593:
2753:
1888:
325:
6373:
have completely characterized universal integral quadratic forms: if all coefficients are integers, then it represents all positive integers if and only if it represents all integers up through 290; if it has an integral matrix, it represents all positive integers if and only if it represents all
5981:
This debate was due to the confusion of quadratic forms (represented by polynomials) and symmetric bilinear forms (represented by matrices), and "twos out" is now the accepted convention; "twos in" is instead the theory of integral symmetric bilinear forms (integral symmetric matrices).
2400:(up to non-zero squares) can also be defined, and for a real quadratic form is a cruder invariant than signature, taking values of only "positive, zero, or negative". Zero corresponds to degenerate, while for a non-degenerate form it is the parity of the number of negative coefficients,
2614:
1359:
1714:
2210:
3314:
3870:
966:
1490:
588:{\displaystyle {\begin{aligned}q(x)&=ax^{2}&&{\textrm {(unary)}}\\q(x,y)&=ax^{2}+bxy+cy^{2}&&{\textrm {(binary)}}\\q(x,y,z)&=ax^{2}+bxy+cy^{2}+dyz+ez^{2}+fxz&&{\textrm {(ternary)}}\end{aligned}}}
281:
1192:
5832:. When the solution set is a paraboloid, whether it is elliptic or hyperbolic is determined by whether all other non-zero eigenvalues are of the same sign: if they are, then it is elliptic; otherwise, it is hyperbolic.
4227:
5644:
1663:
6149:
2081:
6050:
2304:, where these components count the number of 0s, number of 1s, and the number of −1s, respectively. Sylvester's law of inertia shows that this is a well-defined quantity attached to the quadratic form.
4897:
4488:
1994:
2748:{\displaystyle B={\begin{pmatrix}\lambda _{1}&0&\cdots &0\\0&\lambda _{2}&\cdots &0\\\vdots &\vdots &\ddots &0\\0&0&\cdots &\lambda _{n}\end{pmatrix}}}
5491:
3156:
330:
640:, and in the majority of applications of quadratic forms, the coefficients are real or complex numbers. In the algebraic theory of quadratic forms, the coefficients are elements of a certain
3960:
and these two processes are the inverses of each other. As a consequence, over a field of characteristic not equal to 2, the theories of symmetric bilinear forms and of quadratic forms in
2601:
1034:
The study of quadratic forms, in particular the question of whether a given integer can be the value of a quadratic form over the integers, dates back many centuries. One such case is
3719:
3437:
2505:
1883:{\displaystyle A={\begin{bmatrix}a&{\frac {b+d}{2}}&{\frac {c+g}{2}}\\{\frac {b+d}{2}}&e&{\frac {f+h}{2}}\\{\frac {c+g}{2}}&{\frac {f+h}{2}}&k\end{bmatrix}}.}
3577:
3702:
783:
1406:
117:
4349:
3958:
6836:
4072:
5527:
1497:
6082:
232:
1035:
1667:
So, two different matrices define the same quadratic form if and only if they have the same elements on the diagonal and the same values for the sums
1107:
6002:
5766:(we get the equation of an ellipsoid but with imaginary radii); if some eigenvalues are positive and some are negative, then it is a hyperboloid.
6620:
The bilinear form to which a quadratic form is associated is not restricted to being symmetric, which is of significance when 2 is not a unit in
4386:
1930:
6704:
6677:
6656:
6411:
6590:
The theory of quadratic forms over a field of characteristic 2 has important differences and many definitions and theorems must be modified.
5415:
4833:
616:
The theory of quadratic forms and methods used in their study depend in a large measure on the nature of the coefficients, which may be
1354:{\displaystyle q_{A}(x_{1},\ldots ,x_{n})=\sum _{i=1}^{n}\sum _{j=1}^{n}a_{ij}{x_{i}}{x_{j}}=\mathbf {x} ^{\mathsf {T}}A\mathbf {x} ,}
7005:
6920:
6882:
6848:
6802:
6764:
6737:
290:, which has only one variable and includes terms of degree two or less. A quadratic form is one case of the more general concept of
6204:
5826:
degenerates and does not come into play, and the geometric meaning will be determined by other eigenvalues and other components of
6696:
7000:
6900:
226:
1073:
6790:
5977:
a polynomial with integer coefficients (so the associated symmetric matrix may have half-integer coefficients off the diagonal)
2556:
3389:
2457:
202:
3532:
2205:{\displaystyle \lambda _{1}{\tilde {x}}_{1}^{2}+\lambda _{2}{\tilde {x}}_{2}^{2}+\cdots +\lambda _{n}{\tilde {x}}_{n}^{2},}
6974:
6956:
6794:
6521:), but at 2 they are different concepts; this distinction is particularly important for quadratic forms over the integers.
3663:
2268:
2043:
725:
2841:. Although their definition involved a choice of basis and consideration of the corresponding real symmetric matrix
6401:
64:
6969:
6951:
6058:
4294:
2946:
1127:
222:
31:
6530:
6912:
5762:. If all the eigenvalues are positive, then it is an ellipsoid; if all the eigenvalues are negative, then it is an
2067:
3910:
2071:
2027:
6168:
point of view, which was generally adopted by the experts in the arithmetic of quadratic forms during the 1950s;
6990:
5316:
5211:
4235:
formulas show that the property of being a quadratic form does not depend on the choice of a specific basis in
2908:
2337:
2051:
2023:
163:
2340:(a mix of 1 and −1); equivalently, a nondegenerate quadratic form is one whose associated symmetric form is a
5670:. Classification of all quadratic forms up to equivalence can thus be reduced to the case of diagonal forms.
2275:
of the quadratic form, in the sense that any other diagonalization will contain the same number of each. The
6995:
6518:
6514:
5322:
3886:
3309:{\displaystyle q(x_{1},\ldots ,x_{n})=\sum _{i=1}^{n}\sum _{j=1}^{n}a_{ij}{x_{i}}{x_{j}},\quad a_{ij}\in K.}
2272:
2047:
707:
194:
159:
6964:
6541:
4768:. The notion of a quadratic space is a coordinate-free version of the notion of quadratic form. Sometimes,
6510:
4012:
3582:
3138:
2320:
1141:
291:
58:
50:
6599:
This alternating form associated with a quadratic form in characteristic 2 is of interest related to the
5679:
2057:
1133:
1115:
674:
314:
198:
186:
6866:
6640:
6560:
5948:
This is the current use of the term; in the past it was sometimes used differently, as detailed below.
6690:
5409:
5326:. This terminology also applies to vectors and subspaces of a quadratic space. If the restriction of
4956:
4264:
3516:
1122:
6946:
6383:
6370:
6176:
6161:
better understanding of the 2-adic theory of quadratic forms, the 'local' source of the difficulty;
5494:
2779:
can be made to have only 0, 1, and −1 on the diagonal, and the number of the entries of each type (
2019:
741:
158:, and the quadratic form equals zero only when all variables are simultaneously zero, then it is a
130:
6449:
dictates the use of manifestly even coefficients for the products of distinct variables, that is,
6665:
6365:
There are also forms whose image consists of all but one of the positive integers. For example,
6199:
An integral quadratic form whose image consists of all the positive integers is sometimes called
4592:
3865:{\displaystyle b_{q}(x,y)={\tfrac {1}{2}}(q(x+y)-q(x)-q(y))=x^{\mathsf {T}}Ay=y^{\mathsf {T}}Ax.}
2341:
2063:
2031:
1061:
700:
686:
287:
54:
1092:
6916:
6878:
6844:
6798:
6760:
6733:
6700:
6673:
6652:
6426:
5518:
2945:. This stands in contrast with the case of isotropic forms, when the corresponding group, the
2533:
2253:
637:
190:
961:{\displaystyle q(x,y,z)=d((x,y,z),(0,0,0))^{2}=\left\|(x,y,z)\right\|^{2}=x^{2}+y^{2}+z^{2}.}
6926:
6888:
6854:
6816:
6770:
6743:
6721:
6686:
6552:
5708:
4940:
3604:
2994:
1997:
1709:
1485:{\displaystyle A={\begin{bmatrix}a&b&c\\d&e&f\\g&h&k\end{bmatrix}}.}
753:
737:
645:
182:
6812:
6930:
6892:
6874:
6858:
6840:
6820:
6808:
6774:
6747:
6648:
6187:
6165:
5849:
4232:
2213:
749:
682:
641:
625:
17:
2911:. The theorems of Jacobi and Sylvester show that any positive definite quadratic form in
6905:
6729:
4066:
2928:
2847:, Sylvester's law of inertia means that they are invariants of the quadratic form
1111:
757:
745:
655:
633:
621:
174:
141:
2318:
have the same sign is especially important: in this case the quadratic form is called
6984:
6782:
6600:
5853:
3975:
3325:
2933:
2923:
squares by a suitable invertible linear transformation: geometrically, there is only
2075:
1145:
690:
678:
210:
170:
169:
Quadratic forms occupy a central place in various branches of mathematics, including
6416:
6396:
5712:
4595:, it is still possible to use a quadratic form to define a symmetric bilinear form
3987:
2386:
993:
276:{\displaystyle -\mathbf {x} ^{\mathsf {T}}{\boldsymbol {\Sigma }}^{-1}\mathbf {x} }
218:
178:
644:. In the arithmetic theory of quadratic forms, the coefficients belong to a fixed
5956:
Historically there was some confusion and controversy over whether the notion of
3630:
is uniquely determined by the corresponding quadratic form. Under an equivalence
6828:
5759:
3465:
2812:. This is one of the formulations of Sylvester's law of inertia and the numbers
2423:
2246:
1068:
617:
137:
38:
2348:. A real vector space with an indefinite nondegenerate quadratic form of index
6759:. Cambridge Tracts in Mathematics. Vol. 106. Cambridge University Press.
6564:
6421:
6391:
5805:
5745:
5236:
4677:(and is thus alternating). Alternatively, there always exists a bilinear form
214:
46:
6513:, that is, if 2 is invertible in the ring, quadratic forms are equivalent to
5969:
the quadratic form associated to a symmetric matrix with integer coefficients
1106:, and found a method for its solution. In Europe this problem was studied by
6226:
6183:
5755:
4222:{\displaystyle q(v_{1},\ldots ,v_{n})=Q()\quad {\text{for}}\quad \in K^{n}.}
2998:
2379:
2056:
A fundamental problem is the classification of real quadratic forms under a
1077:, which includes, among many other things, a study of equations of the form
6672:, Carus Mathematical Monographs, The Mathematical Association of America,
5639:{\displaystyle q(x)=a_{1}x_{1}^{2}+a_{2}x_{2}^{2}+\cdots +a_{n}x_{n}^{2}.}
2256:
that is not necessarily orthogonal, one can suppose that all coefficients
1658:{\displaystyle q_{A}(x,y,z)=ax^{2}+ey^{2}+kz^{2}+(b+d)xy+(c+g)xz+(f+h)yz.}
6172:
5857:
1140:. Since then, the concept has been generalized, and the connections with
761:
308:
variables. In the cases of one, two, and three variables they are called
206:
3618:
with the same quadratic form, so it may be assumed from the outset that
6575:
If a non-strict inequality (with ≥ or ≤) holds then the quadratic form
6406:
1137:
629:
2389:, concretely the class of the determinant of a representing matrix in
744:. In this way one may visualize 3-dimensional real quadratic forms as
6907:
Quadratic Forms with
Applications to Algebraic Geometry and Topology
6873:. Die Grundlehren der mathematischen Wissenschaften. Vol. 117.
4250:
A finite-dimensional vector space with a quadratic form is called a
6144:{\displaystyle {\begin{pmatrix}a&b/2\\b/2&c\end{pmatrix}}.}
6911:. London Mathematical Society lecture note series. Vol. 217.
6446:
6053:
2242:
5265:
consists of the elements that are orthogonal to every element of
4267:
of degree 2, which means that it has the property that, for all
1060:
are integers. This problem is related to the problem of finding
1038:, which determines when an integer may be expressed in the form
1026:
below for the definition of a quadratic form on a vector space.
2328:(all −1). If none of the terms are 0, then the form is called
1148:, and other areas of mathematics have been further elucidated.
6045:{\displaystyle {\begin{pmatrix}a&b\\b&c\end{pmatrix}}}
4028:
that has the following property: for some basis, the function
2927:
positive definite real quadratic form of every dimension. Its
5863:
An integral quadratic form has integer coefficients, such as
4762:
as defined here is the associated symmetric bilinear form of
1132:
a major portion of which was devoted to a complete theory of
6157:
has been adopted as the standard convention. Those include:
6501:
in ternary forms. Both conventions occur in the literature.
5517:
variables over a field of characteristic not equal to 2 is
2336:; this includes positive definite, negative definite, and
4483:{\displaystyle B(v,w)={\tfrac {1}{2}}(Q(v+w)-Q(v)-Q(w)).}
2517:
in the chosen basis. Under a change of basis, the column
2407:
These results are reformulated in a different way below.
1989:{\displaystyle B=\left({\frac {a_{ij}+a_{ji}}{2}}\right)}
302:
Quadratic forms are homogeneous quadratic polynomials in
6728:. London Mathematical Society Monographs. Vol. 13.
2066:
proved that, for every real quadratic form, there is an
1892:
This generalizes to any number of variables as follows.
1385:
Consider the case of quadratic forms in three variables
6171:
the actual needs for integral quadratic form theory in
5106:
may be characterized in the following equivalent ways:
2767:
are uniquely determined – this is Jacobi's theorem. If
2018:
So, over the real numbers (and, more generally, over a
970:
A closely related notion with geometric overtones is a
6561:§ 45 Reduction of a quadratic form to a sum of squares
6267:
that can each generate all positive integers, namely,
6091:
6065:
In "twos out", binary quadratic forms are of the form
6011:
5808:(either elliptic or hyperbolic); if the corresponding
5357:
The orthogonal group of a non-singular quadratic form
4684:(not in general either unique or symmetric) such that
4412:
3752:
2629:
1729:
1421:
6085:
6005:
5985:
In "twos in", binary quadratic forms are of the form
5840:
Quadratic forms over the ring of integers are called
5530:
5486:{\displaystyle \forall x,y\in A\quad Q(xy)=Q(x)Q(y),}
5418:
4836:
4389:
4297:
4075:
3913:
3722:
3666:
3535:
3392:
3159:
2617:
2559:
2460:
2084:
1933:
1717:
1500:
1409:
1195:
786:
328:
235:
67:
4892:{\displaystyle Q(v)=Q'(Tv){\text{ for all }}v\in V.}
4811:
if there exists an invertible linear transformation
3510:
if there exists a nonsingular linear transformation
2547:
is transformed into another symmetric square matrix
6904:
6143:
6044:
5638:
5485:
4891:
4482:
4343:
4221:
3952:
3864:
3696:
3571:
3431:
3318:This formula may be rewritten using matrices: let
3308:
2960:, is non-compact. Further, the isometry groups of
2747:
2595:
2499:
2204:
2004:, and is the unique symmetric matrix that defines
1988:
1882:
1657:
1484:
1353:
960:
587:
275:
111:
5281:if the kernel of its associated bilinear form is
2271:states that the numbers of each 0, 1, and −1 are
144:numbers, and one speaks of a quadratic form over
6837:Ergebnisse der Mathematik und ihrer Grenzgebiete
5711:3-by-3 matrix. Then the geometric nature of the
4722:consisting of a finite-dimensional vector space
1064:, which appeared in the second millennium BCE.
748:. An example is given by the three-dimensional
5781:, then the shape depends on the corresponding
3976:Bilinear form § Associated quadratic form
286:Quadratic forms are not to be confused with a
5887:(over a field with characteristic 0, such as
2755:by a suitable choice of an orthogonal matrix
129:. The coefficients usually belong to a fixed
8:
5363:is the group of the linear automorphisms of
3591:be different from 2. The coefficient matrix
2773:is allowed to be any invertible matrix then
693:. The theory of integral quadratic forms in
6787:Introduction to Quadratic Forms over Fields
3624:is symmetric. Moreover, a symmetric matrix
2901:assumes both positive and negative values,
5754:are non-zero, then the solution set is an
3901:. Conversely, any symmetric bilinear form
2611:can be transformed into a diagonal matrix
2553:of the same size according to the formula
2252:If the change of variables is given by an
6117:
6102:
6086:
6084:
6006:
6004:
5735:depends on the eigenvalues of the matrix
5648:Such a diagonal form is often denoted by
5627:
5622:
5612:
5593:
5588:
5578:
5565:
5560:
5550:
5529:
5417:
4913:correspond to the equivalence classes of
4872:
4835:
4411:
4388:
4320:
4296:
4210:
4194:
4175:
4162:
4149:
4130:
4105:
4086:
4074:
3912:
3846:
3845:
3825:
3824:
3751:
3727:
3721:
3678:
3677:
3665:
3534:
3413:
3412:
3391:
3288:
3273:
3268:
3261:
3256:
3247:
3237:
3226:
3216:
3205:
3189:
3170:
3158:
2731:
2670:
2636:
2624:
2616:
2596:{\displaystyle A\to B=S^{\mathsf {T}}AS.}
2577:
2576:
2558:
2481:
2480:
2459:
2212:where the associated symmetric matrix is
2193:
2188:
2177:
2176:
2169:
2150:
2145:
2134:
2133:
2126:
2113:
2108:
2097:
2096:
2089:
2083:
1967:
1951:
1944:
1932:
1846:
1826:
1804:
1779:
1757:
1737:
1724:
1716:
1574:
1558:
1542:
1505:
1499:
1416:
1408:
1343:
1333:
1332:
1327:
1316:
1311:
1304:
1299:
1290:
1280:
1269:
1259:
1248:
1232:
1213:
1200:
1194:
949:
936:
923:
910:
869:
785:
575:
574:
553:
525:
497:
449:
448:
439:
411:
369:
368:
359:
329:
327:
268:
259:
254:
246:
245:
240:
234:
103:
75:
66:
6369:has 15 as the exception. Recently, the
5844:, whereas the corresponding modules are
699:variables has important applications to
6438:
5912:if and only if it is integer-valued on
5769:If there exist one or more eigenvalues
5375:: that is, the group of isometries of
4055:is a quadratic form. In particular, if
3432:{\displaystyle q(x)=x^{\mathsf {T}}Ax.}
2917:variables can be brought to the sum of
2511:is the column vector of coordinates of
2500:{\displaystyle q(v)=x^{\mathsf {T}}Ax,}
2378:particularly in the physical theory of
1036:Fermat's theorem on sums of two squares
27:Polynomial with all terms of degree two
6079:, represented by the symmetric matrix
5999:, represented by the symmetric matrix
3847:
3826:
3679:
3572:{\displaystyle \psi (x)=\varphi (Cx).}
3414:
3380:whose entries are the coefficients of
2578:
2482:
1334:
322:and have the following explicit form:
247:
4649:can no longer be recovered from this
2000:, defines the same quadratic form as
121:is a quadratic form in the variables
7:
4247:depends on the choice of the basis.
3966:variables are essentially the same.
3697:{\displaystyle B=C^{\mathsf {T}}AC.}
2432:be the matrix of the quadratic form
2279:of the quadratic form is the triple
1697:. In particular, the quadratic form
4907:-dimensional quadratic spaces over
2074:that puts the quadratic form in a "
1091:. He considered what is now called
225:(where the exponent of a zero-mean
6412:Ramanujan's ternary quadratic form
5852:). They play an important role in
5419:
2806:for −1) depends only on
2541:, and the symmetric square matrix
2438:in a given basis. This means that
2416:be a quadratic form defined on an
740:. This is a basic construction in
681:, in particular, in the theory of
25:
6153:Several points of view mean that
5145:is the associated quadratic form.
4774:is also called a quadratic form.
2873:(similarly, negative definite if
1067:In 628, the Indian mathematician
764:between a point with coordinates
677:have been extensively studied in
112:{\displaystyle 4x^{2}+2xy-3y^{2}}
30:For the usage in statistics, see
6397:Discriminant of a quadratic form
5877:; equivalently, given a lattice
4344:{\displaystyle Q(av)=a^{2}Q(v).}
2523:is multiplied on the left by an
2387:discriminant of a quadratic form
2026:different from two), there is a
1344:
1328:
1022:
269:
255:
241:
227:multivariate normal distribution
6871:Introduction to quadratic forms
6791:Graduate Studies in Mathematics
6645:Introduction to Quadratic Forms
5437:
4167:
4161:
3283:
3147:variables with coefficients in
710:, a non-zero quadratic form in
6668:; Fung, Francis Y. C. (1997),
6557:Introduction to Higher Algebra
6205:Lagrange's four-square theorem
5540:
5534:
5477:
5471:
5465:
5459:
5450:
5441:
4869:
4860:
4846:
4840:
4783:-dimensional quadratic spaces
4474:
4471:
4465:
4456:
4450:
4441:
4429:
4423:
4405:
4393:
4335:
4329:
4310:
4301:
4241:, although the quadratic form
4200:
4168:
4158:
4155:
4123:
4120:
4111:
4079:
3944:
3932:
3923:
3917:
3814:
3811:
3805:
3796:
3790:
3781:
3769:
3763:
3745:
3733:
3563:
3554:
3545:
3539:
3402:
3396:
3195:
3163:
2761:, and the diagonal entries of
2563:
2470:
2464:
2182:
2139:
2102:
2072:orthogonal change of variables
1643:
1631:
1619:
1607:
1595:
1583:
1529:
1511:
1238:
1206:
906:
902:
884:
880:
866:
862:
844:
838:
820:
817:
808:
790:
480:
462:
394:
382:
342:
336:
1:
6795:American Mathematical Society
6757:Arithmetic of quadratic forms
5804:, then the solution set is a
5285:. If there exists a non-zero
4034:that maps the coordinates of
2216:. Moreover, the coefficients
6670:The Sensual (Quadratic) Form
3953:{\displaystyle q(x)=b(x,x),}
2030:between quadratic forms and
1172:determines a quadratic form
6970:Encyclopedia of Mathematics
6952:Encyclopedia of Mathematics
6755:Kitaoka, Yoshiyuki (1993).
6692:Linear Algebra and Geometry
6607:Linear Algebra and Geometry
6059:Disquisitiones Arithmeticae
4585:When the characteristic of
4359:is not 2, the bilinear map
4353:When the characteristic of
2947:indefinite orthogonal group
2884:) for every nonzero vector
1152:Associated symmetric matrix
1128:Disquisitiones Arithmeticae
32:Quadratic form (statistics)
7022:
6913:Cambridge University Press
6555:(with E.P.R. DuVal)(1907)
6445:A tradition going back to
5344:is identically zero, then
5209:
4983:-bilinear form. A mapping
4919:-ary quadratic forms over
3973:
3033:from a finite-dimensional
2269:Sylvester's law of inertia
2068:orthogonal diagonalization
2058:linear change of variables
2044:Sylvester's law of inertia
2041:
648:, frequently the integers
29:
18:Isometry (quadratic forms)
6831:; Husemoller, D. (1973).
6689:; Remizov, A. O. (2012).
6605:Irving Kaplansky (1974),
6195:Universal quadratic forms
5682:in three dimensions, let
5016:associated quadratic form
4591:is 2, so that 2 is not a
3907:defines a quadratic form
3648:and the symmetric matrix
2372:−1s) is often denoted as
2028:one-to-one correspondence
7006:Squares in number theory
6833:Symmetric Bilinear Forms
6726:Rational Quadratic Forms
6515:symmetric bilinear forms
6374:integers up through 15.
5842:integral quadratic forms
5836:Integral quadratic forms
5212:Isotropic quadratic form
5189:, and the polar form of
4901:The isometry classes of
3708:associated bilinear form
3660:are related as follows:
2909:isotropic quadratic form
2862:is positive definite if
2338:isotropic quadratic form
2241:are determined uniquely
2052:Isotropic quadratic form
1911:, defined by the matrix
1494:The above formula gives
164:isotropic quadratic form
57:" is another name for a
7001:Real algebraic geometry
6965:"Binary quadratic form"
6581:is called semidefinite.
6519:polarization identities
6402:Hasse–Minkowski theorem
6247:and found 54 multisets
6052:This is the convention
5958:integral quadratic form
5792:. If the corresponding
4656:in the same way, since
4496:is symmetric. That is,
3887:symmetric bilinear form
3636:, the symmetric matrix
3603:may be replaced by the
2048:Definite quadratic form
1900:Given a quadratic form
1708:is defined by a unique
1142:quadratic number fields
708:homogeneous coordinates
292:homogeneous polynomials
229:has the quadratic form
195:second fundamental form
160:definite quadratic form
6963:A.V.Malyshev (2001) ,
6945:A.V.Malyshev (2001) ,
6145:
6046:
5640:
5487:
5402:has a product so that
4893:
4484:
4345:
4223:
3954:
3866:
3698:
3573:
3433:
3310:
3242:
3221:
3139:homogeneous polynomial
2749:
2597:
2501:
2206:
1990:
1884:
1659:
1486:
1355:
1285:
1264:
1134:binary quadratic forms
962:
675:Binary quadratic forms
589:
277:
113:
59:homogeneous polynomial
6542:Brahmagupta biography
6531:Babylonian Pythagoras
6146:
6047:
5820:, then the dimension
5680:Cartesian coordinates
5641:
5505:Every quadratic form
5488:
5390:If a quadratic space
5308:, the quadratic form
4894:
4485:
4346:
4224:
3955:
3867:
3699:
3574:
3488:-ary quadratic forms
3434:
3311:
3222:
3201:
2993:, but the associated
2750:
2605:Any symmetric matrix
2598:
2502:
2207:
2034:that determine them.
1991:
1885:
1660:
1487:
1356:
1265:
1244:
1074:Brāhmasphuṭasiddhānta
963:
716:variables defines an
590:
278:
199:differential topology
187:differential geometry
162:; otherwise it is an
114:
6647:, Berlin, New York:
6462:in binary forms and
6083:
6003:
5899:), a quadratic form
5528:
5501:Equivalence of forms
5416:
5410:algebra over a field
4834:
4734:and a quadratic map
4541:, and it determines
4387:
4295:
4265:homogeneous function
4073:
3911:
3720:
3710:of a quadratic form
3664:
3533:
3390:
3157:
3121:More concretely, an
2615:
2557:
2458:
2082:
2038:Real quadratic forms
1931:
1715:
1498:
1407:
1193:
1016:a quadratic form on
784:
326:
233:
65:
6666:Conway, John Horton
6371:15 and 290 theorems
6177:intersection theory
5848:(sometimes, simply
5764:imaginary ellipsoid
5632:
5598:
5570:
5495:composition algebra
5259:of a bilinear form
4874: for all
4490:This bilinear form
2856:The quadratic form
2198:
2155:
2118:
1062:Pythagorean triples
742:projective geometry
687:continued fractions
6687:Shafarevich, I. R.
6141:
6132:
6042:
6036:
5881:in a vector space
5846:quadratic lattices
5636:
5618:
5584:
5556:
5483:
5320:, otherwise it is
4889:
4480:
4421:
4341:
4219:
3950:
3862:
3761:
3694:
3569:
3429:
3306:
2839:indices of inertia
2745:
2739:
2593:
2497:
2426:vector space. Let
2307:The case when all
2202:
2175:
2132:
2095:
2032:symmetric matrices
1986:
1880:
1871:
1655:
1482:
1473:
1351:
1023:§ Definitions
974:, which is a pair
958:
701:algebraic topology
585:
583:
288:quadratic equation
273:
203:intersection forms
109:
49:with terms all of
6901:Pfister, Albrecht
6706:978-3-642-30993-9
6679:978-0-88385-030-5
6658:978-3-540-66564-9
6229:generalized this
5674:Geometric meaning
5092:A quadratic form
4875:
4420:
4165:
3760:
3087:and the function
3039:-vector space to
3001:) are different.
2995:Clifford algebras
2937:orthogonal group
2534:invertible matrix
2454:matrix such that
2326:negative definite
2321:positive definite
2267:are 0, 1, or −1.
2254:invertible matrix
2185:
2142:
2105:
1980:
1862:
1842:
1820:
1795:
1773:
1753:
638:analytic geometry
578:
452:
372:
191:Riemannian metric
183:orthogonal groups
61:). For example,
16:(Redirected from
7013:
6977:
6959:
6947:"Quadratic form"
6934:
6910:
6896:
6862:
6839:. Vol. 73.
6824:
6793:. Vol. 67.
6778:
6751:
6710:
6682:
6661:
6627:
6625:
6618:
6612:
6610:
6597:
6591:
6588:
6582:
6580:
6573:
6567:
6550:
6544:
6539:
6533:
6528:
6522:
6508:
6502:
6500:
6494:
6488:
6482:
6475:
6468:
6461:
6455:
6443:
6368:
6360:
6347:
6334:
6321:
6308:
6295:
6282:
6266:
6246:
6224:
6150:
6148:
6147:
6142:
6137:
6136:
6121:
6106:
6078:
6051:
6049:
6048:
6043:
6041:
6040:
5998:
5944:
5933:
5915:
5911:
5904:
5898:
5892:
5886:
5880:
5876:
5831:
5825:
5819:
5803:
5791:
5780:
5753:
5740:
5734:
5715:of the equation
5706:
5700:
5669:
5668:
5645:
5643:
5642:
5637:
5631:
5626:
5617:
5616:
5597:
5592:
5583:
5582:
5569:
5564:
5555:
5554:
5516:
5510:
5492:
5490:
5489:
5484:
5412:, and satisfies
5407:
5401:
5386:
5374:
5368:
5362:
5352:totally singular
5349:
5343:
5337:
5331:
5313:
5307:
5296:
5290:
5284:
5276:
5270:
5264:
5254:
5233:
5227:
5221:
5206:Related concepts
5200:
5194:
5188:
5178:
5168:
5144:
5133:
5115:
5110:There exists an
5105:
5088:
5078:
5023:
5013:
4982:
4976:
4954:
4948:
4941:commutative ring
4938:
4924:
4918:
4912:
4906:
4898:
4896:
4895:
4890:
4876:
4873:
4859:
4825:
4806:
4794:
4782:
4773:
4767:
4761:
4751:
4745:
4739:
4733:
4727:
4721:
4706:
4683:
4676:
4670:
4655:
4648:
4637:
4590:
4581:
4575:
4569:
4546:
4540:
4534:
4528:
4522:
4495:
4489:
4487:
4486:
4481:
4422:
4413:
4382:
4376:
4358:
4350:
4348:
4347:
4342:
4325:
4324:
4290:
4284:
4278:
4272:
4262:
4246:
4240:
4228:
4226:
4225:
4220:
4215:
4214:
4199:
4198:
4180:
4179:
4166:
4163:
4154:
4153:
4135:
4134:
4110:
4109:
4091:
4090:
4064:
4054:
4043:
4033:
4027:
4010:
4000:
3994:
3985:
3965:
3959:
3957:
3956:
3951:
3906:
3900:
3894:
3884:
3871:
3869:
3868:
3863:
3852:
3851:
3850:
3831:
3830:
3829:
3762:
3753:
3732:
3731:
3715:
3703:
3701:
3700:
3695:
3684:
3683:
3682:
3659:
3653:
3647:
3641:
3635:
3629:
3623:
3617:
3605:symmetric matrix
3602:
3596:
3590:
3578:
3576:
3575:
3570:
3528:
3505:
3499:
3493:
3487:
3478:
3463:
3438:
3436:
3435:
3430:
3419:
3418:
3417:
3385:
3379:
3373:
3363:
3347:
3336:
3328:with components
3323:
3315:
3313:
3312:
3307:
3296:
3295:
3279:
3278:
3277:
3267:
3266:
3265:
3255:
3254:
3241:
3236:
3220:
3215:
3194:
3193:
3175:
3174:
3152:
3146:
3136:
3126:
3117:
3086:
3076:
3066:
3044:
3038:
3032:
3018:
2992:
2972:
2965:
2959:
2944:
2922:
2916:
2906:
2900:
2889:
2883:
2872:
2861:
2852:
2846:
2829:
2820:
2811:
2805:
2796:
2787:
2778:
2772:
2766:
2760:
2754:
2752:
2751:
2746:
2744:
2743:
2736:
2735:
2675:
2674:
2641:
2640:
2610:
2602:
2600:
2599:
2594:
2583:
2582:
2581:
2552:
2546:
2540:
2532:
2522:
2516:
2506:
2504:
2503:
2498:
2487:
2486:
2485:
2453:
2443:
2437:
2431:
2421:
2415:
2403:
2399:
2377:
2371:
2365:
2359:
2334:
2333:
2317:
2303:
2266:
2240:
2211:
2209:
2208:
2203:
2197:
2192:
2187:
2186:
2178:
2174:
2173:
2154:
2149:
2144:
2143:
2135:
2131:
2130:
2117:
2112:
2107:
2106:
2098:
2094:
2093:
2014:
2003:
1995:
1993:
1992:
1987:
1985:
1981:
1976:
1975:
1974:
1959:
1958:
1945:
1926:
1910:
1889:
1887:
1886:
1881:
1876:
1875:
1863:
1858:
1847:
1843:
1838:
1827:
1821:
1816:
1805:
1796:
1791:
1780:
1774:
1769:
1758:
1754:
1749:
1738:
1710:symmetric matrix
1707:
1696:
1686:
1676:
1664:
1662:
1661:
1656:
1579:
1578:
1563:
1562:
1547:
1546:
1510:
1509:
1491:
1489:
1488:
1483:
1478:
1477:
1402:
1398:
1376:
1360:
1358:
1357:
1352:
1347:
1339:
1338:
1337:
1331:
1322:
1321:
1320:
1310:
1309:
1308:
1298:
1297:
1284:
1279:
1263:
1258:
1237:
1236:
1218:
1217:
1205:
1204:
1188:
1182:
1171:
1165:
1105:
1090:
1059:
1053:
1047:
1015:
1001:
991:
985:
967:
965:
964:
959:
954:
953:
941:
940:
928:
927:
915:
914:
909:
905:
874:
873:
780:and the origin:
779:
738:projective space
735:
723:
715:
698:
683:quadratic fields
672:
660:
653:
646:commutative ring
626:rational numbers
608:
602:
594:
592:
591:
586:
584:
580:
579:
576:
572:
558:
557:
530:
529:
502:
501:
454:
453:
450:
446:
444:
443:
416:
415:
374:
373:
370:
366:
364:
363:
307:
282:
280:
279:
274:
272:
267:
266:
258:
252:
251:
250:
244:
157:
147:
135:
128:
124:
118:
116:
115:
110:
108:
107:
80:
79:
21:
7021:
7020:
7016:
7015:
7014:
7012:
7011:
7010:
6991:Quadratic forms
6981:
6980:
6962:
6944:
6941:
6923:
6899:
6885:
6875:Springer-Verlag
6865:
6851:
6841:Springer-Verlag
6827:
6805:
6781:
6767:
6754:
6740:
6722:Cassels, J.W.S.
6720:
6717:
6715:Further reading
6707:
6685:
6680:
6664:
6659:
6649:Springer-Verlag
6639:
6636:
6631:
6630:
6621:
6619:
6615:
6604:
6598:
6594:
6589:
6585:
6576:
6574:
6570:
6551:
6547:
6540:
6536:
6529:
6525:
6509:
6505:
6496:
6490:
6484:
6477:
6470:
6463:
6457:
6450:
6444:
6440:
6435:
6387:-quadratic form
6380:
6366:
6363:
6350:
6337:
6324:
6311:
6298:
6285:
6272:
6248:
6230:
6208:
6197:
6188:algebraic group
6131:
6130:
6125:
6111:
6110:
6097:
6087:
6081:
6080:
6066:
6035:
6034:
6029:
6023:
6022:
6017:
6007:
6001:
6000:
5986:
5954:
5935:
5917:
5913:
5909:
5907:with respect to
5900:
5894:
5888:
5882:
5878:
5864:
5838:
5827:
5821:
5817:
5809:
5801:
5793:
5790:
5782:
5778:
5770:
5749:
5736:
5716:
5702:
5683:
5676:
5666:
5657:
5650:
5649:
5608:
5574:
5546:
5526:
5525:
5512:
5506:
5503:
5414:
5413:
5403:
5391:
5376:
5370:
5364:
5358:
5345:
5339:
5333:
5327:
5309:
5298:
5292:
5286:
5282:
5272:
5266:
5260:
5241:
5229:
5223:
5217:
5214:
5208:
5196:
5190:
5180:
5170:
5148:
5135:
5117:
5116:-bilinear form
5111:
5093:
5084:
5025:
5019:
4984:
4978:
4960:
4950:
4944:
4934:
4931:
4920:
4914:
4908:
4902:
4852:
4832:
4831:
4812:
4796:
4784:
4778:
4769:
4763:
4757:
4754:quadratic space
4747:
4741:
4735:
4729:
4723:
4711:
4685:
4678:
4672:
4657:
4650:
4639:
4596:
4586:
4577:
4571:
4548:
4542:
4536:
4530:
4524:
4497:
4491:
4385:
4384:
4378:
4360:
4354:
4316:
4293:
4292:
4286:
4280:
4274:
4268:
4258:
4252:quadratic space
4242:
4236:
4233:change of basis
4206:
4190:
4171:
4145:
4126:
4101:
4082:
4071:
4070:
4056:
4045:
4035:
4029:
4015:
4006:
3996:
3990:
3981:
3978:
3972:
3970:Quadratic space
3961:
3909:
3908:
3902:
3896:
3890:
3883:
3875:
3841:
3820:
3723:
3718:
3717:
3711:
3673:
3662:
3661:
3655:
3649:
3643:
3637:
3631:
3625:
3619:
3607:
3598:
3592:
3586:
3531:
3530:
3511:
3501:
3495:
3489:
3483:
3469:
3461:
3452:
3442:
3408:
3388:
3387:
3381:
3375:
3365:
3361:
3349:
3346:
3338:
3335:
3329:
3319:
3284:
3269:
3257:
3243:
3185:
3166:
3155:
3154:
3148:
3142:
3141:of degree 2 in
3132:
3122:
3088:
3078:
3068:
3046:
3040:
3034:
3020:
3014:
3007:
2974:
2967:
2961:
2949:
2938:
2918:
2912:
2902:
2891:
2885:
2874:
2863:
2857:
2848:
2842:
2830:are called the
2828:
2822:
2819:
2813:
2807:
2804:
2798:
2795:
2789:
2786:
2780:
2774:
2768:
2762:
2756:
2738:
2737:
2727:
2725:
2720:
2715:
2709:
2708:
2703:
2698:
2693:
2687:
2686:
2681:
2676:
2666:
2664:
2658:
2657:
2652:
2647:
2642:
2632:
2625:
2613:
2612:
2606:
2572:
2555:
2554:
2548:
2542:
2536:
2524:
2518:
2512:
2476:
2456:
2455:
2445:
2444:is a symmetric
2439:
2433:
2427:
2417:
2411:
2401:
2390:
2373:
2367:
2361:
2349:
2331:
2330:
2316:
2308:
2301:
2294:
2287:
2280:
2265:
2257:
2239:
2230:
2223:
2217:
2165:
2122:
2085:
2080:
2079:
2054:
2040:
2013:
2005:
2001:
1963:
1947:
1946:
1940:
1929:
1928:
1924:
1912:
1909:
1901:
1898:
1870:
1869:
1864:
1848:
1844:
1828:
1823:
1822:
1806:
1802:
1797:
1781:
1776:
1775:
1759:
1755:
1739:
1735:
1725:
1713:
1712:
1706:
1698:
1688:
1678:
1668:
1570:
1554:
1538:
1501:
1496:
1495:
1472:
1471:
1466:
1461:
1455:
1454:
1449:
1444:
1438:
1437:
1432:
1427:
1417:
1405:
1404:
1400:
1386:
1383:
1374:
1362:
1326:
1312:
1300:
1286:
1228:
1209:
1196:
1191:
1190:
1184:
1181:
1173:
1167:
1157:
1154:
1096:
1093:Pell's equation
1078:
1055:
1049:
1039:
1032:
1003:
997:
987:
975:
972:quadratic space
945:
932:
919:
883:
879:
878:
865:
782:
781:
765:
760:expressing the
750:Euclidean space
729:
717:
711:
694:
671:
663:
656:
649:
622:complex numbers
604:
598:
582:
581:
571:
549:
521:
493:
483:
456:
455:
445:
435:
407:
397:
376:
375:
365:
355:
345:
324:
323:
303:
300:
253:
239:
231:
230:
149:
145:
133:
126:
122:
99:
71:
63:
62:
35:
28:
23:
22:
15:
12:
11:
5:
7019:
7017:
7009:
7008:
7003:
6998:
6996:Linear algebra
6993:
6983:
6982:
6979:
6978:
6960:
6940:
6939:External links
6937:
6936:
6935:
6921:
6897:
6883:
6863:
6849:
6825:
6803:
6783:Lam, Tsit-Yuen
6779:
6765:
6752:
6738:
6730:Academic Press
6716:
6713:
6712:
6711:
6705:
6683:
6678:
6662:
6657:
6635:
6632:
6629:
6628:
6613:
6592:
6583:
6568:
6545:
6534:
6523:
6503:
6437:
6436:
6434:
6431:
6430:
6429:
6427:Witt's theorem
6424:
6419:
6414:
6409:
6404:
6399:
6394:
6389:
6379:
6376:
6362:
6361:
6348:
6335:
6322:
6309:
6296:
6283:
6269:
6225:is universal.
6196:
6193:
6192:
6191:
6180:
6169:
6162:
6140:
6135:
6129:
6126:
6124:
6120:
6116:
6113:
6112:
6109:
6105:
6101:
6098:
6096:
6093:
6092:
6090:
6039:
6033:
6030:
6028:
6025:
6024:
6021:
6018:
6016:
6013:
6012:
6010:
5979:
5978:
5975:
5970:
5967:
5953:
5952:Historical use
5950:
5837:
5834:
5813:
5797:
5786:
5774:
5675:
5672:
5662:
5655:
5635:
5630:
5625:
5621:
5615:
5611:
5607:
5604:
5601:
5596:
5591:
5587:
5581:
5577:
5573:
5568:
5563:
5559:
5553:
5549:
5545:
5542:
5539:
5536:
5533:
5502:
5499:
5482:
5479:
5476:
5473:
5470:
5467:
5464:
5461:
5458:
5455:
5452:
5449:
5446:
5443:
5440:
5436:
5433:
5430:
5427:
5424:
5421:
5369:that preserve
5332:to a subspace
5207:
5204:
5203:
5202:
5146:
4930:
4929:Generalization
4927:
4888:
4885:
4882:
4879:
4871:
4868:
4865:
4862:
4858:
4855:
4851:
4848:
4845:
4842:
4839:
4479:
4476:
4473:
4470:
4467:
4464:
4461:
4458:
4455:
4452:
4449:
4446:
4443:
4440:
4437:
4434:
4431:
4428:
4425:
4419:
4416:
4410:
4407:
4404:
4401:
4398:
4395:
4392:
4340:
4337:
4334:
4331:
4328:
4323:
4319:
4315:
4312:
4309:
4306:
4303:
4300:
4218:
4213:
4209:
4205:
4202:
4197:
4193:
4189:
4186:
4183:
4178:
4174:
4170:
4160:
4157:
4152:
4148:
4144:
4141:
4138:
4133:
4129:
4125:
4122:
4119:
4116:
4113:
4108:
4104:
4100:
4097:
4094:
4089:
4085:
4081:
4078:
4067:standard basis
4003:quadratic form
3971:
3968:
3949:
3946:
3943:
3940:
3937:
3934:
3931:
3928:
3925:
3922:
3919:
3916:
3879:
3861:
3858:
3855:
3849:
3844:
3840:
3837:
3834:
3828:
3823:
3819:
3816:
3813:
3810:
3807:
3804:
3801:
3798:
3795:
3792:
3789:
3786:
3783:
3780:
3777:
3774:
3771:
3768:
3765:
3759:
3756:
3750:
3747:
3744:
3741:
3738:
3735:
3730:
3726:
3716:is defined by
3693:
3690:
3687:
3681:
3676:
3672:
3669:
3583:characteristic
3568:
3565:
3562:
3559:
3556:
3553:
3550:
3547:
3544:
3541:
3538:
3457:
3450:
3428:
3425:
3422:
3416:
3411:
3407:
3404:
3401:
3398:
3395:
3357:
3342:
3333:
3305:
3302:
3299:
3294:
3291:
3287:
3282:
3276:
3272:
3264:
3260:
3253:
3250:
3246:
3240:
3235:
3232:
3229:
3225:
3219:
3214:
3211:
3208:
3204:
3200:
3197:
3192:
3188:
3184:
3181:
3178:
3173:
3169:
3165:
3162:
3129:quadratic form
3011:quadratic form
3006:
3003:
2973:are the same (
2929:isometry group
2826:
2817:
2802:
2793:
2784:
2742:
2734:
2730:
2726:
2724:
2721:
2719:
2716:
2714:
2711:
2710:
2707:
2704:
2702:
2699:
2697:
2694:
2692:
2689:
2688:
2685:
2682:
2680:
2677:
2673:
2669:
2665:
2663:
2660:
2659:
2656:
2653:
2651:
2648:
2646:
2643:
2639:
2635:
2631:
2630:
2628:
2623:
2620:
2592:
2589:
2586:
2580:
2575:
2571:
2568:
2565:
2562:
2496:
2493:
2490:
2484:
2479:
2475:
2472:
2469:
2466:
2463:
2342:nondegenerate
2312:
2299:
2292:
2285:
2261:
2235:
2228:
2221:
2201:
2196:
2191:
2184:
2181:
2172:
2168:
2164:
2161:
2158:
2153:
2148:
2141:
2138:
2129:
2125:
2121:
2116:
2111:
2104:
2101:
2092:
2088:
2070:; that is, an
2039:
2036:
2024:characteristic
2009:
1984:
1979:
1973:
1970:
1966:
1962:
1957:
1954:
1950:
1943:
1939:
1936:
1920:
1905:
1897:
1894:
1879:
1874:
1868:
1865:
1861:
1857:
1854:
1851:
1845:
1841:
1837:
1834:
1831:
1825:
1824:
1819:
1815:
1812:
1809:
1803:
1801:
1798:
1794:
1790:
1787:
1784:
1778:
1777:
1772:
1768:
1765:
1762:
1756:
1752:
1748:
1745:
1742:
1736:
1734:
1731:
1730:
1728:
1723:
1720:
1702:
1654:
1651:
1648:
1645:
1642:
1639:
1636:
1633:
1630:
1627:
1624:
1621:
1618:
1615:
1612:
1609:
1606:
1603:
1600:
1597:
1594:
1591:
1588:
1585:
1582:
1577:
1573:
1569:
1566:
1561:
1557:
1553:
1550:
1545:
1541:
1537:
1534:
1531:
1528:
1525:
1522:
1519:
1516:
1513:
1508:
1504:
1481:
1476:
1470:
1467:
1465:
1462:
1460:
1457:
1456:
1453:
1450:
1448:
1445:
1443:
1440:
1439:
1436:
1433:
1431:
1428:
1426:
1423:
1422:
1420:
1415:
1412:
1382:
1379:
1370:
1350:
1346:
1342:
1336:
1330:
1325:
1319:
1315:
1307:
1303:
1296:
1293:
1289:
1283:
1278:
1275:
1272:
1268:
1262:
1257:
1254:
1251:
1247:
1243:
1240:
1235:
1231:
1227:
1224:
1221:
1216:
1212:
1208:
1203:
1199:
1177:
1153:
1150:
1031:
1028:
957:
952:
948:
944:
939:
935:
931:
926:
922:
918:
913:
908:
904:
901:
898:
895:
892:
889:
886:
882:
877:
872:
868:
864:
861:
858:
855:
852:
849:
846:
843:
840:
837:
834:
831:
828:
825:
822:
819:
816:
813:
810:
807:
804:
801:
798:
795:
792:
789:
758:Euclidean norm
746:conic sections
667:
661:-adic integers
634:linear algebra
573:
570:
567:
564:
561:
556:
552:
548:
545:
542:
539:
536:
533:
528:
524:
520:
517:
514:
511:
508:
505:
500:
496:
492:
489:
486:
484:
482:
479:
476:
473:
470:
467:
464:
461:
458:
457:
447:
442:
438:
434:
431:
428:
425:
422:
419:
414:
410:
406:
403:
400:
398:
396:
393:
390:
387:
384:
381:
378:
377:
367:
362:
358:
354:
351:
348:
346:
344:
341:
338:
335:
332:
331:
299:
296:
271:
265:
262:
257:
249:
243:
238:
211:four-manifolds
175:linear algebra
136:, such as the
106:
102:
98:
95:
92:
89:
86:
83:
78:
74:
70:
43:quadratic form
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
7018:
7007:
7004:
7002:
6999:
6997:
6994:
6992:
6989:
6988:
6986:
6976:
6972:
6971:
6966:
6961:
6958:
6954:
6953:
6948:
6943:
6942:
6938:
6932:
6928:
6924:
6922:0-521-46755-1
6918:
6914:
6909:
6908:
6902:
6898:
6894:
6890:
6886:
6884:3-540-66564-1
6880:
6876:
6872:
6868:
6867:O'Meara, O.T.
6864:
6860:
6856:
6852:
6850:3-540-06009-X
6846:
6842:
6838:
6834:
6830:
6826:
6822:
6818:
6814:
6810:
6806:
6804:0-8218-1095-2
6800:
6796:
6792:
6788:
6784:
6780:
6776:
6772:
6768:
6766:0-521-40475-4
6762:
6758:
6753:
6749:
6745:
6741:
6739:0-12-163260-1
6735:
6731:
6727:
6723:
6719:
6718:
6714:
6708:
6702:
6698:
6694:
6693:
6688:
6684:
6681:
6675:
6671:
6667:
6663:
6660:
6654:
6650:
6646:
6642:
6641:O'Meara, O.T.
6638:
6637:
6633:
6624:
6617:
6614:
6608:
6602:
6601:Arf invariant
6596:
6593:
6587:
6584:
6579:
6572:
6569:
6566:
6562:
6558:
6554:
6553:Maxime Bôcher
6549:
6546:
6543:
6538:
6535:
6532:
6527:
6524:
6520:
6516:
6512:
6507:
6504:
6499:
6493:
6487:
6481:
6474:
6467:
6460:
6454:
6448:
6442:
6439:
6432:
6428:
6425:
6423:
6420:
6418:
6415:
6413:
6410:
6408:
6405:
6403:
6400:
6398:
6395:
6393:
6390:
6388:
6386:
6382:
6381:
6377:
6375:
6372:
6358:
6354:
6349:
6345:
6341:
6336:
6332:
6328:
6323:
6319:
6315:
6310:
6306:
6302:
6297:
6293:
6289:
6284:
6280:
6276:
6271:
6270:
6268:
6264:
6260:
6256:
6252:
6245:
6241:
6237:
6233:
6228:
6223:
6219:
6215:
6211:
6206:
6202:
6194:
6189:
6185:
6181:
6178:
6174:
6170:
6167:
6163:
6160:
6159:
6158:
6156:
6151:
6138:
6133:
6127:
6122:
6118:
6114:
6107:
6103:
6099:
6094:
6088:
6077:
6073:
6069:
6063:
6061:
6060:
6055:
6037:
6031:
6026:
6019:
6014:
6008:
5997:
5993:
5989:
5983:
5976:
5974:
5971:
5968:
5966:
5963:
5962:
5961:
5960:should mean:
5959:
5951:
5949:
5946:
5942:
5938:
5932:
5928:
5924:
5920:
5908:
5903:
5897:
5891:
5885:
5875:
5871:
5867:
5861:
5859:
5855:
5854:number theory
5851:
5847:
5843:
5835:
5833:
5830:
5824:
5816:
5812:
5807:
5800:
5796:
5789:
5785:
5777:
5773:
5767:
5765:
5761:
5757:
5752:
5747:
5742:
5739:
5732:
5729:
5725:
5722:
5719:
5714:
5710:
5705:
5698:
5694:
5690:
5686:
5681:
5673:
5671:
5665:
5661:
5654:
5646:
5633:
5628:
5623:
5619:
5613:
5609:
5605:
5602:
5599:
5594:
5589:
5585:
5579:
5575:
5571:
5566:
5561:
5557:
5551:
5547:
5543:
5537:
5531:
5524:
5523:diagonal form
5520:
5515:
5509:
5500:
5498:
5496:
5493:then it is a
5480:
5474:
5468:
5462:
5456:
5453:
5447:
5444:
5438:
5434:
5431:
5428:
5425:
5422:
5411:
5406:
5399:
5395:
5388:
5387:into itself.
5384:
5380:
5373:
5367:
5361:
5355:
5353:
5348:
5342:
5336:
5330:
5325:
5324:
5319:
5318:
5312:
5305:
5301:
5295:
5289:
5280:
5275:
5269:
5263:
5258:
5252:
5248:
5244:
5239:
5238:
5232:
5226:
5220:
5216:Two elements
5213:
5205:
5199:
5193:
5187:
5183:
5177:
5173:
5166:
5162:
5159:
5155:
5151:
5147:
5142:
5138:
5132:
5128:
5124:
5120:
5114:
5109:
5108:
5107:
5104:
5100:
5096:
5090:
5087:
5082:
5076:
5072:
5068:
5064:
5060:
5056:
5052:
5048:
5044:
5040:
5036:
5032:
5028:
5022:
5017:
5011:
5007:
5003:
4999:
4995:
4991:
4987:
4981:
4975:
4971:
4967:
4963:
4958:
4953:
4947:
4942:
4937:
4928:
4926:
4923:
4917:
4911:
4905:
4899:
4886:
4883:
4880:
4877:
4866:
4863:
4856:
4853:
4849:
4843:
4837:
4829:
4823:
4819:
4815:
4810:
4804:
4800:
4792:
4788:
4781:
4775:
4772:
4766:
4760:
4755:
4750:
4744:
4738:
4732:
4726:
4719:
4715:
4708:
4704:
4700:
4696:
4692:
4688:
4681:
4675:
4668:
4664:
4660:
4653:
4646:
4642:
4635:
4631:
4627:
4623:
4619:
4615:
4611:
4607:
4603:
4599:
4594:
4589:
4583:
4580:
4574:
4567:
4563:
4559:
4555:
4551:
4545:
4539:
4533:
4527:
4520:
4516:
4512:
4508:
4504:
4500:
4494:
4477:
4468:
4462:
4459:
4453:
4447:
4444:
4438:
4435:
4432:
4426:
4417:
4414:
4408:
4402:
4399:
4396:
4390:
4381:
4375:
4371:
4367:
4363:
4357:
4351:
4338:
4332:
4326:
4321:
4317:
4313:
4307:
4304:
4298:
4289:
4283:
4277:
4271:
4266:
4261:
4255:
4253:
4248:
4245:
4239:
4234:
4229:
4216:
4211:
4207:
4203:
4195:
4191:
4187:
4184:
4181:
4176:
4172:
4150:
4146:
4142:
4139:
4136:
4131:
4127:
4117:
4114:
4106:
4102:
4098:
4095:
4092:
4087:
4083:
4076:
4068:
4063:
4059:
4052:
4048:
4042:
4038:
4032:
4026:
4022:
4018:
4014:
4009:
4004:
3999:
3995:over a field
3993:
3989:
3986:-dimensional
3984:
3977:
3969:
3967:
3964:
3947:
3941:
3938:
3935:
3929:
3926:
3920:
3914:
3905:
3899:
3893:
3888:
3882:
3878:
3872:
3859:
3856:
3853:
3842:
3838:
3835:
3832:
3821:
3817:
3808:
3802:
3799:
3793:
3787:
3784:
3778:
3775:
3772:
3766:
3757:
3754:
3748:
3742:
3739:
3736:
3728:
3724:
3714:
3709:
3704:
3691:
3688:
3685:
3674:
3670:
3667:
3658:
3652:
3646:
3640:
3634:
3628:
3622:
3615:
3611:
3606:
3601:
3595:
3589:
3584:
3579:
3566:
3560:
3557:
3551:
3548:
3542:
3536:
3526:
3522:
3518:
3514:
3509:
3504:
3498:
3492:
3486:
3480:
3476:
3472:
3467:
3460:
3456:
3449:
3445:
3439:
3426:
3423:
3420:
3409:
3405:
3399:
3393:
3384:
3378:
3372:
3368:
3360:
3356:
3352:
3345:
3341:
3332:
3327:
3326:column vector
3322:
3316:
3303:
3300:
3297:
3292:
3289:
3285:
3280:
3274:
3270:
3262:
3258:
3251:
3248:
3244:
3238:
3233:
3230:
3227:
3223:
3217:
3212:
3209:
3206:
3202:
3198:
3190:
3186:
3182:
3179:
3176:
3171:
3167:
3160:
3151:
3145:
3140:
3135:
3131:over a field
3130:
3125:
3119:
3118:is bilinear.
3115:
3111:
3107:
3103:
3099:
3095:
3091:
3085:
3081:
3075:
3071:
3064:
3060:
3057:
3053:
3049:
3043:
3037:
3031:
3027:
3023:
3017:
3013:over a field
3012:
3004:
3002:
3000:
2996:
2990:
2986:
2982:
2978:
2971:
2964:
2957:
2953:
2948:
2942:
2936:
2935:
2930:
2926:
2921:
2915:
2910:
2905:
2898:
2894:
2888:
2881:
2877:
2870:
2866:
2860:
2854:
2851:
2845:
2840:
2837:
2833:
2825:
2816:
2810:
2801:
2792:
2783:
2777:
2771:
2765:
2759:
2740:
2732:
2728:
2722:
2717:
2712:
2705:
2700:
2695:
2690:
2683:
2678:
2671:
2667:
2661:
2654:
2649:
2644:
2637:
2633:
2626:
2621:
2618:
2609:
2603:
2590:
2587:
2584:
2573:
2569:
2566:
2560:
2551:
2545:
2539:
2535:
2531:
2527:
2521:
2515:
2510:
2494:
2491:
2488:
2477:
2473:
2467:
2461:
2452:
2448:
2442:
2436:
2430:
2425:
2422:-dimensional
2420:
2414:
2408:
2405:
2397:
2393:
2388:
2383:
2381:
2376:
2370:
2364:
2357:
2353:
2347:
2345:
2339:
2335:
2332:nondegenerate
2327:
2323:
2322:
2315:
2311:
2305:
2298:
2291:
2284:
2278:
2274:
2270:
2264:
2260:
2255:
2250:
2248:
2244:
2238:
2234:
2227:
2220:
2215:
2199:
2194:
2189:
2179:
2170:
2166:
2162:
2159:
2156:
2151:
2146:
2136:
2127:
2123:
2119:
2114:
2109:
2099:
2090:
2086:
2077:
2076:diagonal form
2073:
2069:
2065:
2061:
2059:
2053:
2049:
2045:
2037:
2035:
2033:
2029:
2025:
2021:
2016:
2012:
2008:
1999:
1982:
1977:
1971:
1968:
1964:
1960:
1955:
1952:
1948:
1941:
1937:
1934:
1927:, the matrix
1923:
1919:
1915:
1908:
1904:
1895:
1893:
1890:
1877:
1872:
1866:
1859:
1855:
1852:
1849:
1839:
1835:
1832:
1829:
1817:
1813:
1810:
1807:
1799:
1792:
1788:
1785:
1782:
1770:
1766:
1763:
1760:
1750:
1746:
1743:
1740:
1732:
1726:
1721:
1718:
1711:
1705:
1701:
1695:
1691:
1685:
1681:
1675:
1671:
1665:
1652:
1649:
1646:
1640:
1637:
1634:
1628:
1625:
1622:
1616:
1613:
1610:
1604:
1601:
1598:
1592:
1589:
1586:
1580:
1575:
1571:
1567:
1564:
1559:
1555:
1551:
1548:
1543:
1539:
1535:
1532:
1526:
1523:
1520:
1517:
1514:
1506:
1502:
1492:
1479:
1474:
1468:
1463:
1458:
1451:
1446:
1441:
1434:
1429:
1424:
1418:
1413:
1410:
1403:has the form
1399:. The matrix
1397:
1393:
1389:
1380:
1378:
1373:
1369:
1365:
1348:
1340:
1323:
1317:
1313:
1305:
1301:
1294:
1291:
1287:
1281:
1276:
1273:
1270:
1266:
1260:
1255:
1252:
1249:
1245:
1241:
1233:
1229:
1225:
1222:
1219:
1214:
1210:
1201:
1197:
1189:variables by
1187:
1180:
1176:
1170:
1164:
1160:
1151:
1149:
1147:
1146:modular group
1143:
1139:
1135:
1131:
1129:
1124:
1119:
1117:
1113:
1109:
1103:
1099:
1094:
1089:
1085:
1081:
1076:
1075:
1070:
1065:
1063:
1058:
1052:
1046:
1042:
1037:
1029:
1027:
1025:
1024:
1019:
1014:
1010:
1006:
1000:
996:over a field
995:
990:
983:
979:
973:
968:
955:
950:
946:
942:
937:
933:
929:
924:
920:
916:
911:
899:
896:
893:
890:
887:
875:
870:
859:
856:
853:
850:
847:
841:
835:
832:
829:
826:
823:
814:
811:
805:
802:
799:
796:
793:
787:
777:
773:
769:
763:
759:
755:
751:
747:
743:
739:
736:-dimensional
733:
727:
724:-dimensional
721:
714:
709:
704:
702:
697:
692:
691:modular forms
688:
684:
680:
679:number theory
676:
670:
666:
662:
659:
652:
647:
643:
639:
635:
631:
627:
623:
619:
614:
612:
607:
601:
595:
568:
565:
562:
559:
554:
550:
546:
543:
540:
537:
534:
531:
526:
522:
518:
515:
512:
509:
506:
503:
498:
494:
490:
487:
485:
477:
474:
471:
468:
465:
459:
440:
436:
432:
429:
426:
423:
420:
417:
412:
408:
404:
401:
399:
391:
388:
385:
379:
360:
356:
352:
349:
347:
339:
333:
321:
317:
316:
311:
306:
297:
295:
293:
289:
284:
263:
260:
236:
228:
224:
220:
216:
212:
209:, especially
208:
204:
200:
196:
192:
188:
184:
180:
176:
172:
171:number theory
167:
165:
161:
156:
152:
143:
139:
132:
119:
104:
100:
96:
93:
90:
87:
84:
81:
76:
72:
68:
60:
56:
52:
48:
44:
40:
33:
19:
6968:
6950:
6906:
6870:
6832:
6786:
6756:
6725:
6691:
6669:
6644:
6622:
6616:
6609:, p. 27
6606:
6595:
6586:
6577:
6571:
6556:
6548:
6537:
6526:
6506:
6497:
6491:
6485:
6483:in place of
6479:
6472:
6465:
6458:
6456:in place of
6452:
6441:
6417:Square class
6384:
6367:{1, 2, 5, 5}
6364:
6356:
6352:
6343:
6339:
6330:
6326:
6317:
6313:
6304:
6300:
6291:
6287:
6278:
6274:
6262:
6258:
6254:
6250:
6243:
6239:
6235:
6231:
6221:
6217:
6213:
6209:
6200:
6198:
6154:
6152:
6075:
6071:
6067:
6064:
6057:
5995:
5991:
5987:
5984:
5980:
5972:
5964:
5957:
5955:
5947:
5940:
5936:
5930:
5926:
5922:
5918:
5906:
5905:is integral
5901:
5895:
5889:
5883:
5873:
5869:
5865:
5862:
5845:
5841:
5839:
5828:
5822:
5814:
5810:
5798:
5794:
5787:
5783:
5775:
5771:
5768:
5763:
5750:
5743:
5737:
5730:
5727:
5723:
5720:
5717:
5713:solution set
5703:
5696:
5692:
5688:
5684:
5677:
5663:
5659:
5652:
5647:
5522:
5513:
5507:
5504:
5404:
5397:
5393:
5389:
5382:
5378:
5371:
5365:
5359:
5356:
5351:
5346:
5340:
5334:
5328:
5321:
5315:
5310:
5303:
5299:
5293:
5287:
5279:non-singular
5278:
5273:
5267:
5261:
5256:
5250:
5246:
5242:
5235:
5230:
5224:
5218:
5215:
5197:
5191:
5185:
5181:
5175:
5171:
5164:
5160:
5157:
5153:
5149:
5140:
5136:
5130:
5126:
5122:
5118:
5112:
5102:
5098:
5094:
5091:
5085:
5080:
5074:
5070:
5066:
5062:
5058:
5054:
5050:
5046:
5042:
5038:
5034:
5030:
5026:
5020:
5015:
5009:
5005:
5001:
4997:
4993:
4989:
4985:
4979:
4973:
4969:
4965:
4961:
4951:
4945:
4935:
4932:
4921:
4915:
4909:
4903:
4900:
4830:) such that
4827:
4821:
4817:
4813:
4808:
4802:
4798:
4790:
4786:
4779:
4776:
4770:
4764:
4758:
4753:
4752:is called a
4748:
4742:
4736:
4730:
4724:
4717:
4713:
4709:
4702:
4698:
4694:
4690:
4686:
4679:
4673:
4666:
4662:
4658:
4651:
4644:
4640:
4633:
4629:
4625:
4621:
4617:
4613:
4609:
4605:
4601:
4597:
4587:
4584:
4578:
4572:
4565:
4561:
4557:
4553:
4549:
4543:
4537:
4531:
4525:
4518:
4514:
4510:
4506:
4502:
4498:
4492:
4383:is defined:
4379:
4373:
4369:
4365:
4361:
4355:
4352:
4287:
4281:
4275:
4269:
4259:
4256:
4251:
4249:
4243:
4237:
4230:
4061:
4057:
4050:
4046:
4040:
4036:
4030:
4024:
4020:
4016:
4007:
4002:
3997:
3991:
3988:vector space
3982:
3979:
3962:
3903:
3897:
3895:with matrix
3891:
3880:
3876:
3873:
3712:
3707:
3705:
3656:
3650:
3644:
3638:
3632:
3626:
3620:
3613:
3609:
3599:
3593:
3587:
3580:
3524:
3520:
3512:
3507:
3502:
3496:
3490:
3484:
3481:
3474:
3470:
3458:
3454:
3447:
3443:
3440:
3382:
3376:
3374:matrix over
3370:
3366:
3358:
3354:
3350:
3343:
3339:
3330:
3320:
3317:
3149:
3143:
3133:
3128:
3123:
3120:
3113:
3109:
3105:
3101:
3097:
3093:
3089:
3083:
3079:
3073:
3069:
3062:
3058:
3055:
3051:
3047:
3041:
3035:
3029:
3025:
3021:
3015:
3010:
3008:
2988:
2984:
2980:
2976:
2969:
2962:
2955:
2951:
2940:
2932:
2924:
2919:
2913:
2903:
2896:
2892:
2886:
2879:
2875:
2868:
2864:
2858:
2855:
2849:
2843:
2838:
2835:
2831:
2823:
2814:
2808:
2799:
2790:
2781:
2775:
2769:
2763:
2757:
2607:
2604:
2549:
2543:
2537:
2529:
2525:
2519:
2513:
2508:
2450:
2446:
2440:
2434:
2428:
2418:
2412:
2409:
2406:
2395:
2391:
2384:
2374:
2368:
2362:
2355:
2351:
2343:
2329:
2325:
2319:
2313:
2309:
2306:
2296:
2289:
2282:
2276:
2262:
2258:
2251:
2236:
2232:
2225:
2218:
2062:
2055:
2017:
2010:
2006:
1921:
1917:
1913:
1906:
1902:
1899:
1896:General case
1891:
1703:
1699:
1693:
1689:
1683:
1679:
1673:
1669:
1666:
1493:
1395:
1391:
1387:
1384:
1371:
1367:
1363:
1185:
1178:
1174:
1168:
1162:
1158:
1155:
1126:
1120:
1101:
1097:
1087:
1083:
1079:
1072:
1066:
1056:
1050:
1044:
1040:
1033:
1021:
1017:
1012:
1008:
1004:
998:
994:vector space
988:
981:
977:
971:
969:
775:
771:
767:
731:
719:
712:
705:
695:
668:
664:
657:
650:
615:
611:coefficients
610:
605:
599:
596:
319:
313:
309:
304:
301:
298:Introduction
285:
219:Killing form
179:group theory
168:
154:
150:
120:
42:
36:
6511:away from 2
6207:shows that
5760:hyperboloid
5746:eigenvalues
5234:are called
4638:. However,
4069:, one has
3466:null vector
3005:Definitions
2997:(and hence
2797:for 1, and
2324:(all 1) or
2247:permutation
1069:Brahmagupta
39:mathematics
6985:Categories
6931:0847.11014
6893:0259.10018
6859:0292.10016
6829:Milnor, J.
6821:1068.11023
6775:0785.11021
6748:0395.10029
6634:References
6565:HathiTrust
6422:Witt group
6392:Cubic form
6351:{1, 2, 5,
6338:{1, 2, 4,
6325:{1, 2, 3,
6312:{1, 2, 2,
6299:{1, 1, 3,
6286:{1, 1, 2,
6273:{1, 1, 1,
5916:, meaning
5806:paraboloid
5701:, and let
5519:equivalent
5297:such that
5237:orthogonal
5210:See also:
5201:-bilinear.
5134:such that
5081:polar form
3974:See also:
3529:such that
3508:equivalent
3045:such that
2999:pin groups
2360:(denoting
2273:invariants
2042:See also:
1125:published
223:statistics
215:Lie theory
47:polynomial
6975:EMS Press
6957:EMS Press
6227:Ramanujan
6201:universal
6184:Lie group
5756:ellipsoid
5709:symmetric
5603:⋯
5432:∈
5420:∀
5317:isotropic
5041: : (
4881:∈
4809:isometric
4710:The pair
4460:−
4445:−
4204:∈
4185:…
4140:…
4096:…
4065:with its
3980:Given an
3800:−
3785:−
3552:φ
3537:ψ
3441:A vector
3298:∈
3224:∑
3203:∑
3180:…
3019:is a map
2729:λ
2723:⋯
2701:⋱
2696:⋮
2691:⋮
2679:⋯
2668:λ
2650:⋯
2634:λ
2564:→
2380:spacetime
2277:signature
2183:~
2167:λ
2160:⋯
2140:~
2124:λ
2103:~
2087:λ
1998:symmetric
1267:∑
1246:∑
1223:…
1136:over the
1108:Brouncker
577:(ternary)
261:−
256:Σ
237:−
207:manifolds
94:−
6903:(1995).
6869:(1973).
6785:(2005).
6724:(1978).
6697:Springer
6643:(2000),
6517:(by the
6378:See also
6190:aspects.
6173:topology
6155:twos out
6056:uses in
5973:twos out
5858:topology
5850:lattices
5667:⟩
5651:⟨
5323:definite
5169:for all
5121: :
5097: :
5029: :
4996: :
4988: :
4964: :
4857:′
4828:isometry
4816: :
4671:for all
4570:for all
4523:for all
4364: :
4257:The map
4019: :
4013:function
3581:Let the
3067:for all
3024: :
2882:) < 0
2871:) > 0
2836:negative
2832:positive
2344:bilinear
2214:diagonal
1138:integers
1121:In 1801
1116:Lagrange
1048:, where
1007: :
907:‖
881:‖
762:distance
752:and the
630:integers
609:are the
451:(binary)
6813:2104929
6407:Quadric
6355:}, 6 ≤
6342:}, 4 ≤
6329:}, 3 ≤
6316:}, 2 ≤
6303:}, 3 ≤
6290:}, 2 ≤
6277:}, 1 ≤
6166:lattice
5965:twos in
5744:If all
5658:, ...,
5079:is the
5014:is the
3453:, ...,
3386:. Then
3364:be the
3337:, ...,
3324:be the
2934:compact
2890:. When
2788:for 0,
2366:1s and
2231:, ...,
1381:Example
1166:matrix
1030:History
986:, with
756:of the
728:in the
726:quadric
654:or the
603:, ...,
371:(unary)
320:ternary
221:), and
142:complex
6929:
6919:
6891:
6881:
6857:
6847:
6819:
6811:
6801:
6773:
6763:
6746:
6736:
6703:
6676:
6655:
5678:Using
5408:is an
5257:kernel
5255:. The
5024:, and
4977:be an
4959:, and
4957:module
4949:be an
4756:, and
3874:Thus,
2983:) ≈ O(
2907:is an
2507:where
2064:Jacobi
2050:, and
1361:where
1144:, the
1071:wrote
1020:. See
1002:, and
754:square
706:Using
689:, and
597:where
318:, and
315:binary
193:, the
53:two ("
51:degree
6447:Gauss
6433:Notes
6054:Gauss
5758:or a
5707:be a
5521:to a
5306:) = 0
5253:) = 0
4939:be a
4740:from
4728:over
4669:) = 0
4377:over
4263:is a
4011:is a
3889:over
3885:is a
3500:over
3477:) = 0
3464:is a
3137:is a
3127:-ary
2931:is a
2243:up to
2020:field
1123:Gauss
1112:Euler
642:field
632:. In
628:, or
310:unary
217:(the
189:(the
148:. If
131:field
45:is a
6917:ISBN
6879:ISBN
6845:ISBN
6799:ISBN
6761:ISBN
6734:ISBN
6701:ISBN
6674:ISBN
6653:ISBN
6563:via
6359:≤ 10
6346:≤ 14
6333:≤ 10
6294:≤ 14
6186:and
6182:the
6175:for
6164:the
5929:) ∈
5856:and
5222:and
5179:and
5156:) =
5069:) −
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