Category: Quadratic forms

Perfect lattice
In mathematics, a perfect lattice (or perfect form) is a lattice in a Euclidean vector space, that is completely determined by the set S of its minimal vectors in the sense that there is only one posi
15 and 290 theorems
In mathematics, the 15 theorem or Conway–Schneeberger Fifteen Theorem, proved by John H. Conway and W. A. Schneeberger in 1993, states that if a positive definite quadratic form with integer matrix re
Clifford algebra
In mathematics, a Clifford algebra is an algebra generated by a vector space with a quadratic form, and is a unital associative algebra. As K-algebras, they generalize the real numbers, complex number
In mathematics, the universal invariant or u-invariant of a field describes the structure of quadratic forms over the field. The universal invariant u(F) of a field F is the largest dimension of an an
Witt group
In mathematics, a Witt group of a field, named after Ernst Witt, is an abelian group whose elements are represented by symmetric bilinear forms over the field.
Isotropic quadratic form
In mathematics, a quadratic form over a field F is said to be isotropic if there is a non-zero vector on which the form evaluates to zero. Otherwise the quadratic form is anisotropic. More precisely,
Bhargava cube
In mathematics, in number theory, a Bhargava cube (also called Bhargava's cube) is a configuration consisting of eight integers placed at the eight corners of a cube. This configuration was extensivel
Kaplansky's theorem on quadratic forms
In mathematics, Kaplansky's theorem on quadratic forms is a result on simultaneous representation of primes by quadratic forms. It was proved in 2003 by Irving Kaplansky.
Universal quadratic form
In mathematics, a universal quadratic form is a quadratic form over a ring that represents every element of the ring. A non-singular form over a field which represents zero non-trivially is universal.
Gerbaldi's theorem
In linear algebra and projective geometry, Gerbaldi's theorem, proved by Gerbaldi, states that one can find six pairwise apolar linearly independent nondegenerate ternary quadratic forms. These are pe
Linked field
In mathematics, a linked field is a field for which the quadratic forms attached to quaternion algebras have a common property.
Projective orthogonal group
In projective geometry and linear algebra, the projective orthogonal group PO is the induced action of the orthogonal group of a quadratic space V = (V,Q) on the associated projective space P(V). Expl
Ε-quadratic form
In mathematics, specifically the theory of quadratic forms, an ε-quadratic form is a generalization of quadratic forms to skew-symmetric settings and to *-rings; ε = ±1, accordingly for symmetric or s
Hasse–Minkowski theorem
The Hasse–Minkowski theorem is a fundamental result in number theory which states that two quadratic forms over a number field are equivalent if and only if they are equivalent locally at all places,
Unimodular lattice
In geometry and mathematical group theory, a unimodular lattice is an integral lattice of determinant 1 or −1. For a lattice in n-dimensional Euclidean space, this is equivalent to requiring that the
Sylvester's law of inertia
Sylvester's law of inertia is a theorem in matrix algebra about certain properties of the coefficient matrix of a real quadratic form that remain invariant under a change of basis. Namely, if A is the
E8 lattice
In mathematics, the E8 lattice is a special lattice in R8. It can be characterized as the unique positive-definite, even, unimodular lattice of rank 8. The name derives from the fact that it is the ro
Signature (topology)
In the field of topology, the signature is an integer invariant which is defined for an oriented manifold M of dimension divisible by four. This invariant of a manifold has been studied in detail, sta
Hilbert symbol
In mathematics, the Hilbert symbol or norm-residue symbol is a function (–, –) from K× × K× to the group of nth roots of unity in a local field K such as the fields of reals or p-adic numbers . It is
Witt's theorem
In mathematics, Witt's theorem, named after Ernst Witt, is a basic result in the algebraic theory of quadratic forms: any isometry between two subspaces of a nonsingular quadratic space over a field k
Definite quadratic form
In mathematics, a definite quadratic form is a quadratic form over some real vector space V that has the same sign (always positive or always negative) for every non-zero vector of V. According to tha
Ramanujan's ternary quadratic form
In number theory, a branch of mathematics, Ramanujan's ternary quadratic form is the algebraic expression x2 + y2 + 10z2 with integral values for x, y and z. Srinivasa Ramanujan considered this expres
Surgery structure set
In mathematics, the surgery structure set is the basic object in the study of manifolds which are homotopy equivalent to a closed manifold X. It is a concept which helps to answer the question whether
Binary quadratic form
In mathematics, a binary quadratic form is a quadratic homogeneous polynomial in two variables where a, b, c are the coefficients. When the coefficients can be arbitrary complex numbers, most results
Hurwitz problem
In mathematics, the Hurwitz problem (named after Adolf Hurwitz) is the problem of finding multiplicative relations between quadratic forms which generalise those known to exist between sums of squares
Eutactic lattice
In mathematics, a eutactic lattice (or eutactic form) is a lattice in Euclidean space whose minimal vectors form a eutactic star. This means they have a set of positive eutactic coefficients ci such t
Meyer's theorem
In number theory, Meyer's theorem on quadratic forms states that an indefinite quadratic form Q in five or more variables over the field of rational numbers nontrivially represents zero. In other word
Spinor genus
In mathematics, the spinor genus is a classification of quadratic forms and lattices over the ring of integers, introduced by Martin Eichler. It refines the genus but may be coarser than proper equiva
In mathematics, the discriminant of a polynomial is a quantity that depends on the coefficients and allows deducing some properties of the roots without computing them. More precisely, it is a polynom
Markov spectrum
In mathematics, the Markov spectrum devised by Andrey Markov is a complicated set of real numbers arising in Markov Diophantine equation and also in the theory of Diophantine approximation.
Leech lattice
In mathematics, the Leech lattice is an even unimodular lattice Λ24 in 24-dimensional Euclidean space, which is one of the best models for the kissing number problem. It was discovered by John Leech.
Quaternionic structure
In mathematics, a quaternionic structure or Q-structure is an axiomatic system that abstracts the concept of a quaternion algebra over a field. A quaternionic structure is a triple (G, Q, q) where G i
Brauer–Wall group
In mathematics, the Brauer–Wall group or super Brauer group or graded Brauer group for a field F is a group BW(F) classifying finite-dimensional graded central division algebras over the field. It was
Büchi's problem
In number theory, Büchi's problem, also known as the n squares' problem, is an open problem named after the Swiss mathematician Julius Richard Büchi. It asks whether there is a positive integer M such
Null vector
In mathematics, given a vector space X with an associated quadratic form q, written (X, q), a null vector or isotropic vector is a non-zero element x of X for which q(x) = 0. In the theory of real bil
Genus of a quadratic form
In mathematics, the genus is a classification of quadratic forms and lattices over the ring of integers. An integral quadratic form is a quadratic form on Zn, or equivalently a free Z-module of finite
Coxeter–Todd lattice
In mathematics, the Coxeter–Todd lattice K12, discovered by Coxeter and Todd, is a 12-dimensional even integral lattice of discriminant 36 with no norm-2 vectors. It is the sublattice of the Leech lat
Hilbert's eleventh problem
Hilbert's eleventh problem is one of David Hilbert's list of open mathematical problems posed at the Second International Congress of Mathematicians in Paris in 1900. A furthering of the theory of qua
Barnes–Wall lattice
In mathematics, the Barnes–Wall lattice Λ16, discovered by Eric Stephen Barnes and G. E. (Tim) Wall, is the 16-dimensional positive-definite even integral lattice of discriminant 28 with no norm-2 vec
Arf invariant
In mathematics, the Arf invariant of a nonsingular quadratic form over a field of characteristic 2 was defined by Turkish mathematician Cahit Arf when he started the systematic study of quadratic form
Hurwitz's theorem (composition algebras)
In mathematics, Hurwitz's theorem is a theorem of Adolf Hurwitz (1859–1919), published posthumously in 1923, solving the Hurwitz problem for finite-dimensional unital real non-associative algebras end
Point reflection
In geometry, a point reflection (point inversion, central inversion, or inversion through a point) is a type of isometry of Euclidean space. An object that is invariant under a point reflection is sai
Generalized Clifford algebra
In mathematics, a Generalized Clifford algebra (GCA) is a unital associative algebra that generalizes the Clifford algebra, and goes back to the work of Hermann Weyl, who utilized and formalized these
Quadratic form (statistics)
In multivariate statistics, if is a vector of random variables, and is an -dimensional symmetric matrix, then the scalar quantity is known as a quadratic form in .
In mathematics, II25,1 is the even 26-dimensional Lorentzian unimodular lattice. It has several unusual properties, arising from Conway's discovery that it has a norm zero Weyl vector. In particular i
Isotropic line
In the geometry of quadratic forms, an isotropic line or null line is a line for which the quadratic form applied to the displacement vector between any pair of its points is zero. An isotropic line o
Negative definiteness
In mathematics, negative definiteness is a property of any object to which a bilinear form may be naturally associated, which is negative-definite. See, in particular: * Negative-definite bilinear fo
Smith–Minkowski–Siegel mass formula
In mathematics, the Smith–Minkowski–Siegel mass formula (or Minkowski–Siegel mass formula) is a formula for the sum of the weights of the lattices (quadratic forms) in a genus, weighted by the recipro
Donaldson's theorem
In mathematics, and especially differential topology and gauge theory, Donaldson's theorem states that a definite intersection form of a compact, oriented, smooth manifold of dimension 4 is diagonalis
Tensor product of quadratic forms
In mathematics, the tensor product of quadratic forms is most easily understood when one views the quadratic forms as quadratic spaces. If R is a commutative ring where 2 is invertible (that is, R has
Pfister form
In mathematics, a Pfister form is a particular kind of quadratic form, introduced by Albrecht Pfister in 1965. In what follows, quadratic forms are considered over a field F of characteristic not 2. F
Quadratic form
In mathematics, a quadratic form is a polynomial with terms all of degree two ("form" is another name for a homogeneous polynomial). For example, is a quadratic form in the variables x and y. The coef
Hasse invariant of a quadratic form
In mathematics, the Hasse invariant (or Hasse–Witt invariant) of a quadratic form Q over a field K takes values in the Brauer group Br(K). The name "Hasse–Witt" comes from Helmut Hasse and Ernst Witt.
Orthogonal group
In mathematics, the orthogonal group in dimension n, denoted O(n), is the group of distance-preserving transformations of a Euclidean space of dimension n that preserve a fixed point, where the group
Class number formula
In number theory, the class number formula relates many important invariants of a number field to a special value of its Dedekind zeta function.
In mathematics, algebraic L-theory is the K-theory of quadratic forms; the term was coined by C. T. C. Wall, with L being used as the letter after K. Algebraic L-theory, also known as "Hermitian K-the
Oppenheim conjecture
In Diophantine approximation, the Oppenheim conjecture concerns representations of numbers by real quadratic forms in several variables. It was formulated in 1929 by Alexander Oppenheim and later the
Positive definiteness
In mathematics, positive definiteness is a property of any object to which a bilinear form or a sesquilinear form may be naturally associated, which is positive-definite. See, in particular: * Positi
Composition algebra
In mathematics, a composition algebra A over a field K is a not necessarily associative algebra over K together with a nondegenerate quadratic form N that satisfies for all x and y in A. A composition
Niemeier lattice
In mathematics, a Niemeier lattice is one of the 24 positive definite even unimodular lattices of rank 24,which were classified by Hans-Volker Niemeier. gave a simplified proof of the classification.