was coined in 1851 by the
British mathematician
334:
More generally, the discriminant of a univariate polynomial of positive
11705:
11409:
Solving polynomial equations: foundations, algorithms, and applications
10781:
10777:
10773:
9706:
9227:
depends on three variables, and consists of the terms of degree two of
5683:
in the roots may be expressed as a quasi-homogeneous polynomial in the
2359:
144:
11669:
847:. The nonzero entries of the first column of the Sylvester matrix are
766:{\displaystyle A'(x)=na_{n}x^{n-1}+(n-1)a_{n-1}x^{n-2}+\cdots +a_{1},}
11835:
9887:
9454:
8938:
5362:
908:
619:
3266:(i.e., it is divisible by the square of a non-constant polynomial).
38:
11825:
10266:. It is a generalization of the absolute discriminant allowing for
9763:
More specifically, it is proportional to the squared volume of the
9517:
8934:
8663:, and thus two discriminants may be associated to a conic section.
8497:
8173:
5797:
512:{\displaystyle A(x)=a_{n}x^{n}+a_{n-1}x^{n-1}+\cdots +a_{1}x+a_{0}}
3240:
is zero if and only if the polynomial has a multiple root in some
11258:
10995:, there exists a unique (up to isomorphism) quadratic field with
8926:
7401:
5640:. It is also quasi-homogeneous of the same degree, if, for every
2197:{\displaystyle b^{2}c^{2}-4ac^{3}-4b^{3}d-27a^{2}d^{2}+18abcd\,.}
823:
1889:{\displaystyle x_{1,2}={\frac {-b\pm {\sqrt {b^{2}-4ac}}}{2a}}.}
10841:
A quadratic field is a field extension of the rational numbers
9593:
if not 0, does not provide any useful information, as changing
8930:
8298:
More precisely, a quadratic forms on may be expressed as a sum
8163:
5435:
does not change the roots, but multiplies the leading term by
8138:
3315:
9741:
9709:, which is elliptic or hyperbolic, depending on the sign of
7465:
In the case of a homogeneous bivariate polynomial of degree
7291:
that is parallel to the axis of the selected indeterminate.
6319:
satisfies the ruless without appearing in the discriminant.
11371:
Discriminants, resultants and multidimensional determinants
8493:
7345:-axis. In other words, the computation of the roots of the
3262:
0, this is equivalent to saying that the polynomial is not
1689:
1685:
7373:, is the discriminant of a polynomial defining the field.
7167:. This means that the discriminants must be computed with
5796:
It follows from what precedes that the exponents in every
5786:{\displaystyle P=a_{n}x^{n}+a_{n-1}x^{n-1}+\cdots +a_{0}.}
3309:
345:
Several generalizations are also called discriminant: the
11356:
10824:
The first eleven negative fundamental discriminants are:
10769:
The first eleven positive fundamental discriminants are:
6147:{\displaystyle ni_{0}+(n-1)i_{1}+\cdots +i_{n-1}=n(n-1),}
4871:{\displaystyle \operatorname {Disc} _{x}(A^{\varphi })=0}
4162:{\displaystyle A=a_{n}x^{n}+a_{n-1}x^{n-1}+\cdots +a_{0}}
11704:
11551:. London Mathematical Society Monographs. Vol. 13.
11465:"Cubic Discriminant | Brilliant Math & Science Wiki"
9516:. If there is only one singular point, the surface is a
8484:
A quadratic form in four variables is the equation of a
8194:, a discriminant is equivalent to −1, 0, or 1. Over the
7271:
in the space of the other indeterminates. The points of
7226:
5869:
appearing in the discriminant satisfy the two equations
4991:{\displaystyle \varphi (\operatorname {Disc} _{x}(A))=0}
4819:{\displaystyle \varphi (\operatorname {Disc} _{x}(A))=0}
11610:
11443:. Springer-Verlag New York, Inc. ch. 10.3 pp. 153–154.
11204:{\displaystyle S=\{-8,-4,8,-3,5,-7,-11,13,17,-19,...\}}
11217:
11036:
11001:
10968:
10937:
10902:
10872:
10847:
10720:
10700:
10671:
10651:
10631:
10586:
10566:
10542:
10462:
10442:
10422:
10402:
8803:
8775:{\displaystyle ax^{2}+2bxy+cy^{2}+2dxz+2eyz+fz^{2}=0.}
8288:{\displaystyle a_{1}x_{1}^{2}+\cdots +a_{n}x_{n}^{2}.}
4593:{\displaystyle \varphi (\operatorname {Disc} _{x}(A))}
4024:
11329:
Elimination practice: software tools and applications
11243:
11101:
11056:
10497:
10320:
9960:
9715:
9675:
9642:
9612:
9606:
does not change the surface, but changes the sign of
9569:
9533:
9478:
9423:
9368:
9336:
9242:
9203:
9171:
9050:
9008:
8967:
8889:
8797:
8675:
8533:
8414:
8393:
may be zero). Equivalently, for any symmetric matrix
8307:
8222:
8098:
8039:
7969:
7943:
7885:
7859:
7817:
7789:
7733:
7547:
7505:
7477:
7200:
7173:
7112:
7085:
7029:
6934:
6836:
6791:
6691:
6661:
6634:
6439:
6292:
6215:
6172:
6053:
5958:
5878:
5805:
5699:
5652:
5609:
5514:
5340:
5066:
5004:
4950:
4884:
4832:
4778:
4653:
4606:
4558:
4513:
4424:
4376:
4356:
4213:
4184:
4078:
4035:
3906:
3868:
3794:
3752:
3717:
3619:
3491:
3385:
3345:
3285:
3070:
3008:
2489:
2408:
2262:
2217:
2087:
2022:
1814:
1761:
1712:
1574:
1332:
1255:
1191:
1160:
1125:
989:
959:
917:
880:
853:
782:
641:
570:
537:
403:
284:
231:
183:
8941:, or, if degenerated, a pair of intersecting lines.
7341:-axis and of some of the asymptotes parallel to the
1323:
In terms of the roots, the discriminant is equal to
11091:the prime numbers congruent to 1 modulo 4, and the
10235:can be referred to as the absolute discriminant of
9155:{\displaystyle Q_{4}(x,y,z,t)=t^{2}P(x/t,y/t,z/t).}
8190:, a discriminant is equivalent to 0 or 1. Over the
7453:, and it is better to take, as a discriminant, the
6425:
6034:{\displaystyle i_{1}+2i_{2}+\cdots +ni_{n}=n(n-1),}
5570:{\displaystyle {\binom {n}{2}}={\frac {n(n-1)}{2}}}
5439:. In terms of its expression as a determinant of a
1801:The square root of the discriminant appears in the
1234:
374:(these three concepts are essentially equivalent).
63:. Unsourced material may be challenged and removed.
11249:
11229:
11203:
11083:
11042:
11014:
10987:
10950:
10921:
10885:
10866:There exists a fundamental connection: an integer
10855:
10754:
10706:
10686:
10657:
10637:
10614:
10572:
10548:
10528:
10483:
10448:
10428:
10408:
10385:
10219:
9731:
9697:
9661:
9628:
9585:
9555:
9500:
9445:
9390:
9352:
9319:
9219:
9187:
9154:
9024:
8994:
8914:
8866:
8774:
8619:
8436:
8356:
8287:
8123:
8052:
7984:
7955:
7929:
7871:
7845:
7801:
7772:
7713:
7496:
7213:
7186:
7125:
7098:
7054:
7012:
6914:
6819:
6774:
6674:
6647:
6604:
6311:
6278:
6197:
6146:
6033:
5938:
5862:{\displaystyle a_{0}^{i_{0}},\dots ,a_{n}^{i_{n}}}
5861:
5785:
5665:
5622:
5569:
5353:
5323:
5017:
4990:
4933:
4870:
4818:
4754:
4637:
4592:
4544:
4493:
4407:
4362:
4330:
4190:
4161:
4053:
4011:
3890:
3854:
3770:
3738:
3699:
3587:
3459:
3358:
3298:
3211:
3053:
2991:
2472:
2301:
2245:
2196:
2070:
1888:
1790:
1744:
1658:and considering that one of the two blocks of the
1601:
1553:
1300:
1223:
1173:
1138:
1108:
972:
933:
899:
866:
814:
765:
602:
556:
511:
299:
259:
214:
11402:
8176:of the nonzero squares (that is, two elements of
5531:
5518:
4934:{\displaystyle \deg(A)-\deg(A^{\varphi })\geq 2.}
3920:
3919:
3766:
3765:
3649:
3648:
1181:by 1 in the first column of the Sylvester matrix—
11919:
9974:
9398:and the surface has real points, it is either a
7225:this computation. Equivalently, the formulas of
155:without computing them. More precisely, it is a
8620:{\displaystyle ax^{2}+2bxy+cy^{2}+2dx+2ey+f=0,}
1926:and the square of the difference of the roots.
342:of 4 (including none), and negative otherwise.
11727:Zero polynomial (degree undefined or −1 or −∞)
10893:is a fundamental discriminant if and only if:
9032:depends on four variables, and is obtained by
8448:. Then the discriminant is the product of the
5939:{\displaystyle i_{0}+i_{1}+\cdots +i_{n}=2n-2}
3855:{\displaystyle P(x)=a_{n}x^{n}+\cdots +a_{0},}
3251:is zero if and only if the polynomial and its
11690:
11095:of the prime numbers congruent to 3 modulo 4:
8357:{\displaystyle \sum _{i=1}^{n}a_{i}L_{i}^{2}}
8137:. Thus the discriminant is well defined only
7404:the degree of the polynomial. The polynomial
6820:{\displaystyle \operatorname {Disc} ^{h}(A),}
1154:. Such a problem may be avoided by replacing
11198:
11108:
10305:
9320:{\displaystyle Q_{3}(x,y,z)=Q_{4}(x,y,z,0).}
8382:is the number of the variables (some of the
7460:Euler's identity for homogeneous polynomials
7277:are exactly the projection of the points of
7232:
7055:{\displaystyle \operatorname {Disc} ^{h}(A)}
2318:formulas for the roots of a cubic polynomial
1626:fundamental theorem of symmetric polynomials
1239:When the above polynomial is defined over a
327:is zero if and only if the polynomial has a
11587:Math refresher for scientists and engineers
11312:Sylvester coins the word "discriminant" on
8198:, a discriminant is equivalent to a unique
7329:, then the discriminant is a polynomial in
5580:The discriminant of a polynomial of degree
5491:The discriminant of a polynomial of degree
5465:, the determinant is homogeneous of degree
5417:The discriminant of a polynomial of degree
2473:{\displaystyle ax^{4}+bx^{3}+cx^{2}+dx+e\,}
2363:The discriminant of the quartic polynomial
1805:for the roots of the quadratic polynomial:
151:and allows deducing some properties of the
11697:
11683:
11665:Wolfram Mathworld: Polynomial Discriminant
10836:
10298:generated by the absolute discriminant of
7227:§ Invariance under ring homomorphisms
7133:are permitted to be zero, the polynomials
7023:Because of these properties, the quantity
4769:), these properties may be summarised as:
3921:
3767:
3650:
1965:The zero set of discriminant of the cubic
11300:
10958:is the discriminant of a quadratic field.
10849:
9748:Discriminant of an algebraic number field
9742:Discriminant of an algebraic number field
8430:
8114:
7367:discriminant of an algebraic number field
7313:is the implicit equation of a real plane
6279:{\displaystyle ax^{4}+bx^{3}+cx^{2}+dx+e}
5036:
4944:This is often interpreted as saying that
3247:The discriminant of a polynomial over an
3201:
3050:
2981:
2469:
2295:
2190:
2067:
1784:
1741:
348:discriminant of an algebraic number field
123:Learn how and when to remove this message
11590:. John Wiley and Sons. sec. 3.2, p. 45.
10386:{\displaystyle Q(x,y)=ax^{2}+bxy+cy^{2}}
8944:
7773:{\displaystyle Q(X)=XAX^{\mathrm {T} },}
6353:if and only if its discriminant is zero.
5354:{\displaystyle \operatorname {Res} _{x}}
3236:The discriminant of a polynomial over a
2358:
1960:
1301:{\displaystyle r_{1},r_{2},\dots ,r_{n}}
323:roots. Similarly, the discriminant of a
11543:
9818:, and the subject of current research.
7930:{\displaystyle X=(x_{1},\ldots ,x_{n})}
7506:§ Homogeneous bivariate polynomial
7287:), which either are singular or have a
3310:Invariance under change of the variable
1308:, not necessarily all distinct, in any
11920:
11583:
11503:
11436:
11025:
9825:be an algebraic number field, and let
8459:, which is well-defined as a class in
7437:. This is the case if and only if the
6327:In this section, all polynomials have
4408:{\displaystyle \varphi (a_{n})\neq 0,}
3771:{\displaystyle P^{\mathrm {r} }\!\!\;}
1702:Quadratic equation § Discriminant
11678:
8124:{\displaystyle S^{\mathrm {T} }A\,S,}
7306:with real coefficients, so that
6410:pairs of complex conjugate roots and
4054:{\displaystyle \varphi \colon R\to S}
3314:The discriminant of a polynomial is,
2320:. Specifically, this quantity can be
270:the quantity which appears under the
11325:
10314:, which are expressions of the form:
8437:{\displaystyle S^{\mathrm {T} }A\,S}
7237:The typical use of discriminants in
4350:The discriminant is invariant under
3255:have a non-constant common divisor.
3231:
2071:{\displaystyle ax^{3}+bx^{2}+cx+d\,}
1632:by noting that this expression is a
1622:fundamental theorem of Galois theory
61:adding citations to reliable sources
32:
10755:{\textstyle m\equiv 2,3{\pmod {4}}}
10744:
10737:
10604:
10597:
10250:of number fields. The latter is an
5597:in the coefficients, if, for every
4025:Invariance under ring homomorphisms
3054:{\displaystyle x^{4}+cx^{2}+dx+e\,}
1920:The discriminant is the product of
1224:{\displaystyle a_{0},\ldots ,a_{n}}
815:{\displaystyle a_{0},\ldots ,a_{n}}
603:{\displaystyle a_{0},\ldots ,a_{n}}
13:
11485:"Discriminant of a cubic equation"
10962:For each fundamental discriminant
10645:is equal to four times an integer
10456:are integers. The discriminant of
10239:to distinguish it from the of an
9962:
9717:
9677:
9644:
9614:
9571:
9556:{\displaystyle \Delta _{4}\neq 0,}
9535:
9480:
9465:. In all cases, it has a positive
9425:
9370:
9338:
9330:Let us denote its discriminant by
9173:
9165:Let us denote its discriminant by
8421:
8105:
7976:
7761:
7511:
7504:times the discriminant defined in
7360:
5522:
4638:{\displaystyle \varphi (a_{n})=0,}
4600:may be zero or not. One has, when
4545:{\displaystyle \varphi (a_{n})=0,}
3913:
3759:
3642:
2302:{\displaystyle -4p^{3}-27q^{2}\,.}
1957:Cubic equation § Discriminant
147:is a quantity that depends on the
14:
11959:
11658:
10714:is congruent to 2 or 3 modulo 4 (
10615:{\textstyle D\equiv 1{\pmod {4}}}
9760:of the) algebraic number field.
9662:{\displaystyle \Delta _{4}\neq 0}
9446:{\displaystyle \Delta _{4}<0,}
9391:{\displaystyle \Delta _{4}>0,}
8507:
8378:are independent linear forms and
7369:, which, in some cases including
7255:be such a curve or hypersurface;
3002:The depressed quartic polynomial
2381:. The surface represents points (
2253:, the discriminant simplifies to
11507:Integers, polynomials, and rings
11440:Integers, polynomials, and rings
7985:{\displaystyle X^{\mathrm {T} }}
7261:is defined as the zero set of a
6628:Supposing, for the moment, that
6426:Homogeneous bivariate polynomial
6335:
5473:in the entries, and dividing by
1235:Expression in terms of the roots
37:
11604:
11577:
11537:
11524:
10282:, the relative discriminant of
7471:, this general discriminant is
7392:indeterminates over a field of
7386:be a homogeneous polynomial in
7325:with coefficients depending on
7155:may have a degree smaller than
5586:is quasi-homogeneous of degree
5051:is a product of polynomials in
3333:denotes a polynomial of degree
3318:a scaling, invariant under any
1146:may not be well defined if the
48:needs additional citations for
11497:
11477:
11457:
11430:
11396:
11350:
11319:
11294:
11270:
10748:
10738:
10608:
10598:
10478:
10466:
10336:
10324:
10197:
10184:
10159:
10146:
10100:
10087:
10070:
10057:
10037:
10024:
10009:
9996:
9878:} be the set of embeddings of
9698:{\displaystyle \Delta _{3}=0,}
9501:{\displaystyle \Delta _{4}=0,}
9311:
9287:
9271:
9253:
9146:
9104:
9085:
9061:
8989:
8971:
8479:algebraically closed extension
8064:of the partial derivatives of
7924:
7892:
7840:
7824:
7743:
7737:
7583:
7551:
7321:as a univariate polynomial in
7049:
7043:
7004:
6998:
6980:
6968:
6954:
6948:
6906:
6900:
6882:
6870:
6856:
6850:
6811:
6805:
6766:
6763:
6751:
6745:
6726:
6723:
6711:
6705:
6455:
6443:
6138:
6126:
6082:
6070:
6025:
6013:
5685:elementary symmetric functions
5577:squared differences of roots.
5558:
5546:
5401:
5383:are the respective degrees of
5311:
5305:
5289:
5277:
5261:
5249:
5233:
5227:
5202:
5192:
5176:
5170:
5148:
5135:
5119:
5113:
5090:
5084:
4979:
4976:
4970:
4954:
4922:
4909:
4897:
4891:
4859:
4846:
4807:
4804:
4798:
4782:
4746:
4733:
4711:
4691:
4682:
4679:
4673:
4657:
4623:
4610:
4587:
4584:
4578:
4562:
4530:
4517:
4485:
4482:
4476:
4460:
4451:
4438:
4393:
4380:
4325:
4312:
4281:
4262:
4243:
4230:
4045:
3961:
3947:
3928:
3922:
3804:
3798:
3727:
3721:
3694:
3691:
3685:
3679:
3660:
3657:
3651:
3633:
3582:
3579:
3573:
3567:
3549:
3537:
3523:
3520:
3511:
3505:
3454:
3451:
3445:
3439:
3420:
3417:
3405:
3399:
1645:
1545:
1519:
1466:
1454:
1447:
1437:
1425:
1398:
1352:
1346:
1318:fundamental theorem of algebra
1310:algebraically closed extension
1103:
1086:
1047:
1035:
1028:
1018:
1009:
1003:
706:
694:
656:
650:
564:), such that the coefficients
413:
407:
1:
11887:Horner's method of evaluation
11406:; Emiris, Ioannis Z. (2005).
11264:
9800:analytic class number formula
7530:, which is defined over some
7400:characteristic that does not
7298:be a bivariate polynomial in
6322:
6286:, in which case the monomial
5365:with respect to the variable
4204:for producing the polynomial
3226:
2343:of the cubic equation is the
1745:{\displaystyle ax^{2}+bx+c\,}
1564:It is thus the square of the
1150:of the coefficients contains
389:
28:Discriminant (disambiguation)
11278:"Discriminant | mathematics"
10580:is congruent to 1 modulo 4 (
9732:{\displaystyle \Delta _{4}.}
9629:{\displaystyle \Delta _{3}.}
9586:{\displaystyle \Delta _{3},}
9353:{\displaystyle \Delta _{3}.}
9188:{\displaystyle \Delta _{4}.}
8666:The first quadratic form is
8060:times its discriminant. The
5018:{\displaystyle A^{\varphi }}
3891:{\displaystyle a_{0}\neq 0,}
1791:{\displaystyle b^{2}-4ac\,.}
1680:has 59 terms, and that of a
1602:{\displaystyle a_{n}^{2n-2}}
7:
11892:Polynomial identity testing
10529:{\displaystyle D=b^{2}-4ac}
10254:in the ring of integers of
9406:. In both cases, this is a
9197:The second quadratic form,
4370:in the following sense. If
3739:{\displaystyle P(0)\neq 0.}
2354:
1950:
1695:
1231:with integer coefficients.
826:coefficients, which is the
614:, or, more generally, to a
557:{\displaystyle a_{n}\neq 0}
215:{\displaystyle ax^{2}+bx+c}
10:
11964:
11504:Irving, Ronald S. (2004).
11437:Irving, Ronald S. (2004).
9756:measures the size of the (
9745:
8475:quadratic projective curve
7846:{\displaystyle A=(a_{ij})}
7515:
6785:Denoting this quantity by
6682:are both nonzero, one has
3269:In nonzero characteristic
2246:{\displaystyle x^{3}+px+q}
1954:
1699:
358:; and more generally, the
307:this discriminant is zero
260:{\displaystyle b^{2}-4ac,}
25:
18:
11864:
11803:
11716:
10856:{\textstyle \mathbb {Q} }
10816:(sequence A003658 in the
10306:Fundamental discriminants
9771:, and it regulates which
9453:the surface is either an
8915:{\displaystyle b^{2}-ac,}
7956:{\displaystyle n\times 1}
7872:{\displaystyle 1\times n}
7802:{\displaystyle n\times n}
7433:have a nontrivial common
7233:Use in algebraic geometry
6198:{\displaystyle b^{2}-4ac}
5497:is homogeneous of degree
5423:is homogeneous of degree
3371:Invariance by translation
3320:projective transformation
2207:In the special case of a
1983:, i.e. points satisfying
1917:roots if it is negative.
1706:The quadratic polynomial
377:
11670:Planetmath: Discriminant
11584:Fanchi, John R. (2006).
11549:Rational Quadratic Forms
11084:{\displaystyle -8,8,-4,}
10988:{\textstyle D_{0}\neq 1}
8995:{\displaystyle P(x,y,z)}
8785:Its discriminant is the
7518:Fundamental discriminant
7068:homogeneous discriminant
5690:Consider the polynomial
4363:{\displaystyle \varphi }
4191:{\displaystyle \varphi }
3366:as leading coefficient.
1676:has 16 terms, that of a
300:{\displaystyle a\neq 0,}
174:The discriminant of the
19:Not to be confused with
11948:Algebraic number theory
11877:Greatest common divisor
11624:10.1007/3-540-45455-1_7
11326:Wang, Dongming (2004).
11282:Encyclopedia Britannica
10837:Quadratic number fields
9752:The discriminant of an
7497:{\displaystyle d^{d-2}}
7412:projective hypersurface
7263:multivariate polynomial
7247:algebraic hypersurfaces
6625:in two indeterminates.
6312:{\displaystyle bc^{4}d}
3605:Invariance by inversion
3477:Invariance by homothety
1684:has 246 terms. This is
900:{\displaystyle na_{n},}
372:projective hypersurface
11749:Quadratic function (2)
11334:Imperial College Press
11303:Philosophical Magazine
11251:
11231:
11205:
11085:
11044:
11016:
10989:
10952:
10923:
10887:
10857:
10756:
10708:
10688:
10659:
10639:
10616:
10574:
10550:
10530:
10485:
10450:
10430:
10410:
10387:
10312:binary quadratic forms
10221:
9792:Dedekind zeta function
9754:algebraic number field
9733:
9699:
9663:
9630:
9587:
9557:
9502:
9447:
9392:
9354:
9321:
9221:
9220:{\displaystyle Q_{3},}
9189:
9156:
9026:
9025:{\displaystyle Q_{4},}
8996:
8916:
8868:
8776:
8621:
8438:
8358:
8328:
8289:
8162:of the multiplicative
8125:
8062:multivariate resultant
8054:
7996:different from 2, the
7986:
7957:
7931:
7873:
7847:
7803:
7774:
7715:
7615:
7536:homogeneous polynomial
7498:
7439:multivariate resultant
7349:-discriminant and the
7241:is for studying plane
7215:
7188:
7127:
7100:
7056:
7014:
6916:
6821:
6776:
6676:
6649:
6617:homogeneous polynomial
6606:
6565:
6313:
6280:
6199:
6148:
6044:and also the equation
6035:
5940:
5863:
5787:
5667:
5624:
5571:
5408:homogeneous polynomial
5406:The discriminant is a
5355:
5325:
5037:Product of polynomials
5019:
4992:
4935:
4872:
4826:if and only if either
4820:
4756:
4639:
4594:
4546:
4495:
4409:
4364:
4332:
4192:
4163:
4069:. Given a polynomial
4055:
4013:
3892:
3856:
3772:
3740:
3701:
3589:
3461:
3360:
3300:
3213:
3055:
2993:
2474:
2396:
2303:
2247:
2198:
2072:
2013:
1890:
1792:
1746:
1650:The discriminant of a
1603:
1566:Vandermonde polynomial
1555:
1302:
1225:
1175:
1140:
1110:
974:
935:
934:{\displaystyle a_{n}.}
911:is thus a multiple of
901:
868:
816:
767:
604:
558:
513:
384:James Joseph Sylvester
368:homogeneous polynomial
301:
261:
216:
11732:Constant function (0)
11366:Zelevinsky, Andrei V.
11252:
11232:
11206:
11086:
11045:
11017:
10990:
10953:
10924:
10888:
10858:
10762:) and is square-free.
10757:
10709:
10689:
10660:
10640:
10617:
10575:
10551:
10531:
10486:
10451:
10431:
10411:
10388:
10222:
9786:formulas such as the
9734:
9700:
9664:
9631:
9588:
9558:
9503:
9459:two-sheet hyperboloid
9448:
9404:one-sheet hyperboloid
9400:hyperbolic paraboloid
9393:
9355:
9322:
9222:
9190:
9157:
9027:
8997:
8945:Real quadric surfaces
8917:
8869:
8777:
8622:
8439:
8359:
8308:
8290:
8126:
8086:, changes the matrix
8055:
8053:{\displaystyle 2^{n}}
7987:
7958:
7932:
7874:
7848:
7804:
7775:
7716:
7595:
7526:is a function over a
7499:
7245:, and more generally
7216:
7214:{\displaystyle a_{n}}
7189:
7187:{\displaystyle a_{0}}
7128:
7126:{\displaystyle a_{n}}
7101:
7099:{\displaystyle a_{0}}
7057:
7015:
6917:
6822:
6777:
6677:
6675:{\displaystyle a_{n}}
6650:
6648:{\displaystyle a_{0}}
6607:
6545:
6349:The polynomial has a
6314:
6281:
6200:
6149:
6036:
5941:
5864:
5788:
5668:
5666:{\displaystyle x^{i}}
5646:, the coefficient of
5625:
5623:{\displaystyle x^{i}}
5603:, the coefficient of
5572:
5445:− 1) × (2
5414:in the coefficients.
5356:
5326:
5020:
4993:
4936:
4873:
4821:
4757:
4640:
4595:
4547:
4496:
4410:
4365:
4333:
4193:
4164:
4056:
4014:
3893:
3857:
3780:reciprocal polynomial
3773:
3741:
3702:
3590:
3462:
3361:
3359:{\displaystyle a_{n}}
3301:
3299:{\displaystyle x^{p}}
3214:
3056:
2994:
2475:
2362:
2304:
2248:
2199:
2073:
2016:The cubic polynomial
1964:
1891:
1793:
1747:
1604:
1556:
1303:
1226:
1176:
1174:{\displaystyle a_{n}}
1141:
1139:{\displaystyle a_{n}}
1111:
975:
973:{\displaystyle a_{n}}
936:
902:
869:
867:{\displaystyle a_{n}}
817:
768:
605:
559:
514:
311:the polynomial has a
302:
262:
217:
11865:Tools and algorithms
11785:Quintic function (5)
11773:Quartic function (4)
11710:polynomial functions
11362:Kapranov, Mikhail M.
11241:
11230:{\textstyle D\neq 1}
11215:
11099:
11054:
11034:
10999:
10966:
10935:
10922:{\textstyle D_{0}=1}
10900:
10870:
10845:
10718:
10698:
10669:
10649:
10629:
10584:
10564:
10540:
10495:
10460:
10440:
10420:
10400:
10318:
10231:The discriminant of
9958:
9713:
9673:
9640:
9610:
9567:
9531:
9476:
9421:
9410:that has a negative
9366:
9334:
9240:
9201:
9169:
9048:
9006:
8965:
8887:
8795:
8673:
8531:
8488:. The surface has a
8412:
8305:
8220:
8096:
8037:
7967:
7941:
7883:
7857:
7815:
7787:
7731:
7724:or, in matrix form,
7545:
7475:
7333:whose roots are the
7198:
7171:
7110:
7083:
7027:
6932:
6834:
6789:
6689:
6659:
6632:
6437:
6402:such that there are
6367:such that there are
6334:It has been seen in
6290:
6213:
6170:
6051:
5956:
5876:
5803:
5697:
5673:is given the weight
5650:
5630:is given the weight
5607:
5512:
5338:
5064:
5002:
4948:
4882:
4830:
4776:
4651:
4604:
4556:
4511:
4422:
4374:
4354:
4211:
4182:
4076:
4033:
3904:
3866:
3792:
3750:
3715:
3617:
3489:
3383:
3343:
3283:
3068:
3006:
2487:
2406:
2260:
2215:
2085:
2020:
1812:
1759:
1710:
1634:symmetric polynomial
1572:
1330:
1253:
1189:
1158:
1123:
987:
957:
915:
878:
851:
780:
639:
568:
535:
401:
282:
229:
181:
176:quadratic polynomial
161:polynomial factoring
57:improve this article
26:For other uses, see
11795:Septic equation (7)
11790:Sextic equation (6)
11737:Linear function (1)
11404:Dickenstein, Alicia
11026:Prime factorization
10484:{\textstyle Q(x,y)}
9861:(i.e. a basis as a
9788:functional equation
9463:elliptic paraboloid
8353:
8281:
8247:
8200:square-free integer
8025:Hessian determinant
7643:
7425:partial derivatives
5858:
5827:
4178:, the homomorphism
1598:
1502:
1381:
776:is a polynomial in
522:be a polynomial of
157:polynomial function
11761:Cubic function (3)
11754:Quadratic equation
11358:Gelfand, Israel M.
11257:that are pairwise
11247:
11227:
11201:
11081:
11040:
11015:{\textstyle D_{0}}
11012:
10985:
10951:{\textstyle D_{0}}
10948:
10919:
10886:{\textstyle D_{0}}
10883:
10853:
10752:
10704:
10684:
10655:
10635:
10612:
10570:
10546:
10526:
10481:
10446:
10426:
10406:
10383:
10270:to be bigger than
10217:
10202:
9891:ring homomorphisms
9765:fundamental domain
9729:
9695:
9659:
9626:
9583:
9553:
9508:the surface has a
9498:
9467:Gaussian curvature
9443:
9412:Gaussian curvature
9388:
9350:
9317:
9217:
9185:
9152:
9022:
8992:
8912:
8864:
8855:
8772:
8656:are real numbers.
8617:
8502:Gaussian curvature
8486:projective surface
8434:
8354:
8339:
8285:
8267:
8233:
8121:
8079:nonsingular matrix
8050:
7982:
7953:
7927:
7869:
7843:
7799:
7770:
7711:
7674:
7629:
7494:
7289:tangent hyperplane
7285:points at infinity
7239:algebraic geometry
7211:
7184:
7123:
7096:
7052:
7010:
6912:
6817:
6772:
6672:
6645:
6602:
6336:§ Low degrees
6309:
6276:
6195:
6144:
6031:
5936:
5859:
5837:
5806:
5783:
5663:
5620:
5567:
5351:
5321:
5319:
5015:
4988:
4931:
4868:
4816:
4767:algebraic geometry
4752:
4635:
4590:
4542:
4491:
4405:
4360:
4328:
4188:
4159:
4051:
4009:
3888:
3852:
3768:
3736:
3697:
3585:
3457:
3356:
3296:
3277:irreducible factor
3209:
3207:
3051:
2989:
2987:
2470:
2401:quartic polynomial
2397:
2299:
2243:
2194:
2068:
2014:
1886:
1788:
1742:
1599:
1575:
1551:
1518:
1479:
1397:
1358:
1298:
1221:
1171:
1136:
1106:
970:
931:
897:
864:
812:
763:
600:
554:
509:
353:discriminant of a
297:
257:
212:
169:algebraic geometry
11915:
11914:
11856:Quasi-homogeneous
11633:978-3-540-43863-2
11545:Cassels, J. W. S.
11336:. ch. 10 p. 180.
11250:{\displaystyle S}
11093:additive inverses
10687:{\textstyle D=4m}
9951:). Symbolically,
9705:the surface is a
8522:implicit equation
8401:elementary matrix
8184:equivalence class
8147:is an element of
7647:
7594:
7588:
7355:inflection points
7294:For example, let
6376:complex conjugate
5565:
5529:
5480:makes the degree
5412:quasi-homogeneous
4067:commutative rings
3232:Zero discriminant
3061:has discriminant
2480:has discriminant
2078:has discriminant
1915:complex conjugate
1881:
1870:
1803:quadratic formula
1752:has discriminant
1652:linear polynomial
1503:
1382:
1071:
321:complex conjugate
315:. In the case of
276:quadratic formula
133:
132:
125:
107:
11955:
11778:Quartic equation
11699:
11692:
11685:
11676:
11675:
11653:
11652:
11608:
11602:
11601:
11581:
11575:
11574:
11541:
11535:
11528:
11522:
11521:
11501:
11495:
11494:
11492:
11491:
11481:
11475:
11474:
11472:
11471:
11461:
11455:
11454:
11434:
11428:
11427:
11400:
11394:
11393:
11388:. Archived from
11354:
11348:
11347:
11323:
11317:
11310:
11298:
11292:
11291:
11289:
11288:
11274:
11256:
11254:
11253:
11248:
11236:
11234:
11233:
11228:
11210:
11208:
11207:
11202:
11090:
11088:
11087:
11082:
11049:
11047:
11046:
11041:
11021:
11019:
11018:
11013:
11011:
11010:
10994:
10992:
10991:
10986:
10978:
10977:
10957:
10955:
10954:
10949:
10947:
10946:
10928:
10926:
10925:
10920:
10912:
10911:
10892:
10890:
10889:
10884:
10882:
10881:
10862:
10860:
10859:
10854:
10852:
10761:
10759:
10758:
10753:
10751:
10713:
10711:
10710:
10705:
10693:
10691:
10690:
10685:
10664:
10662:
10661:
10656:
10644:
10642:
10641:
10636:
10621:
10619:
10618:
10613:
10611:
10579:
10577:
10576:
10571:
10555:
10553:
10552:
10547:
10535:
10533:
10532:
10527:
10513:
10512:
10490:
10488:
10487:
10482:
10455:
10453:
10452:
10447:
10435:
10433:
10432:
10427:
10415:
10413:
10412:
10407:
10392:
10390:
10389:
10384:
10382:
10381:
10354:
10353:
10274:; in fact, when
10226:
10224:
10223:
10218:
10213:
10212:
10207:
10203:
10196:
10195:
10183:
10182:
10158:
10157:
10145:
10144:
10122:
10109:
10099:
10098:
10086:
10085:
10069:
10068:
10056:
10055:
10036:
10035:
10023:
10022:
10008:
10007:
9995:
9994:
9970:
9969:
9834:ring of integers
9769:ring of integers
9758:ring of integers
9738:
9736:
9735:
9730:
9725:
9724:
9704:
9702:
9701:
9696:
9685:
9684:
9668:
9666:
9665:
9660:
9652:
9651:
9635:
9633:
9632:
9627:
9622:
9621:
9605:
9598:
9592:
9590:
9589:
9584:
9579:
9578:
9562:
9560:
9559:
9554:
9543:
9542:
9507:
9505:
9504:
9499:
9488:
9487:
9469:at every point.
9452:
9450:
9449:
9444:
9433:
9432:
9414:at every point.
9397:
9395:
9394:
9389:
9378:
9377:
9359:
9357:
9356:
9351:
9346:
9345:
9326:
9324:
9323:
9318:
9286:
9285:
9252:
9251:
9232:
9226:
9224:
9223:
9218:
9213:
9212:
9194:
9192:
9191:
9186:
9181:
9180:
9161:
9159:
9158:
9153:
9142:
9128:
9114:
9100:
9099:
9060:
9059:
9040:
9031:
9029:
9028:
9023:
9018:
9017:
9001:
8999:
8998:
8993:
8921:
8919:
8918:
8913:
8899:
8898:
8873:
8871:
8870:
8865:
8860:
8859:
8781:
8779:
8778:
8773:
8765:
8764:
8719:
8718:
8688:
8687:
8655:
8626:
8624:
8623:
8618:
8577:
8576:
8546:
8545:
8469:
8458:
8443:
8441:
8440:
8435:
8426:
8425:
8424:
8407:
8398:
8392:
8381:
8377:
8363:
8361:
8360:
8355:
8352:
8347:
8338:
8337:
8327:
8322:
8294:
8292:
8291:
8286:
8280:
8275:
8266:
8265:
8246:
8241:
8232:
8231:
8205:By a theorem of
8196:rational numbers
8182:are in the same
8181:
8171:
8157:
8146:
8136:
8130:
8128:
8127:
8122:
8110:
8109:
8108:
8091:
8085:
8069:
8059:
8057:
8056:
8051:
8049:
8048:
8032:
8019:
8009:
7991:
7989:
7988:
7983:
7981:
7980:
7979:
7962:
7960:
7959:
7954:
7936:
7934:
7933:
7928:
7923:
7922:
7904:
7903:
7878:
7876:
7875:
7870:
7852:
7850:
7849:
7844:
7839:
7838:
7810:symmetric matrix
7808:
7806:
7805:
7800:
7779:
7777:
7776:
7771:
7766:
7765:
7764:
7720:
7718:
7717:
7712:
7707:
7706:
7697:
7696:
7687:
7686:
7673:
7642:
7637:
7628:
7627:
7614:
7609:
7592:
7586:
7582:
7581:
7563:
7562:
7503:
7501:
7500:
7495:
7493:
7492:
7470:
7452:
7446:
7432:
7423:
7418:if and only the
7409:
7391:
7385:
7371:quadratic fields
7352:
7348:
7344:
7340:
7336:
7332:
7328:
7324:
7320:
7312:
7305:
7301:
7297:
7282:
7276:
7270:
7260:
7254:
7243:algebraic curves
7220:
7218:
7217:
7212:
7210:
7209:
7193:
7191:
7190:
7185:
7183:
7182:
7166:
7160:
7154:
7143:
7132:
7130:
7129:
7124:
7122:
7121:
7105:
7103:
7102:
7097:
7095:
7094:
7075:
7061:
7059:
7058:
7053:
7039:
7038:
7019:
7017:
7016:
7011:
6994:
6993:
6984:
6983:
6944:
6943:
6921:
6919:
6918:
6913:
6896:
6895:
6886:
6885:
6846:
6845:
6826:
6824:
6823:
6818:
6801:
6800:
6781:
6779:
6778:
6773:
6741:
6740:
6701:
6700:
6681:
6679:
6678:
6673:
6671:
6670:
6654:
6652:
6651:
6646:
6644:
6643:
6624:
6611:
6609:
6608:
6603:
6601:
6600:
6591:
6590:
6575:
6574:
6564:
6559:
6541:
6540:
6531:
6530:
6509:
6508:
6493:
6492:
6480:
6479:
6470:
6469:
6420:
6409:
6401:
6387:
6373:
6366:
6344:
6318:
6316:
6315:
6310:
6305:
6304:
6285:
6283:
6282:
6277:
6260:
6259:
6244:
6243:
6228:
6227:
6204:
6202:
6201:
6196:
6182:
6181:
6162:
6153:
6151:
6150:
6145:
6119:
6118:
6094:
6093:
6066:
6065:
6040:
6038:
6037:
6032:
6006:
6005:
5984:
5983:
5968:
5967:
5945:
5943:
5942:
5937:
5920:
5919:
5901:
5900:
5888:
5887:
5868:
5866:
5865:
5860:
5857:
5856:
5855:
5845:
5826:
5825:
5824:
5814:
5792:
5790:
5789:
5784:
5779:
5778:
5760:
5759:
5744:
5743:
5725:
5724:
5715:
5714:
5678:
5672:
5670:
5669:
5664:
5662:
5661:
5645:
5639:
5629:
5627:
5626:
5621:
5619:
5618:
5602:
5596:
5585:
5576:
5574:
5573:
5568:
5566:
5561:
5541:
5536:
5535:
5534:
5521:
5507:
5496:
5487:
5479:
5472:
5464:
5456:Sylvester matrix
5450:
5438:
5434:
5430:
5422:
5394:
5388:
5382:
5376:
5370:
5360:
5358:
5357:
5352:
5350:
5349:
5330:
5328:
5327:
5322:
5320:
5301:
5300:
5273:
5272:
5245:
5244:
5223:
5222:
5213:
5212:
5184:
5166:
5165:
5156:
5155:
5131:
5130:
5109:
5108:
5080:
5079:
5056:
5050:
5024:
5022:
5021:
5016:
5014:
5013:
4997:
4995:
4994:
4989:
4966:
4965:
4940:
4938:
4937:
4932:
4921:
4920:
4877:
4875:
4874:
4869:
4858:
4857:
4842:
4841:
4825:
4823:
4822:
4817:
4794:
4793:
4761:
4759:
4758:
4753:
4745:
4744:
4729:
4728:
4719:
4718:
4709:
4708:
4669:
4668:
4644:
4642:
4641:
4636:
4622:
4621:
4599:
4597:
4596:
4591:
4574:
4573:
4551:
4549:
4548:
4543:
4529:
4528:
4500:
4498:
4497:
4492:
4472:
4471:
4450:
4449:
4434:
4433:
4414:
4412:
4411:
4406:
4392:
4391:
4369:
4367:
4366:
4361:
4346:
4337:
4335:
4334:
4329:
4324:
4323:
4299:
4298:
4280:
4279:
4255:
4254:
4242:
4241:
4223:
4222:
4203:
4197:
4195:
4194:
4189:
4177:
4168:
4166:
4165:
4160:
4158:
4157:
4139:
4138:
4123:
4122:
4104:
4103:
4094:
4093:
4060:
4058:
4057:
4052:
4018:
4016:
4015:
4010:
4005:
4004:
3986:
3985:
3976:
3975:
3957:
3943:
3942:
3918:
3917:
3916:
3897:
3895:
3894:
3889:
3878:
3877:
3861:
3859:
3858:
3853:
3848:
3847:
3829:
3828:
3819:
3818:
3787:
3777:
3775:
3774:
3769:
3764:
3763:
3762:
3745:
3743:
3742:
3737:
3706:
3704:
3703:
3698:
3675:
3674:
3647:
3646:
3645:
3629:
3628:
3594:
3592:
3591:
3586:
3563:
3562:
3553:
3552:
3501:
3500:
3466:
3464:
3463:
3458:
3435:
3434:
3395:
3394:
3365:
3363:
3362:
3357:
3355:
3354:
3338:
3332:
3305:
3303:
3302:
3297:
3295:
3294:
3274:
3218:
3216:
3215:
3210:
3208:
3200:
3199:
3184:
3183:
3165:
3164:
3146:
3145:
3136:
3135:
3120:
3119:
3110:
3109:
3091:
3090:
3076:
3060:
3058:
3057:
3052:
3034:
3033:
3018:
3017:
2998:
2996:
2995:
2990:
2988:
2980:
2979:
2970:
2969:
2960:
2959:
2944:
2943:
2934:
2933:
2918:
2917:
2908:
2907:
2892:
2887:
2874:
2873:
2858:
2857:
2848:
2847:
2832:
2831:
2822:
2821:
2800:
2799:
2781:
2780:
2756:
2751:
2741:
2740:
2716:
2715:
2706:
2705:
2687:
2686:
2674:
2673:
2655:
2654:
2645:
2644:
2629:
2624:
2617:
2616:
2604:
2603:
2588:
2587:
2578:
2577:
2568:
2567:
2552:
2551:
2536:
2535:
2520:
2519:
2510:
2509:
2495:
2479:
2477:
2476:
2471:
2453:
2452:
2437:
2436:
2421:
2420:
2394:
2380:
2327:
2323:
2308:
2306:
2305:
2300:
2294:
2293:
2278:
2277:
2252:
2250:
2249:
2244:
2227:
2226:
2203:
2201:
2200:
2195:
2171:
2170:
2161:
2160:
2142:
2141:
2126:
2125:
2107:
2106:
2097:
2096:
2077:
2075:
2074:
2069:
2051:
2050:
2035:
2034:
2011:
1982:
1945:rational numbers
1942:
1925:
1912:
1895:
1893:
1892:
1887:
1882:
1880:
1872:
1871:
1857:
1856:
1847:
1835:
1830:
1829:
1797:
1795:
1794:
1789:
1771:
1770:
1751:
1749:
1748:
1743:
1725:
1724:
1660:Sylvester matrix
1641:
1636:in the roots of
1630:Vieta's formulas
1608:
1606:
1605:
1600:
1597:
1583:
1560:
1558:
1557:
1552:
1544:
1543:
1531:
1530:
1517:
1501:
1487:
1478:
1477:
1473:
1433:
1432:
1423:
1422:
1410:
1409:
1396:
1380:
1366:
1342:
1341:
1307:
1305:
1304:
1299:
1297:
1296:
1278:
1277:
1265:
1264:
1248:
1230:
1228:
1227:
1222:
1220:
1219:
1201:
1200:
1180:
1178:
1177:
1172:
1170:
1169:
1145:
1143:
1142:
1137:
1135:
1134:
1115:
1113:
1112:
1107:
1102:
1082:
1081:
1072:
1070:
1069:
1060:
1059:
1058:
1054:
1016:
999:
998:
979:
977:
976:
971:
969:
968:
952:
946:
940:
938:
937:
932:
927:
926:
906:
904:
903:
898:
893:
892:
873:
871:
870:
865:
863:
862:
846:
839:
832:Sylvester matrix
821:
819:
818:
813:
811:
810:
792:
791:
772:
770:
769:
764:
759:
758:
740:
739:
724:
723:
690:
689:
674:
673:
649:
627:
616:commutative ring
609:
607:
606:
601:
599:
598:
580:
579:
563:
561:
560:
555:
547:
546:
530:
518:
516:
515:
510:
508:
507:
492:
491:
473:
472:
457:
456:
438:
437:
428:
427:
325:cubic polynomial
306:
304:
303:
298:
266:
264:
263:
258:
241:
240:
221:
219:
218:
213:
196:
195:
128:
121:
117:
114:
108:
106:
65:
41:
33:
11963:
11962:
11958:
11957:
11956:
11954:
11953:
11952:
11938:Quadratic forms
11918:
11917:
11916:
11911:
11860:
11799:
11742:Linear equation
11712:
11703:
11661:
11656:
11634:
11609:
11605:
11598:
11582:
11578:
11563:
11542:
11538:
11529:
11525:
11518:
11502:
11498:
11489:
11487:
11483:
11482:
11478:
11469:
11467:
11463:
11462:
11458:
11451:
11435:
11431:
11424:
11416:. ch. 1 p. 26.
11401:
11397:
11386:
11355:
11351:
11344:
11324:
11320:
11311:
11299:
11295:
11286:
11284:
11276:
11275:
11271:
11267:
11242:
11239:
11238:
11216:
11213:
11212:
11100:
11097:
11096:
11055:
11052:
11051:
11035:
11032:
11031:
11028:
11006:
11002:
11000:
10997:
10996:
10973:
10969:
10967:
10964:
10963:
10942:
10938:
10936:
10933:
10932:
10907:
10903:
10901:
10898:
10897:
10877:
10873:
10871:
10868:
10867:
10848:
10846:
10843:
10842:
10839:
10736:
10719:
10716:
10715:
10699:
10696:
10695:
10670:
10667:
10666:
10650:
10647:
10646:
10630:
10627:
10626:
10596:
10585:
10582:
10581:
10565:
10562:
10561:
10541:
10538:
10537:
10508:
10504:
10496:
10493:
10492:
10461:
10458:
10457:
10441:
10438:
10437:
10421:
10418:
10417:
10401:
10398:
10397:
10395:
10377:
10373:
10349:
10345:
10319:
10316:
10315:
10308:
10292:principal ideal
10230:
10208:
10201:
10200:
10191:
10187:
10178:
10174:
10172:
10167:
10162:
10153:
10149:
10140:
10136:
10133:
10132:
10127:
10121:
10115:
10114:
10108:
10103:
10094:
10090:
10081:
10077:
10074:
10073:
10064:
10060:
10051:
10047:
10045:
10040:
10031:
10027:
10018:
10014:
10012:
10003:
9999:
9990:
9986:
9982:
9978:
9977:
9965:
9961:
9959:
9956:
9955:
9949:
9943:
9884:complex numbers
9877:
9871:
9859:
9848:
9842:
9830:
9750:
9744:
9720:
9716:
9714:
9711:
9710:
9680:
9676:
9674:
9671:
9670:
9647:
9643:
9641:
9638:
9637:
9617:
9613:
9611:
9608:
9607:
9600:
9594:
9574:
9570:
9568:
9565:
9564:
9538:
9534:
9532:
9529:
9528:
9483:
9479:
9477:
9474:
9473:
9428:
9424:
9422:
9419:
9418:
9373:
9369:
9367:
9364:
9363:
9341:
9337:
9335:
9332:
9331:
9281:
9277:
9247:
9243:
9241:
9238:
9237:
9228:
9208:
9204:
9202:
9199:
9198:
9176:
9172:
9170:
9167:
9166:
9138:
9124:
9110:
9095:
9091:
9055:
9051:
9049:
9046:
9045:
9036:
9013:
9009:
9007:
9004:
9003:
8966:
8963:
8962:
8955:Euclidean space
8951:quadric surface
8947:
8894:
8890:
8888:
8885:
8884:
8854:
8853:
8848:
8843:
8837:
8836:
8831:
8826:
8820:
8819:
8814:
8809:
8799:
8798:
8796:
8793:
8792:
8760:
8756:
8714:
8710:
8683:
8679:
8674:
8671:
8670:
8661:quadratic forms
8631:
8572:
8568:
8541:
8537:
8532:
8529:
8528:
8510:
8481:of the field).
8460:
8457:
8449:
8446:diagonal matrix
8420:
8419:
8415:
8413:
8410:
8409:
8403:
8394:
8391:
8383:
8379:
8376:
8368:
8348:
8343:
8333:
8329:
8323:
8312:
8306:
8303:
8302:
8276:
8271:
8261:
8257:
8242:
8237:
8227:
8223:
8221:
8218:
8217:
8188:complex numbers
8177:
8167:
8148:
8142:
8132:
8104:
8103:
8099:
8097:
8094:
8093:
8087:
8081:
8075:change of basis
8065:
8044:
8040:
8038:
8035:
8034:
8028:
8015:
8005:
7975:
7974:
7970:
7968:
7965:
7964:
7942:
7939:
7938:
7918:
7914:
7899:
7895:
7884:
7881:
7880:
7858:
7855:
7854:
7831:
7827:
7816:
7813:
7812:
7788:
7785:
7784:
7760:
7759:
7755:
7732:
7729:
7728:
7702:
7698:
7692:
7688:
7679:
7675:
7651:
7638:
7633:
7620:
7616:
7610:
7599:
7577:
7573:
7558:
7554:
7546:
7543:
7542:
7520:
7514:
7512:Quadratic forms
7482:
7478:
7476:
7473:
7472:
7466:
7448:
7442:
7428:
7419:
7416:singular points
7405:
7387:
7381:
7363:
7361:Generalizations
7350:
7346:
7342:
7338:
7334:
7330:
7326:
7322:
7318:
7315:algebraic curve
7307:
7303:
7299:
7295:
7283:(including the
7278:
7272:
7266:
7256:
7250:
7235:
7205:
7201:
7199:
7196:
7195:
7178:
7174:
7172:
7169:
7168:
7162:
7156:
7145:
7134:
7117:
7113:
7111:
7108:
7107:
7090:
7086:
7084:
7081:
7080:
7071:
7034:
7030:
7028:
7025:
7024:
6989:
6985:
6964:
6960:
6939:
6935:
6933:
6930:
6929:
6891:
6887:
6866:
6862:
6841:
6837:
6835:
6832:
6831:
6796:
6792:
6790:
6787:
6786:
6736:
6732:
6696:
6692:
6690:
6687:
6686:
6666:
6662:
6660:
6657:
6656:
6639:
6635:
6633:
6630:
6629:
6620:
6596:
6592:
6580:
6576:
6570:
6566:
6560:
6549:
6536:
6532:
6526:
6522:
6498:
6494:
6488:
6484:
6475:
6471:
6465:
6461:
6438:
6435:
6434:
6428:
6411:
6403:
6392:
6379:
6368:
6357:
6340:
6325:
6300:
6296:
6291:
6288:
6287:
6255:
6251:
6239:
6235:
6223:
6219:
6214:
6211:
6210:
6177:
6173:
6171:
6168:
6167:
6158:
6108:
6104:
6089:
6085:
6061:
6057:
6052:
6049:
6048:
6001:
5997:
5979:
5975:
5963:
5959:
5957:
5954:
5953:
5915:
5911:
5896:
5892:
5883:
5879:
5877:
5874:
5873:
5851:
5847:
5846:
5841:
5820:
5816:
5815:
5810:
5804:
5801:
5800:
5774:
5770:
5749:
5745:
5733:
5729:
5720:
5716:
5710:
5706:
5698:
5695:
5694:
5674:
5657:
5653:
5651:
5648:
5647:
5641:
5631:
5614:
5610:
5608:
5605:
5604:
5598:
5587:
5581:
5542:
5540:
5530:
5517:
5516:
5515:
5513:
5510:
5509:
5498:
5492:
5481:
5478:
5474:
5466:
5463:
5459:
5440:
5436:
5432:
5424:
5418:
5404:
5390:
5384:
5378:
5372:
5366:
5345:
5341:
5339:
5336:
5335:
5318:
5317:
5296:
5292:
5268:
5264:
5240:
5236:
5218:
5214:
5205:
5201:
5185:
5183:
5180:
5179:
5161:
5157:
5151:
5147:
5126:
5122:
5104:
5100:
5093:
5075:
5071:
5067:
5065:
5062:
5061:
5052:
5042:
5039:
5009:
5005:
5003:
5000:
4999:
4998:if and only if
4961:
4957:
4949:
4946:
4945:
4916:
4912:
4883:
4880:
4879:
4853:
4849:
4837:
4833:
4831:
4828:
4827:
4789:
4785:
4777:
4774:
4773:
4740:
4736:
4724:
4720:
4714:
4710:
4698:
4694:
4664:
4660:
4652:
4649:
4648:
4617:
4613:
4605:
4602:
4601:
4569:
4565:
4557:
4554:
4553:
4524:
4520:
4512:
4509:
4508:
4467:
4463:
4445:
4441:
4429:
4425:
4423:
4420:
4419:
4387:
4383:
4375:
4372:
4371:
4355:
4352:
4351:
4342:
4319:
4315:
4288:
4284:
4269:
4265:
4250:
4246:
4237:
4233:
4218:
4214:
4212:
4209:
4208:
4199:
4183:
4180:
4179:
4173:
4153:
4149:
4128:
4124:
4112:
4108:
4099:
4095:
4089:
4085:
4077:
4074:
4073:
4034:
4031:
4030:
4027:
4000:
3996:
3981:
3977:
3971:
3967:
3953:
3938:
3934:
3912:
3911:
3907:
3905:
3902:
3901:
3873:
3869:
3867:
3864:
3863:
3843:
3839:
3824:
3820:
3814:
3810:
3793:
3790:
3789:
3783:
3758:
3757:
3753:
3751:
3748:
3747:
3716:
3713:
3712:
3670:
3666:
3641:
3640:
3636:
3624:
3620:
3618:
3615:
3614:
3558:
3554:
3533:
3529:
3496:
3492:
3490:
3487:
3486:
3430:
3426:
3390:
3386:
3384:
3381:
3380:
3350:
3346:
3344:
3341:
3340:
3334:
3323:
3312:
3290:
3286:
3284:
3281:
3280:
3270:
3249:integral domain
3242:field extension
3234:
3229:
3206:
3205:
3195:
3191:
3179:
3175:
3160:
3156:
3141:
3137:
3131:
3127:
3115:
3111:
3105:
3101:
3086:
3082:
3077:
3075:
3071:
3069:
3066:
3065:
3029:
3025:
3013:
3009:
3007:
3004:
3003:
2986:
2985:
2975:
2971:
2965:
2961:
2955:
2951:
2939:
2935:
2929:
2925:
2913:
2909:
2903:
2899:
2891:
2885:
2884:
2869:
2865:
2853:
2849:
2843:
2839:
2827:
2823:
2817:
2813:
2795:
2791:
2776:
2772:
2755:
2749:
2748:
2736:
2732:
2711:
2707:
2701:
2697:
2682:
2678:
2669:
2665:
2650:
2646:
2640:
2636:
2628:
2622:
2621:
2612:
2608:
2599:
2595:
2583:
2579:
2573:
2569:
2563:
2559:
2547:
2543:
2531:
2527:
2515:
2511:
2505:
2501:
2496:
2494:
2490:
2488:
2485:
2484:
2448:
2444:
2432:
2428:
2416:
2412:
2407:
2404:
2403:
2382:
2364:
2357:
2330:Cardano formula
2328:in the case of
2325:
2321:
2289:
2285:
2273:
2269:
2261:
2258:
2257:
2222:
2218:
2216:
2213:
2212:
2209:depressed cubic
2166:
2162:
2156:
2152:
2137:
2133:
2121:
2117:
2102:
2098:
2092:
2088:
2086:
2083:
2082:
2046:
2042:
2030:
2026:
2021:
2018:
2017:
1984:
1966:
1959:
1953:
1930:
1921:
1900:
1873:
1852:
1848:
1846:
1836:
1834:
1819:
1815:
1813:
1810:
1809:
1766:
1762:
1760:
1757:
1756:
1720:
1716:
1711:
1708:
1707:
1704:
1698:
1648:
1637:
1584:
1579:
1573:
1570:
1569:
1539:
1535:
1526:
1522:
1507:
1488:
1483:
1469:
1450:
1446:
1428:
1424:
1418:
1414:
1405:
1401:
1386:
1367:
1362:
1337:
1333:
1331:
1328:
1327:
1314:complex numbers
1292:
1288:
1273:
1269:
1260:
1256:
1254:
1251:
1250:
1244:
1237:
1215:
1211:
1196:
1192:
1190:
1187:
1186:
1165:
1161:
1159:
1156:
1155:
1130:
1126:
1124:
1121:
1120:
1095:
1077:
1073:
1065:
1061:
1050:
1031:
1027:
1017:
1015:
994:
990:
988:
985:
984:
964:
960:
958:
955:
954:
948:
942:
922:
918:
916:
913:
912:
888:
884:
879:
876:
875:
858:
854:
852:
849:
848:
841:
835:
806:
802:
787:
783:
781:
778:
777:
754:
750:
729:
725:
713:
709:
679:
675:
669:
665:
642:
640:
637:
636:
623:
594:
590:
575:
571:
569:
566:
565:
542:
538:
536:
533:
532:
526:
503:
499:
487:
483:
462:
458:
446:
442:
433:
429:
423:
419:
402:
399:
398:
392:
380:
283:
280:
279:
236:
232:
230:
227:
226:
191:
187:
182:
179:
178:
129:
118:
112:
109:
66:
64:
54:
42:
31:
24:
17:
12:
11:
5:
11961:
11951:
11950:
11945:
11940:
11935:
11933:Conic sections
11930:
11913:
11912:
11910:
11909:
11904:
11899:
11894:
11889:
11884:
11879:
11874:
11868:
11866:
11862:
11861:
11859:
11858:
11853:
11848:
11843:
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11833:
11828:
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11807:
11805:
11801:
11800:
11798:
11797:
11792:
11787:
11782:
11781:
11780:
11770:
11769:
11768:
11766:Cubic equation
11758:
11757:
11756:
11746:
11745:
11744:
11734:
11729:
11723:
11721:
11714:
11713:
11702:
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11694:
11687:
11679:
11673:
11672:
11667:
11660:
11659:External links
11657:
11655:
11654:
11632:
11603:
11596:
11576:
11561:
11553:Academic Press
11536:
11523:
11516:
11496:
11476:
11456:
11449:
11429:
11422:
11395:
11392:on 2013-01-13.
11384:
11349:
11342:
11318:
11305:. 4th series.
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11246:
11226:
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11197:
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11113:
11110:
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11059:
11050:consisting of
11043:{\textstyle S}
11039:
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10833:
10822:
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10764:
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10723:
10707:{\textstyle m}
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10658:{\textstyle m}
10654:
10638:{\textstyle D}
10634:
10623:
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10573:{\textstyle D}
10569:
10549:{\textstyle D}
10545:
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10449:{\textstyle c}
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10429:{\textstyle b}
10425:
10409:{\textstyle a}
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9939:
9873:
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9852:integral basis
9846:
9840:
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9746:Main article:
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8520:defined by an
8509:
8508:Conic sections
8506:
8490:singular point
8453:
8433:
8429:
8423:
8418:
8399:, there is an
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7994:characteristic
7978:
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7963:column vector
7952:
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7524:quadratic form
7513:
7510:
7491:
7488:
7485:
7481:
7455:primitive part
7394:characteristic
7362:
7359:
7234:
7231:
7229:must be used.
7208:
7204:
7181:
7177:
7120:
7116:
7093:
7089:
7062:is called the
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6389:
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6331:coefficients.
6324:
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5748:
5742:
5739:
5736:
5732:
5728:
5723:
5719:
5713:
5709:
5705:
5702:
5687:of the roots.
5660:
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5125:
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3817:
3813:
3809:
3806:
3803:
3800:
3797:
3788:; that is, if
3761:
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3709:
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3696:
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3499:
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3401:
3398:
3393:
3389:
3375:
3374:
3353:
3349:
3311:
3308:
3293:
3289:
3260:characteristic
3233:
3230:
3228:
3225:
3220:
3219:
3204:
3198:
3194:
3190:
3187:
3182:
3178:
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3171:
3168:
3163:
3159:
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3123:
3118:
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3097:
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2658:
2653:
2649:
2643:
2639:
2635:
2632:
2627:
2625:
2623:
2620:
2615:
2611:
2607:
2602:
2598:
2594:
2591:
2586:
2582:
2576:
2572:
2566:
2562:
2558:
2555:
2550:
2546:
2542:
2539:
2534:
2530:
2526:
2523:
2518:
2514:
2508:
2504:
2500:
2497:
2493:
2492:
2468:
2465:
2462:
2459:
2456:
2451:
2447:
2443:
2440:
2435:
2431:
2427:
2424:
2419:
2415:
2411:
2356:
2353:
2310:
2309:
2298:
2292:
2288:
2284:
2281:
2276:
2272:
2268:
2265:
2242:
2239:
2236:
2233:
2230:
2225:
2221:
2205:
2204:
2193:
2189:
2186:
2183:
2180:
2177:
2174:
2169:
2165:
2159:
2155:
2151:
2148:
2145:
2140:
2136:
2132:
2129:
2124:
2120:
2116:
2113:
2110:
2105:
2101:
2095:
2091:
2066:
2063:
2060:
2057:
2054:
2049:
2045:
2041:
2038:
2033:
2029:
2025:
1952:
1949:
1897:
1896:
1885:
1879:
1876:
1869:
1866:
1863:
1860:
1855:
1851:
1845:
1842:
1839:
1833:
1828:
1825:
1822:
1818:
1799:
1798:
1787:
1783:
1780:
1777:
1774:
1769:
1765:
1740:
1737:
1734:
1731:
1728:
1723:
1719:
1715:
1697:
1694:
1647:
1644:
1624:, or from the
1596:
1593:
1590:
1587:
1582:
1578:
1562:
1561:
1550:
1547:
1542:
1538:
1534:
1529:
1525:
1521:
1516:
1513:
1510:
1506:
1500:
1497:
1494:
1491:
1486:
1482:
1476:
1472:
1468:
1465:
1462:
1459:
1456:
1453:
1449:
1445:
1442:
1439:
1436:
1431:
1427:
1421:
1417:
1413:
1408:
1404:
1400:
1395:
1392:
1389:
1385:
1379:
1376:
1373:
1370:
1365:
1361:
1357:
1354:
1351:
1348:
1345:
1340:
1336:
1295:
1291:
1287:
1284:
1281:
1276:
1272:
1268:
1263:
1259:
1236:
1233:
1218:
1214:
1210:
1207:
1204:
1199:
1195:
1168:
1164:
1133:
1129:
1117:
1116:
1105:
1101:
1098:
1094:
1091:
1088:
1085:
1080:
1076:
1068:
1064:
1057:
1053:
1049:
1046:
1043:
1040:
1037:
1034:
1030:
1026:
1023:
1020:
1014:
1011:
1008:
1005:
1002:
997:
993:
967:
963:
930:
925:
921:
896:
891:
887:
883:
861:
857:
809:
805:
801:
798:
795:
790:
786:
774:
773:
762:
757:
753:
749:
746:
743:
738:
735:
732:
728:
722:
719:
716:
712:
708:
705:
702:
699:
696:
693:
688:
685:
682:
678:
672:
668:
664:
661:
658:
655:
652:
648:
645:
597:
593:
589:
586:
583:
578:
574:
553:
550:
545:
541:
520:
519:
506:
502:
498:
495:
490:
486:
482:
479:
476:
471:
468:
465:
461:
455:
452:
449:
445:
441:
436:
432:
426:
422:
418:
415:
412:
409:
406:
391:
388:
379:
376:
355:quadratic form
309:if and only if
296:
293:
290:
287:
268:
267:
256:
253:
250:
247:
244:
239:
235:
211:
208:
205:
202:
199:
194:
190:
186:
131:
130:
72:"Discriminant"
45:
43:
36:
15:
9:
6:
4:
3:
2:
11960:
11949:
11946:
11944:
11941:
11939:
11936:
11934:
11931:
11929:
11926:
11925:
11923:
11908:
11907:Gröbner basis
11905:
11903:
11900:
11898:
11895:
11893:
11890:
11888:
11885:
11883:
11880:
11878:
11875:
11873:
11872:Factorization
11870:
11869:
11867:
11863:
11857:
11854:
11852:
11849:
11847:
11844:
11842:
11839:
11837:
11834:
11832:
11829:
11827:
11824:
11822:
11819:
11817:
11814:
11812:
11809:
11808:
11806:
11804:By properties
11802:
11796:
11793:
11791:
11788:
11786:
11783:
11779:
11776:
11775:
11774:
11771:
11767:
11764:
11763:
11762:
11759:
11755:
11752:
11751:
11750:
11747:
11743:
11740:
11739:
11738:
11735:
11733:
11730:
11728:
11725:
11724:
11722:
11720:
11715:
11711:
11707:
11700:
11695:
11693:
11688:
11686:
11681:
11680:
11677:
11671:
11668:
11666:
11663:
11662:
11651:
11647:
11643:
11639:
11635:
11629:
11625:
11621:
11617:
11613:
11607:
11599:
11597:0-471-75715-2
11593:
11589:
11588:
11580:
11572:
11568:
11564:
11562:0-12-163260-1
11558:
11555:. p. 6.
11554:
11550:
11546:
11540:
11533:
11532:Arf invariant
11527:
11519:
11517:0-387-40397-3
11513:
11509:
11508:
11500:
11486:
11480:
11466:
11460:
11452:
11450:0-387-40397-3
11446:
11442:
11441:
11433:
11425:
11423:3-540-24326-7
11419:
11415:
11411:
11410:
11405:
11399:
11391:
11387:
11385:3-7643-3660-9
11381:
11378:. p. 1.
11377:
11373:
11372:
11367:
11363:
11359:
11353:
11345:
11343:1-86094-438-8
11339:
11335:
11331:
11330:
11322:
11315:
11308:
11304:
11297:
11283:
11279:
11273:
11269:
11262:
11260:
11244:
11224:
11221:
11218:
11195:
11192:
11189:
11186:
11183:
11180:
11177:
11174:
11171:
11168:
11165:
11162:
11159:
11156:
11153:
11150:
11147:
11144:
11141:
11138:
11135:
11132:
11129:
11126:
11123:
11120:
11117:
11114:
11111:
11105:
11102:
11094:
11078:
11075:
11072:
11069:
11066:
11063:
11060:
11057:
11037:
11023:
11007:
11003:
10982:
10979:
10974:
10970:
10943:
10939:
10931:
10916:
10913:
10908:
10904:
10896:
10895:
10894:
10878:
10874:
10864:
10831:
10827:
10826:
10825:
10819:
10815:
10811:
10807:
10803:
10799:
10795:
10791:
10787:
10783:
10779:
10775:
10772:
10771:
10770:
10767:
10745:
10741:
10733:
10730:
10727:
10724:
10721:
10701:
10681:
10678:
10675:
10672:
10652:
10632:
10624:
10605:
10601:
10593:
10590:
10587:
10567:
10559:
10558:
10557:
10543:
10523:
10520:
10517:
10514:
10509:
10505:
10501:
10498:
10475:
10472:
10469:
10463:
10443:
10423:
10403:
10393:
10378:
10374:
10370:
10367:
10364:
10361:
10358:
10355:
10350:
10346:
10342:
10339:
10333:
10330:
10327:
10321:
10313:
10303:
10301:
10297:
10293:
10289:
10285:
10281:
10278: =
10277:
10273:
10269:
10265:
10261:
10257:
10253:
10249:
10245:
10242:
10238:
10234:
10214:
10209:
10204:
10192:
10188:
10179:
10175:
10169:
10164:
10154:
10150:
10141:
10137:
10129:
10124:
10118:
10111:
10105:
10095:
10091:
10082:
10078:
10065:
10061:
10052:
10048:
10042:
10032:
10028:
10019:
10015:
10004:
10000:
9991:
9987:
9979:
9971:
9966:
9954:
9953:
9952:
9950:
9942:
9937:
9933:
9929:
9926:
9923:
9919:
9915:
9911:
9907:
9903:
9899:
9896: →
9895:
9892:
9889:
9885:
9881:
9876:
9868:), and let {σ
9867:
9865:
9860:
9853:
9849:
9839:
9835:
9831:
9824:
9819:
9817:
9813:
9809:
9805:
9801:
9797:
9793:
9789:
9785:
9780:
9778:
9774:
9770:
9766:
9761:
9759:
9755:
9749:
9739:
9726:
9721:
9708:
9692:
9689:
9686:
9681:
9656:
9653:
9648:
9623:
9618:
9604:
9597:
9580:
9575:
9550:
9547:
9544:
9539:
9525:
9523:
9519:
9515:
9511:
9495:
9492:
9489:
9484:
9470:
9468:
9464:
9460:
9456:
9440:
9437:
9434:
9429:
9415:
9413:
9409:
9408:ruled surface
9405:
9401:
9385:
9382:
9379:
9374:
9360:
9347:
9342:
9314:
9308:
9305:
9302:
9299:
9296:
9293:
9290:
9282:
9278:
9274:
9268:
9265:
9262:
9259:
9256:
9248:
9244:
9236:
9235:
9234:
9231:
9214:
9209:
9205:
9195:
9182:
9177:
9149:
9143:
9139:
9135:
9132:
9129:
9125:
9121:
9118:
9115:
9111:
9107:
9101:
9096:
9092:
9088:
9082:
9079:
9076:
9073:
9070:
9067:
9064:
9056:
9052:
9044:
9043:
9042:
9039:
9035:
9019:
9014:
9010:
8986:
8983:
8980:
8977:
8974:
8968:
8959:
8956:
8952:
8942:
8940:
8936:
8932:
8928:
8909:
8906:
8903:
8900:
8895:
8891:
8883:
8882:
8881:
8878:
8861:
8856:
8850:
8845:
8840:
8833:
8828:
8823:
8816:
8811:
8806:
8800:
8791:
8790:
8789:
8788:
8769:
8766:
8761:
8757:
8753:
8750:
8747:
8744:
8741:
8738:
8735:
8732:
8729:
8726:
8723:
8720:
8715:
8711:
8707:
8704:
8701:
8698:
8695:
8692:
8689:
8684:
8680:
8676:
8669:
8668:
8667:
8664:
8662:
8657:
8654:
8650:
8646:
8642:
8638:
8634:
8614:
8611:
8608:
8605:
8602:
8599:
8596:
8593:
8590:
8587:
8584:
8581:
8578:
8573:
8569:
8565:
8562:
8559:
8556:
8553:
8550:
8547:
8542:
8538:
8534:
8527:
8526:
8525:
8523:
8519:
8515:
8514:conic section
8505:
8503:
8499:
8495:
8491:
8487:
8482:
8480:
8476:
8471:
8467:
8463:
8456:
8452:
8447:
8431:
8427:
8416:
8406:
8402:
8397:
8390:
8386:
8375:
8371:
8349:
8344:
8340:
8334:
8330:
8324:
8319:
8316:
8313:
8309:
8301:
8300:
8299:
8282:
8277:
8272:
8268:
8262:
8258:
8254:
8251:
8248:
8243:
8238:
8234:
8228:
8224:
8216:
8215:
8214:
8212:
8211:diagonal form
8208:
8203:
8201:
8197:
8193:
8189:
8185:
8180:
8175:
8170:
8165:
8161:
8155:
8151:
8145:
8140:
8135:
8118:
8115:
8111:
8100:
8090:
8084:
8080:
8076:
8071:
8068:
8063:
8045:
8041:
8031:
8026:
8021:
8018:
8013:
8008:
8003:
7999:
7995:
7971:
7950:
7947:
7944:
7919:
7915:
7911:
7908:
7905:
7900:
7896:
7889:
7886:
7866:
7863:
7860:
7835:
7832:
7828:
7821:
7818:
7811:
7796:
7793:
7790:
7767:
7756:
7752:
7749:
7746:
7740:
7734:
7727:
7726:
7725:
7708:
7703:
7699:
7693:
7689:
7683:
7680:
7676:
7670:
7667:
7664:
7661:
7658:
7655:
7652:
7648:
7644:
7639:
7634:
7630:
7624:
7621:
7617:
7611:
7606:
7603:
7600:
7596:
7589:
7578:
7574:
7570:
7567:
7564:
7559:
7555:
7548:
7541:
7540:
7539:
7538:of degree 2:
7537:
7533:
7529:
7525:
7519:
7509:
7507:
7489:
7486:
7483:
7479:
7469:
7463:
7461:
7456:
7451:
7445:
7440:
7436:
7431:
7426:
7422:
7417:
7413:
7408:
7403:
7399:
7395:
7390:
7384:
7378:
7374:
7372:
7368:
7358:
7356:
7316:
7310:
7292:
7290:
7286:
7281:
7275:
7269:
7264:
7259:
7253:
7248:
7244:
7240:
7230:
7228:
7224:
7206:
7202:
7179:
7175:
7165:
7159:
7152:
7148:
7141:
7137:
7118:
7114:
7091:
7087:
7077:
7074:
7069:
7065:
7046:
7040:
7035:
7031:
7007:
7001:
6995:
6990:
6986:
6977:
6974:
6971:
6965:
6961:
6957:
6951:
6945:
6940:
6936:
6928:
6927:
6926:
6909:
6903:
6897:
6892:
6888:
6879:
6876:
6873:
6867:
6863:
6859:
6853:
6847:
6842:
6838:
6830:
6829:
6828:
6814:
6808:
6802:
6797:
6793:
6769:
6760:
6757:
6754:
6748:
6742:
6737:
6733:
6729:
6720:
6717:
6714:
6708:
6702:
6697:
6693:
6685:
6684:
6683:
6667:
6663:
6640:
6636:
6626:
6623:
6618:
6597:
6593:
6587:
6584:
6581:
6577:
6571:
6567:
6561:
6556:
6553:
6550:
6546:
6542:
6537:
6533:
6527:
6523:
6519:
6516:
6513:
6510:
6505:
6502:
6499:
6495:
6489:
6485:
6481:
6476:
6472:
6466:
6462:
6458:
6452:
6449:
6446:
6440:
6433:
6432:
6431:
6418:
6414:
6407:
6399:
6395:
6390:
6386:
6382:
6377:
6372:
6364:
6360:
6355:
6352:
6351:multiple root
6348:
6347:
6346:
6343:
6337:
6332:
6330:
6320:
6306:
6301:
6297:
6293:
6273:
6270:
6267:
6264:
6261:
6256:
6252:
6248:
6245:
6240:
6236:
6232:
6229:
6224:
6220:
6216:
6207:
6192:
6189:
6186:
6183:
6178:
6174:
6164:
6161:
6141:
6135:
6132:
6129:
6123:
6120:
6115:
6112:
6109:
6105:
6101:
6098:
6095:
6090:
6086:
6079:
6076:
6073:
6067:
6062:
6058:
6054:
6047:
6046:
6045:
6028:
6022:
6019:
6016:
6010:
6007:
6002:
5998:
5994:
5991:
5988:
5985:
5980:
5976:
5972:
5969:
5964:
5960:
5952:
5951:
5950:
5933:
5930:
5927:
5924:
5921:
5916:
5912:
5908:
5905:
5902:
5897:
5893:
5889:
5884:
5880:
5872:
5871:
5870:
5852:
5848:
5842:
5838:
5834:
5831:
5828:
5821:
5817:
5811:
5807:
5799:
5780:
5775:
5771:
5767:
5764:
5761:
5756:
5753:
5750:
5746:
5740:
5737:
5734:
5730:
5726:
5721:
5717:
5711:
5707:
5703:
5700:
5693:
5692:
5691:
5688:
5686:
5682:
5677:
5658:
5654:
5644:
5638:
5634:
5615:
5611:
5601:
5594:
5590:
5584:
5578:
5562:
5555:
5552:
5549:
5543:
5537:
5526:
5523:
5505:
5501:
5495:
5489:
5485:
5470:
5458:) divided by
5457:
5453:
5448:
5444:
5428:
5421:
5415:
5413:
5409:
5399:
5396:
5393:
5387:
5381:
5375:
5369:
5364:
5346:
5342:
5314:
5308:
5302:
5297:
5293:
5286:
5283:
5280:
5274:
5269:
5265:
5258:
5255:
5252:
5246:
5241:
5237:
5230:
5224:
5219:
5215:
5209:
5206:
5198:
5195:
5189:
5187:
5173:
5167:
5162:
5158:
5152:
5144:
5141:
5138:
5132:
5127:
5123:
5116:
5110:
5105:
5101:
5097:
5095:
5087:
5081:
5076:
5072:
5060:
5059:
5058:
5055:
5049:
5045:
5034:
5032:
5028:
5027:multiple root
5010:
5006:
4985:
4982:
4973:
4967:
4962:
4958:
4951:
4928:
4925:
4917:
4913:
4906:
4903:
4900:
4894:
4888:
4885:
4865:
4862:
4854:
4850:
4843:
4838:
4834:
4813:
4810:
4801:
4795:
4790:
4786:
4779:
4772:
4771:
4770:
4768:
4749:
4741:
4737:
4730:
4725:
4721:
4715:
4705:
4702:
4699:
4695:
4688:
4685:
4676:
4670:
4665:
4661:
4654:
4647:
4646:
4645:
4632:
4629:
4626:
4618:
4614:
4607:
4581:
4575:
4570:
4566:
4559:
4539:
4536:
4533:
4525:
4521:
4514:
4505:
4488:
4479:
4473:
4468:
4464:
4457:
4454:
4446:
4442:
4435:
4430:
4426:
4418:
4417:
4416:
4402:
4399:
4396:
4388:
4384:
4377:
4357:
4348:
4345:
4320:
4316:
4309:
4306:
4303:
4300:
4295:
4292:
4289:
4285:
4276:
4273:
4270:
4266:
4259:
4256:
4251:
4247:
4238:
4234:
4227:
4224:
4219:
4215:
4207:
4206:
4205:
4202:
4185:
4176:
4154:
4150:
4146:
4143:
4140:
4135:
4132:
4129:
4125:
4119:
4116:
4113:
4109:
4105:
4100:
4096:
4090:
4086:
4082:
4079:
4072:
4071:
4070:
4068:
4064:
4048:
4042:
4039:
4036:
4006:
4001:
3997:
3993:
3990:
3987:
3982:
3978:
3972:
3968:
3964:
3958:
3954:
3950:
3944:
3939:
3935:
3931:
3925:
3908:
3900:
3899:
3885:
3882:
3879:
3874:
3870:
3849:
3844:
3840:
3836:
3833:
3830:
3825:
3821:
3815:
3811:
3807:
3801:
3795:
3786:
3781:
3754:
3733:
3730:
3724:
3718:
3710:
3688:
3682:
3676:
3671:
3667:
3663:
3654:
3637:
3630:
3625:
3621:
3613:
3612:
3611:
3610:
3606:
3603:
3602:
3598:
3576:
3570:
3564:
3559:
3555:
3546:
3543:
3540:
3534:
3530:
3526:
3517:
3514:
3508:
3502:
3497:
3493:
3485:
3484:
3483:
3482:
3478:
3475:
3474:
3470:
3448:
3442:
3436:
3431:
3427:
3423:
3414:
3411:
3408:
3402:
3396:
3391:
3387:
3379:
3378:
3377:
3376:
3372:
3369:
3368:
3367:
3351:
3347:
3337:
3330:
3326:
3321:
3317:
3307:
3291:
3287:
3278:
3273:
3267:
3265:
3261:
3256:
3254:
3250:
3245:
3243:
3239:
3224:
3202:
3196:
3192:
3188:
3185:
3180:
3176:
3172:
3169:
3166:
3161:
3157:
3153:
3150:
3147:
3142:
3138:
3132:
3128:
3124:
3121:
3116:
3112:
3106:
3102:
3098:
3095:
3092:
3087:
3083:
3079:
3064:
3063:
3062:
3047:
3044:
3041:
3038:
3035:
3030:
3026:
3022:
3019:
3014:
3010:
2982:
2976:
2972:
2966:
2962:
2956:
2952:
2948:
2945:
2940:
2936:
2930:
2926:
2922:
2919:
2914:
2910:
2904:
2900:
2896:
2893:
2889:
2881:
2878:
2875:
2870:
2866:
2862:
2859:
2854:
2850:
2844:
2840:
2836:
2833:
2828:
2824:
2818:
2814:
2810:
2807:
2804:
2801:
2796:
2792:
2788:
2785:
2782:
2777:
2773:
2769:
2766:
2763:
2760:
2757:
2753:
2745:
2742:
2737:
2733:
2729:
2726:
2723:
2720:
2717:
2712:
2708:
2702:
2698:
2694:
2691:
2688:
2683:
2679:
2675:
2670:
2666:
2662:
2659:
2656:
2651:
2647:
2641:
2637:
2633:
2630:
2626:
2618:
2613:
2609:
2605:
2600:
2596:
2592:
2589:
2584:
2580:
2574:
2570:
2564:
2560:
2556:
2553:
2548:
2544:
2540:
2537:
2532:
2528:
2524:
2521:
2516:
2512:
2506:
2502:
2498:
2483:
2482:
2481:
2466:
2463:
2460:
2457:
2454:
2449:
2445:
2441:
2438:
2433:
2429:
2425:
2422:
2417:
2413:
2409:
2402:
2393:
2389:
2385:
2379:
2375:
2371:
2367:
2361:
2352:
2350:
2346:
2342:
2338:
2333:
2331:
2319:
2314:
2296:
2290:
2286:
2282:
2279:
2274:
2270:
2266:
2263:
2256:
2255:
2254:
2240:
2237:
2234:
2231:
2228:
2223:
2219:
2210:
2191:
2187:
2184:
2181:
2178:
2175:
2172:
2167:
2163:
2157:
2153:
2149:
2146:
2143:
2138:
2134:
2130:
2127:
2122:
2118:
2114:
2111:
2108:
2103:
2099:
2093:
2089:
2081:
2080:
2079:
2064:
2061:
2058:
2055:
2052:
2047:
2043:
2039:
2036:
2031:
2027:
2023:
2009:
2005:
2001:
1998:
1994:
1990:
1987:
1981:
1977:
1973:
1969:
1963:
1958:
1948:
1946:
1941:
1937:
1933:
1927:
1924:
1918:
1916:
1911:
1907:
1903:
1883:
1877:
1874:
1867:
1864:
1861:
1858:
1853:
1849:
1843:
1840:
1837:
1831:
1826:
1823:
1820:
1816:
1808:
1807:
1806:
1804:
1785:
1781:
1778:
1775:
1772:
1767:
1763:
1755:
1754:
1753:
1738:
1735:
1732:
1729:
1726:
1721:
1717:
1713:
1703:
1693:
1691:
1687:
1683:
1679:
1675:
1672:
1667:
1665:
1661:
1657:
1656:empty product
1653:
1643:
1640:
1635:
1631:
1627:
1623:
1619:
1615:
1614:multiple root
1610:
1594:
1591:
1588:
1585:
1580:
1576:
1567:
1548:
1540:
1536:
1532:
1527:
1523:
1514:
1511:
1508:
1504:
1498:
1495:
1492:
1489:
1484:
1480:
1474:
1470:
1463:
1460:
1457:
1451:
1443:
1440:
1434:
1429:
1419:
1415:
1411:
1406:
1402:
1393:
1390:
1387:
1383:
1377:
1374:
1371:
1368:
1363:
1359:
1355:
1349:
1343:
1338:
1334:
1326:
1325:
1324:
1321:
1319:
1315:
1311:
1293:
1289:
1285:
1282:
1279:
1274:
1270:
1266:
1261:
1257:
1247:
1242:
1232:
1216:
1212:
1208:
1205:
1202:
1197:
1193:
1184:
1166:
1162:
1153:
1152:zero divisors
1149:
1131:
1127:
1099:
1096:
1092:
1089:
1083:
1078:
1074:
1066:
1062:
1055:
1051:
1044:
1041:
1038:
1032:
1024:
1021:
1012:
1006:
1000:
995:
991:
983:
982:
981:
965:
961:
951:
945:
928:
923:
919:
910:
894:
889:
885:
881:
859:
855:
844:
838:
833:
829:
825:
807:
803:
799:
796:
793:
788:
784:
760:
755:
751:
747:
744:
741:
736:
733:
730:
726:
720:
717:
714:
710:
703:
700:
697:
691:
686:
683:
680:
676:
670:
666:
662:
659:
653:
646:
643:
635:
634:
633:
631:
626:
621:
617:
613:
595:
591:
587:
584:
581:
576:
572:
551:
548:
543:
539:
529:
525:
504:
500:
496:
493:
488:
484:
480:
477:
474:
469:
466:
463:
459:
453:
450:
447:
443:
439:
434:
430:
424:
420:
416:
410:
404:
397:
396:
395:
387:
385:
375:
373:
369:
365:
361:
357:
356:
350:
349:
343:
341:
337:
332:
330:
329:multiple root
326:
322:
318:
314:
310:
294:
291:
288:
285:
277:
273:
254:
251:
248:
245:
242:
237:
233:
225:
224:
223:
209:
206:
203:
200:
197:
192:
188:
184:
177:
172:
170:
166:
165:number theory
162:
158:
154:
150:
146:
142:
138:
127:
124:
116:
113:November 2011
105:
102:
98:
95:
91:
88:
84:
81:
77:
74: –
73:
69:
68:Find sources:
62:
58:
52:
51:
46:This article
44:
40:
35:
34:
29:
22:
11943:Determinants
11902:Discriminant
11901:
11821:Multivariate
11615:
11612:Cohen, Henri
11606:
11586:
11579:
11548:
11539:
11526:
11506:
11499:
11488:. Retrieved
11479:
11468:. Retrieved
11459:
11439:
11432:
11408:
11398:
11390:the original
11370:
11352:
11328:
11321:
11306:
11302:
11296:
11285:. Retrieved
11281:
11272:
11029:
10961:
10865:
10840:
10823:
10768:
10765:
10491:is given by:
10394:
10309:
10299:
10295:
10287:
10283:
10279:
10275:
10271:
10267:
10263:
10259:
10255:
10247:
10243:
10236:
10232:
10229:
9945:
9940:
9938:)-entry is σ
9935:
9931:
9927:
9921:
9917:
9905:
9902:discriminant
9901:
9897:
9893:
9879:
9874:
9863:
9855:
9844:
9837:
9826:
9822:
9820:
9816:open problem
9803:
9795:
9781:
9762:
9751:
9636:However, if
9602:
9595:
9563:the sign of
9526:
9471:
9416:
9361:
9329:
9229:
9196:
9164:
9037:
9034:homogenizing
8960:
8948:
8924:
8879:
8876:
8784:
8665:
8658:
8652:
8648:
8644:
8640:
8636:
8632:
8629:
8524:of the form
8511:
8483:
8472:
8465:
8461:
8454:
8450:
8404:
8395:
8388:
8384:
8373:
8369:
8366:
8297:
8210:
8204:
8192:real numbers
8178:
8168:
8153:
8149:
8143:
8133:
8088:
8082:
8072:
8066:
8029:
8022:
8016:
8006:
8001:
7998:discriminant
7997:
7782:
7723:
7528:vector space
7521:
7467:
7464:
7449:
7443:
7429:
7420:
7414:, which has
7406:
7388:
7382:
7379:
7375:
7364:
7308:
7293:
7279:
7273:
7267:
7257:
7251:
7236:
7222:
7163:
7157:
7150:
7146:
7139:
7135:
7078:
7072:
7067:
7064:discriminant
7063:
7022:
6924:
6784:
6627:
6621:
6614:
6429:
6416:
6412:
6405:
6397:
6393:
6384:
6380:
6370:
6362:
6358:
6341:
6333:
6326:
6208:
6165:
6159:
6156:
6043:
5948:
5795:
5689:
5675:
5642:
5636:
5632:
5599:
5592:
5588:
5582:
5579:
5503:
5499:
5493:
5490:
5483:
5468:
5446:
5442:
5426:
5419:
5416:
5405:
5397:
5391:
5385:
5379:
5373:
5367:
5361:denotes the
5333:
5053:
5047:
5043:
5040:
4943:
4764:
4506:
4503:
4349:
4343:
4340:
4200:
4174:
4171:
4063:homomorphism
4028:
3784:
3778:denotes the
3604:
3476:
3370:
3335:
3328:
3324:
3313:
3271:
3268:
3257:
3246:
3235:
3221:
3001:
2398:
2391:
2387:
2383:
2377:
2373:
2369:
2365:
2345:cyclic group
2341:Galois group
2337:number field
2334:
2315:
2311:
2206:
2015:
2007:
2003:
1999:
1996:
1992:
1988:
1985:
1979:
1975:
1971:
1967:
1939:
1935:
1931:
1928:
1922:
1919:
1909:
1905:
1901:
1898:
1800:
1705:
1668:
1649:
1638:
1611:
1563:
1322:
1316:, where the
1245:
1238:
1182:
1118:
949:
943:
842:
836:
775:
624:
610:belong to a
531:(this means
527:
521:
393:
381:
360:discriminant
359:
352:
346:
344:
333:
269:
173:
149:coefficients
141:discriminant
140:
134:
119:
110:
100:
93:
86:
79:
67:
55:Please help
50:verification
47:
11928:Polynomials
11851:Homogeneous
11846:Square-free
11841:Irreducible
11706:Polynomials
11211:An integer
9914:determinant
9514:at infinity
9512:, possibly
9233:; that is
8787:determinant
8518:plane curve
8012:determinant
8002:determinant
7879:row vector
7396:0, or of a
6421:real roots.
6388:real roots.
6345:, one has:
5402:Homogeneity
5031:at infinity
3264:square-free
2211:polynomial
1646:Low degrees
1320:applies.)
828:determinant
313:double root
272:square root
137:mathematics
21:Determinant
11922:Categories
11811:Univariate
11571:0395.10029
11490:2023-03-21
11470:2023-03-21
11376:Birkhäuser
11309:: 391–410.
11287:2020-08-09
11265:References
9798:, and the
9707:paraboloid
9041:; that is
8408:such that
8367:where the
7937:, and the
7516:See also:
7410:defines a
7317:. Viewing
7311: = 0
6619:of degree
6378:roots and
6323:Real roots
5029:(possibly
3253:derivative
3227:Properties
1955:See also:
1700:See also:
630:derivative
390:Definition
370:, or of a
145:polynomial
83:newspapers
11897:Resultant
11836:Trinomial
11816:Bivariate
11642:0302-9743
11222:≠
11181:−
11160:−
11151:−
11136:−
11121:−
11112:−
11073:−
11058:−
10980:≠
10725:≡
10591:≡
10515:−
10241:extension
10176:σ
10170:⋯
10165:⋯
10138:σ
10130:⋮
10125:⋱
10119:⋮
10112:⋮
10106:⋱
10079:σ
10049:σ
10043:⋯
10016:σ
9988:σ
9963:Δ
9888:injective
9882:into the
9808:A theorem
9718:Δ
9678:Δ
9654:≠
9645:Δ
9615:Δ
9572:Δ
9545:≠
9536:Δ
9481:Δ
9455:ellipsoid
9426:Δ
9371:Δ
9339:Δ
9174:Δ
8939:hyperbola
8901:−
8310:∑
8252:⋯
7948:×
7909:…
7864:×
7794:×
7668:≤
7656:≤
7649:∑
7597:∑
7568:…
7487:−
7041:
6996:
6975:−
6946:
6898:
6877:−
6848:
6803:
6743:
6703:
6585:−
6547:∑
6517:⋯
6503:−
6374:pairs of
6184:−
6133:−
6113:−
6099:⋯
6077:−
6020:−
5989:⋯
5931:−
5906:⋯
5832:…
5765:⋯
5754:−
5738:−
5681:symmetric
5553:−
5363:resultant
5303:
5275:
5247:
5225:
5196:−
5168:
5133:
5111:
5082:
5011:φ
4968:
4952:φ
4926:≥
4918:φ
4907:
4901:−
4889:
4855:φ
4844:
4796:
4780:φ
4742:φ
4731:
4703:−
4689:φ
4671:
4655:φ
4608:φ
4576:
4560:φ
4515:φ
4474:
4458:φ
4447:φ
4436:
4397:≠
4378:φ
4358:φ
4310:φ
4304:⋯
4293:−
4274:−
4260:φ
4228:φ
4220:φ
4186:φ
4144:⋯
4133:−
4117:−
4046:→
4040::
4037:φ
3991:⋯
3880:≠
3834:⋯
3731:≠
3677:
3631:
3565:
3544:−
3531:α
3515:α
3503:
3437:
3415:α
3397:
3170:−
3122:−
3096:−
2920:−
2894:−
2834:−
2805:−
2721:−
2689:−
2631:−
2554:−
2522:−
2280:−
2264:−
2147:−
2128:−
2109:−
1859:−
1844:±
1838:−
1773:−
1688:sequence
1592:−
1533:−
1512:≠
1505:∏
1496:−
1461:−
1441:−
1412:−
1384:∏
1375:−
1344:
1283:…
1243:, it has
1206:…
1084:
1042:−
1022:−
1001:
909:resultant
797:…
745:⋯
734:−
718:−
701:−
684:−
620:resultant
585:…
549:≠
478:⋯
467:−
451:−
289:≠
243:−
11882:Division
11831:Binomial
11826:Monomial
11547:(1978).
11414:Springer
11368:(1994).
11314:page 406
10694:) where
10625:Case 2:
10560:Case 1:
9872:, ..., σ
9784:analytic
9777:ramified
9518:cylinder
8935:parabola
8498:cylinder
8174:subgroup
8160:quotient
7783:for the
6827:one has
5798:monomial
4198:acts on
2355:Degree 4
1951:Degree 3
1696:Degree 2
1100:′
907:and the
647:′
628:and its
340:multiple
11650:2041075
11259:coprime
10290:is the
9930:whose (
9916:of the
9912:of the
9908:is the
9900:). The
9866:-module
9843:, ...,
9832:be its
9812:Hermite
9790:of the
9767:of the
8953:in the
8949:A real
8927:ellipse
8172:by the
8010:is the
7066:or the
5057:, then
3339:, with
2351:three.
1690:A007878
1678:quintic
1674:quartic
1671:general
1249:roots,
830:of the
824:integer
366:, of a
274:in the
97:scholar
11719:degree
11648:
11640:
11630:
11594:
11569:
11559:
11514:
11447:
11420:
11382:
11340:
10436:, and
10396:where
9925:matrix
9910:square
9886:(i.e.
9850:be an
9836:. Let
9773:primes
9461:or an
8931:circle
8630:where
8207:Jacobi
8164:monoid
8158:, the
7853:, the
7593:
7587:
7402:divide
7249:. Let
6400:− 2)/4
5452:matrix
5371:, and
5334:where
5025:has a
4415:then
3746:Here,
1682:sextic
1618:simple
1568:times
1183:before
618:. The
524:degree
378:Origin
351:; the
336:degree
167:, and
139:, the
99:
92:
85:
78:
70:
10252:ideal
9599:into
9527:When
9520:or a
9457:or a
9402:or a
8929:or a
8516:is a
8496:or a
8444:is a
8139:up to
8092:into
7992:. In
7534:by a
7532:basis
7398:prime
7223:after
6615:be a
5454:(the
4552:then
4061:be a
3898:then
3862:and
3711:when
3316:up to
3238:field
2349:order
1664:empty
1241:field
822:with
612:field
394:Let
362:of a
278:. If
153:roots
143:of a
104:JSTOR
90:books
11708:and
11638:ISSN
11628:ISBN
11592:ISBN
11557:ISBN
11512:ISBN
11445:ISBN
11418:ISBN
11380:ISBN
11338:ISBN
10929:, or
10830:OEIS
10818:OEIS
9821:Let
9802:for
9775:are
9669:and
9522:cone
9435:<
9380:>
8961:Let
8659:Two
8494:cone
8023:The
7662:<
7435:zero
7380:Let
7302:and
7194:and
7149:(1,
7144:and
7142:, 1)
7106:and
7032:Disc
6987:Disc
6937:Disc
6925:and
6889:Disc
6839:Disc
6794:Disc
6734:Disc
6694:Disc
6655:and
6430:Let
6329:real
5949:and
5595:− 1)
5506:− 1)
5449:− 1)
5389:and
5377:and
5294:disc
5216:disc
5159:disc
5102:disc
5073:disc
4959:Disc
4835:Disc
4787:Disc
4722:Disc
4662:Disc
4567:Disc
4465:Disc
4427:Disc
4029:Let
3668:Disc
3622:Disc
3556:Disc
3494:Disc
3428:Disc
3388:Disc
2399:The
2326:1/18
2006:+ 18
2002:– 27
1943:are
1686:OEIS
1628:and
1391:<
1335:Disc
1148:ring
992:Disc
947:and
874:and
840:and
364:form
317:real
76:news
11717:By
11620:doi
11567:Zbl
10742:mod
10602:mod
10294:of
9975:det
9920:by
9904:of
9854:of
9810:of
9794:of
9472:If
9417:If
9362:If
8213:as
8166:of
8033:is
8027:of
8014:of
8004:of
8000:or
7462:).
7427:of
7079:If
7070:of
6419:+ 2
6415:− 4
6408:+ 1
6396:≤ (
6383:− 4
5486:− 2
5471:− 1
5429:− 2
5343:Res
5266:Res
5238:Res
5124:Res
5041:If
5033:).
4904:deg
4886:deg
4878:or
4507:If
4341:in
4172:in
4065:of
3782:of
3306:).
3258:In
3189:256
3151:144
3125:128
2660:144
2593:144
2557:128
2525:192
2499:256
2347:of
2010:= 0
2008:bcd
1995:– 4
1991:– 4
1929:If
1662:is
1075:Res
953:by
834:of
632:,
622:of
222:is
135:In
59:by
11924::
11646:MR
11644:,
11636:,
11626:,
11565:.
11412:.
11374:.
11364:;
11360:;
11332:.
11280:.
11261:.
11184:19
11175:17
11169:13
11163:11
10832:).
10820:).
10814:33
10812:,
10810:29
10808:,
10806:28
10804:,
10802:24
10800:,
10798:21
10796:,
10794:17
10792:,
10790:13
10788:,
10786:12
10784:,
10780:,
10776:,
10416:,
10302:.
9806:.
9779:.
8770:0.
8651:,
8647:,
8643:,
8639:,
8635:,
8512:A
8470:.
8464:/(
8202:.
8152:/(
8020:.
7522:A
7357:.
7076:.
6365:/4
6361:≤
6163:.
5635:−
5488:.
5441:(2
5395:.
5048:PQ
5046:=
4929:2.
4347:.
3734:0.
3244:.
3173:27
3080:16
2863:18
2837:27
2786:16
2761:18
2724:80
2634:27
2390:,
2386:,
2376:+
2374:dx
2372:+
2370:cx
2368:+
2332:.
2322:−3
2283:27
2176:18
2150:27
1978:+
1976:cx
1974:+
1972:bx
1970:+
1938:,
1934:,
1908:,
1904:,
1692:.
1642:.
1609:.
980::
950:A'
386:.
171:.
163:,
11698:e
11691:t
11684:v
11622::
11600:.
11573:.
11534:.
11520:.
11493:.
11473:.
11453:.
11426:.
11346:.
11316:.
11307:2
11290:.
11245:S
11225:1
11219:D
11199:}
11196:.
11193:.
11190:.
11187:,
11178:,
11172:,
11166:,
11157:,
11154:7
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11145:5
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11139:3
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11130:8
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11124:4
11118:,
11115:8
11109:{
11106:=
11103:S
11079:,
11076:4
11070:,
11067:8
11064:,
11061:8
11038:S
11008:0
11004:D
10983:1
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10971:D
10944:0
10940:D
10917:1
10914:=
10909:0
10905:D
10879:0
10875:D
10850:Q
10782:8
10778:5
10774:1
10749:)
10746:4
10739:(
10734:3
10731:,
10728:2
10722:m
10702:m
10682:m
10679:4
10676:=
10673:D
10665:(
10653:m
10633:D
10609:)
10606:4
10599:(
10594:1
10588:D
10568:D
10544:D
10524:c
10521:a
10518:4
10510:2
10506:b
10502:=
10499:D
10479:)
10476:y
10473:,
10470:x
10467:(
10464:Q
10444:c
10424:b
10404:a
10379:2
10375:y
10371:c
10368:+
10365:y
10362:x
10359:b
10356:+
10351:2
10347:x
10343:a
10340:=
10337:)
10334:y
10331:,
10328:x
10325:(
10322:Q
10300:K
10296:Z
10288:Q
10286:/
10284:K
10280:Q
10276:L
10272:Q
10268:L
10264:L
10262:/
10260:K
10256:L
10248:L
10246:/
10244:K
10237:K
10233:K
10215:.
10210:2
10205:)
10198:)
10193:n
10189:b
10185:(
10180:n
10160:)
10155:1
10151:b
10147:(
10142:n
10101:)
10096:1
10092:b
10088:(
10083:2
10071:)
10066:n
10062:b
10058:(
10053:1
10038:)
10033:2
10029:b
10025:(
10020:1
10010:)
10005:1
10001:b
9997:(
9992:1
9980:(
9972:=
9967:K
9948:j
9946:b
9944:(
9941:i
9936:j
9934:,
9932:i
9928:B
9922:n
9918:n
9906:K
9898:C
9894:K
9880:K
9875:n
9870:1
9864:Z
9858:K
9856:O
9847:n
9845:b
9841:1
9838:b
9829:K
9827:O
9823:K
9804:K
9796:K
9727:.
9722:4
9693:,
9690:0
9687:=
9682:3
9657:0
9649:4
9624:.
9619:3
9603:P
9601:−
9596:P
9581:,
9576:3
9551:,
9548:0
9540:4
9496:,
9493:0
9490:=
9485:4
9441:,
9438:0
9430:4
9386:,
9383:0
9375:4
9348:.
9343:3
9315:.
9312:)
9309:0
9306:,
9303:z
9300:,
9297:y
9294:,
9291:x
9288:(
9283:4
9279:Q
9275:=
9272:)
9269:z
9266:,
9263:y
9260:,
9257:x
9254:(
9249:3
9245:Q
9230:P
9215:,
9210:3
9206:Q
9183:.
9178:4
9150:.
9147:)
9144:t
9140:/
9136:z
9133:,
9130:t
9126:/
9122:y
9119:,
9116:t
9112:/
9108:x
9105:(
9102:P
9097:2
9093:t
9089:=
9086:)
9083:t
9080:,
9077:z
9074:,
9071:y
9068:,
9065:x
9062:(
9057:4
9053:Q
9038:P
9020:,
9015:4
9011:Q
8990:)
8987:z
8984:,
8981:y
8978:,
8975:x
8972:(
8969:P
8910:,
8907:c
8904:a
8896:2
8892:b
8862:.
8857:|
8851:f
8846:e
8841:d
8834:e
8829:c
8824:b
8817:d
8812:b
8807:a
8801:|
8767:=
8762:2
8758:z
8754:f
8751:+
8748:z
8745:y
8742:e
8739:2
8736:+
8733:z
8730:x
8727:d
8724:2
8721:+
8716:2
8712:y
8708:c
8705:+
8702:y
8699:x
8696:b
8693:2
8690:+
8685:2
8681:x
8677:a
8653:f
8649:e
8645:d
8641:c
8637:b
8633:a
8615:,
8612:0
8609:=
8606:f
8603:+
8600:y
8597:e
8594:2
8591:+
8588:x
8585:d
8582:2
8579:+
8574:2
8570:y
8566:c
8563:+
8560:y
8557:x
8554:b
8551:2
8548:+
8543:2
8539:x
8535:a
8468:)
8466:K
8462:K
8455:i
8451:a
8432:S
8428:A
8422:T
8417:S
8405:S
8396:A
8389:i
8385:a
8380:n
8374:i
8370:L
8350:2
8345:i
8341:L
8335:i
8331:a
8325:n
8320:1
8317:=
8314:i
8283:.
8278:2
8273:n
8269:x
8263:n
8259:a
8255:+
8249:+
8244:2
8239:1
8235:x
8229:1
8225:a
8179:K
8169:K
8156:)
8154:K
8150:K
8144:K
8134:S
8119:,
8116:S
8112:A
8106:T
8101:S
8089:A
8083:S
8067:Q
8046:n
8042:2
8030:Q
8017:A
8007:Q
7977:T
7972:X
7951:1
7945:n
7925:)
7920:n
7916:x
7912:,
7906:,
7901:1
7897:x
7893:(
7890:=
7887:X
7867:n
7861:1
7841:)
7836:j
7833:i
7829:a
7825:(
7822:=
7819:A
7797:n
7791:n
7768:,
7762:T
7757:X
7753:A
7750:X
7747:=
7744:)
7741:X
7738:(
7735:Q
7709:,
7704:j
7700:x
7694:i
7690:x
7684:j
7681:i
7677:a
7671:n
7665:j
7659:i
7653:1
7645:+
7640:2
7635:i
7631:x
7625:i
7622:i
7618:a
7612:n
7607:1
7604:=
7601:i
7590:=
7584:)
7579:n
7575:x
7571:,
7565:,
7560:1
7556:x
7552:(
7549:Q
7490:2
7484:d
7480:d
7468:d
7450:n
7444:A
7430:A
7421:n
7407:A
7389:n
7383:A
7351:X
7347:Y
7343:Y
7339:Y
7335:X
7331:X
7327:X
7323:Y
7319:f
7309:f
7304:Y
7300:X
7296:f
7280:V
7274:W
7268:W
7258:V
7252:V
7207:n
7203:a
7180:0
7176:a
7164:n
7158:n
7153:)
7151:y
7147:A
7140:x
7138:(
7136:A
7119:n
7115:a
7092:0
7088:a
7073:A
7050:)
7047:A
7044:(
7036:h
7008:.
7005:)
7002:A
6999:(
6991:h
6981:)
6978:1
6972:n
6969:(
6966:n
6962:x
6958:=
6955:)
6952:A
6949:(
6941:y
6910:,
6907:)
6904:A
6901:(
6893:h
6883:)
6880:1
6874:n
6871:(
6868:n
6864:y
6860:=
6857:)
6854:A
6851:(
6843:x
6815:,
6812:)
6809:A
6806:(
6798:h
6770:.
6767:)
6764:)
6761:y
6758:,
6755:1
6752:(
6749:A
6746:(
6738:y
6730:=
6727:)
6724:)
6721:1
6718:,
6715:x
6712:(
6709:A
6706:(
6698:x
6668:n
6664:a
6641:0
6637:a
6622:n
6598:i
6594:y
6588:i
6582:n
6578:x
6572:i
6568:a
6562:n
6557:0
6554:=
6551:i
6543:=
6538:n
6534:y
6528:n
6524:a
6520:+
6514:+
6511:y
6506:1
6500:n
6496:x
6490:1
6486:a
6482:+
6477:n
6473:x
6467:0
6463:a
6459:=
6456:)
6453:y
6450:,
6447:x
6444:(
6441:A
6417:k
6413:n
6406:k
6404:2
6398:n
6394:k
6385:k
6381:n
6371:k
6369:2
6363:n
6359:k
6342:n
6307:d
6302:4
6298:c
6294:b
6274:e
6271:+
6268:x
6265:d
6262:+
6257:2
6253:x
6249:c
6246:+
6241:3
6237:x
6233:b
6230:+
6225:4
6221:x
6217:a
6193:c
6190:a
6187:4
6179:2
6175:b
6160:n
6142:,
6139:)
6136:1
6130:n
6127:(
6124:n
6121:=
6116:1
6110:n
6106:i
6102:+
6096:+
6091:1
6087:i
6083:)
6080:1
6074:n
6071:(
6068:+
6063:0
6059:i
6055:n
6029:,
6026:)
6023:1
6017:n
6014:(
6011:n
6008:=
6003:n
5999:i
5995:n
5992:+
5986:+
5981:2
5977:i
5973:2
5970:+
5965:1
5961:i
5934:2
5928:n
5925:2
5922:=
5917:n
5913:i
5909:+
5903:+
5898:1
5894:i
5890:+
5885:0
5881:i
5853:n
5849:i
5843:n
5839:a
5835:,
5829:,
5822:0
5818:i
5812:0
5808:a
5781:.
5776:0
5772:a
5768:+
5762:+
5757:1
5751:n
5747:x
5741:1
5735:n
5731:a
5727:+
5722:n
5718:x
5712:n
5708:a
5704:=
5701:P
5676:i
5659:i
5655:x
5643:i
5637:i
5633:n
5616:i
5612:x
5600:i
5593:n
5591:(
5589:n
5583:n
5563:2
5559:)
5556:1
5550:n
5547:(
5544:n
5538:=
5532:)
5527:2
5524:n
5519:(
5504:n
5502:(
5500:n
5494:n
5484:n
5482:2
5477:n
5475:a
5469:n
5467:2
5462:n
5460:a
5447:n
5443:n
5437:λ
5433:λ
5427:n
5425:2
5420:n
5392:Q
5386:P
5380:q
5374:p
5368:x
5347:x
5315:,
5312:)
5309:Q
5306:(
5298:x
5290:)
5287:P
5284:,
5281:Q
5278:(
5270:x
5262:)
5259:Q
5256:,
5253:P
5250:(
5242:x
5234:)
5231:P
5228:(
5220:x
5210:q
5207:p
5203:)
5199:1
5193:(
5190:=
5177:)
5174:Q
5171:(
5163:x
5153:2
5149:)
5145:Q
5142:,
5139:P
5136:(
5128:x
5120:)
5117:P
5114:(
5106:x
5098:=
5091:)
5088:R
5085:(
5077:x
5054:x
5044:R
5007:A
4986:0
4983:=
4980:)
4977:)
4974:A
4971:(
4963:x
4955:(
4923:)
4914:A
4910:(
4898:)
4895:A
4892:(
4866:0
4863:=
4860:)
4851:A
4847:(
4839:x
4814:0
4811:=
4808:)
4805:)
4802:A
4799:(
4791:x
4783:(
4750:.
4747:)
4738:A
4734:(
4726:x
4716:2
4712:)
4706:1
4700:n
4696:a
4692:(
4686:=
4683:)
4680:)
4677:A
4674:(
4666:x
4658:(
4633:,
4630:0
4627:=
4624:)
4619:n
4615:a
4611:(
4588:)
4585:)
4582:A
4579:(
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4563:(
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4537:0
4534:=
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4526:n
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4518:(
4489:.
4486:)
4483:)
4480:A
4477:(
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4455:=
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4439:(
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4403:,
4400:0
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4225:=
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4083:=
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3965:=
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3955:/
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3940:n
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3932:=
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3808:=
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3802:x
3799:(
3796:P
3785:P
3760:r
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3728:)
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3695:)
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3664:=
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3634:(
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3607::
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3527:=
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3506:(
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3479::
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3440:(
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3424:=
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2297:.
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2024:a
2012:.
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1997:b
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1832:=
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1786:.
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1739:c
1736:+
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1435:=
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1399:(
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1356:=
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1350:A
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1294:n
1290:r
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1087:(
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1056:2
1052:/
1048:)
1045:1
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1036:(
1033:n
1029:)
1025:1
1019:(
1013:=
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1007:A
1004:(
996:x
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962:a
944:A
929:.
924:n
920:a
895:,
890:n
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882:n
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660:=
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417:=
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120:(
115:)
111:(
101:·
94:·
87:·
80:·
53:.
30:.
23:.
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