TheInfoList

OR:

In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a quadratic form is a
polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exampl ...
with terms all of degree two ("
form Form is the shape, visual appearance, or configuration of an object. In a wider sense, the form is the way something happens. Form also refers to: *Form (document), a document (printed or electronic) with spaces in which to write or enter data * ...
" is another name for a
homogeneous polynomial In mathematics, a homogeneous polynomial, sometimes called quantic in older texts, is a polynomial whose nonzero terms all have the same degree. For example, x^5 + 2 x^3 y^2 + 9 x y^4 is a homogeneous polynomial of degree 5, in two variables; ...
). For example, :$4x^2 + 2xy - 3y^2$ is a quadratic form in the variables and . The coefficients usually belong to a fixed field , such as the real or complex numbers, and one speaks of a quadratic form over . If $K=\mathbb R$, and the quadratic form takes zero only when all variables are simultaneously zero, then it is a definite quadratic form, otherwise it is an
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, if ''q'' is a quadratic form on a vector ...
. Quadratic forms occupy a central place in various branches of mathematics, including
number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Ma ...
,
linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrice ...
,
group theory In abstract algebra, group theory studies the algebraic structures known as groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces, can all be seen ...
(
orthogonal group In mathematics, the orthogonal group in dimension , denoted , is the group of distance-preserving transformations of a Euclidean space of dimension that preserve a fixed point, where the group operation is given by composing transformations. ...
),
differential geometry Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and mult ...
( Riemannian metric, second fundamental form), differential topology ( intersection forms of four-manifolds), and Lie theory (the Killing form). Quadratic forms are not to be confused with a
quadratic equation In algebra, a quadratic equation () is any equation that can be rearranged in standard form as ax^2 + bx + c = 0\,, where represents an unknown value, and , , and represent known numbers, where . (If and then the equation is linear, not qu ...
, 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 homogeneous polynomials.

# Introduction

Quadratic forms are homogeneous quadratic polynomials in ''n'' variables. In the cases of one, two, and three variables they are called unary, binary, and ternary and have the following explicit form: :$\begin q\left(x\right) &= ax^2&&\textrm \\ q\left(x,y\right) &= ax^2 + bxy + cy^2&&\textrm \\ q\left(x,y,z\right) &= ax^2 + bxy + cy^2 + dyz + ez^2 + fxz&&\textrm \end$ where ''a'', …, ''f'' are the coefficients. The notation $\langle a_1, \ldots, a_n\rangle$ is often used for the quadratic form : $q\left(x\right) = a_1 x_1^2 + a_2 x_2^2 + \cdots + a_n x_n^2.$ 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 real or
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...
s,
rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ra ...
s, or
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
s. In
linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrice ...
,
analytic geometry In classical mathematics, analytic geometry, also known as coordinate geometry or Cartesian geometry, is the study of geometry using a coordinate system. This contrasts with synthetic geometry. Analytic geometry is used in physics and enginee ...
, 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 field. In the arithmetic theory of quadratic forms, the coefficients belong to a fixed commutative ring, frequently the integers Z or the ''p''-adic integers Z''p''. Binary quadratic forms have been extensively studied in
number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Ma ...
, in particular, in the theory of quadratic fields, continued fractions, and modular forms. The theory of integral quadratic forms in ''n'' variables has important applications to
algebraic topology Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify ...
. Using
homogeneous coordinates In mathematics, homogeneous coordinates or projective coordinates, introduced by August Ferdinand Möbius in his 1827 work , are a system of coordinates used in projective geometry, just as Cartesian coordinates are used in Euclidean geometr ...
, a non-zero quadratic form in ''n'' variables defines an (''n''−2)-dimensional quadric in the (''n''−1)-dimensional projective space. This is a basic construction in projective geometry. In this way one may visualize 3-dimensional real quadratic forms as conic sections. An example is given by the three-dimensional
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidea ...
and the square of the Euclidean norm expressing the
distance Distance is a numerical or occasionally qualitative measurement of how far apart objects or points are. In physics or everyday usage, distance may refer to a physical length or an estimation based on other criteria (e.g. "two counties over"). ...
between a point with coordinates and the origin: : $q\left(x,y,z\right) = d\left(\left(x,y,z\right), \left(0,0,0\right)\right)^2 = \, \left(x,y,z\right)\, ^2 = x^2 + y^2 + z^2.$ A closely related notion with geometric overtones is a quadratic space, which is a pair , with ''V'' a
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
over a field ''K'', and a quadratic form on ''V''. See below for the definition of a quadratic form on a vector space.

# History

The study of particular 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 Fermat's theorem on sums of two squares, which determines when an integer may be expressed in the form , where ''x'', ''y'' are integers. This problem is related to the problem of finding Pythagorean triples, which appeared in the second millennium B.C. In 628, the Indian mathematician Brahmagupta wrote '' Brāhmasphuṭasiddhānta'', which includes, among many other things, a study of equations of the form . In particular he considered what is now called Pell's equation, , and found a method for its solution. In Europe this problem was studied by Brouncker, Euler and Lagrange. In 1801 Gauss published '' Disquisitiones Arithmeticae,'' a major portion of which was devoted to a complete theory of binary quadratic forms over the
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
s. Since then, the concept has been generalized, and the connections with quadratic number fields, the modular group, and other areas of mathematics have been further elucidated.

# Associated symmetric matrix

Any matrix determines a quadratic form in variables by : $q_A\left(x_1,\ldots,x_n\right) = \sum_^\sum_^a_ = \mathbf x^\mathrm A \mathbf x,$ where $A = \left(a_\right)$.

## Example

Consider the case of quadratic forms in three variables $x, y, z.$ The matrix has the form :$A=\begin a&b&c\\d&e&f\\g&h&k \end.$ The above formula gives :$q_A\left(x,y,z\right)=ax^2 + ey^2 +kz^2 + \left(b+d\right)xy + \left(c+g\right)xz + \left(f+h\right)yz.$ 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 $b+d, c+g$ and $f+h.$ In particular, the quadratic form $q_A$ is defined by a unique symmetric matrix :$A=\begin a&\frac2&\frac2\\\frac2&e&\frac2\\\frac2&\frac2&k \end.$ This generalizes to any number of variables as follows.

## General case

Given a quadratic form $q_A,$ defined by the matrix $A=\left\left(a_\right\right),$ the matrix $B = \left(\frac 2\right)$ is symmetric, defines the same quadratic form as , and is the unique symmetric matrix that defines $q_A.$ So, over the real numbers (and, more generally, over a field of characteristic different from two), there is a one-to-one correspondence between quadratic forms and symmetric matrices that determine them.

A fundamental question is the classification of the real quadratic form under linear change of variables.
Jacobi Jacobi may refer to: * People with the surname Jacobi Mathematics: * Jacobi sum, a type of character sum * Jacobi method, a method for determining the solutions of a diagonally dominant system of linear equations * Jacobi eigenvalue algorithm, ...
proved that, for every real quadratic form, there is an orthogonal diagonalization, that is an orthogonal change of variables that puts the quadratic form in a "diagonal form" : $\lambda_1 \tilde x_1^2 + \lambda_2 \tilde x_2^2 + \cdots + \lambda_n \tilde x_n^2,$ where the associated symmetric matrix is diagonal. Moreover, the coefficients are determined uniquely up to a permutation. If the change of variables is given by an invertible matrix, that is not necessarily orthogonal, one can suppose that all coefficients are 0, 1, or −1. Sylvester's law of inertia states that the numbers of each 1 and −1 are invariants of the quadratic form, in the sense that any other diagonalization will contain the same number of each. The signature of the quadratic form is the triple , where ''n''0 is the number of 0s and ''n''± is the number of ±1s. Sylvester's law of inertia shows that this is a well-defined quantity attached to the quadratic form. The case when all ''λ''''i'' have the same sign is especially important: in this case the quadratic form is called
positive definite 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: * Positive-definite bilinear form * Positive-definite fu ...
(all 1) or negative definite (all −1). If none of the terms are 0, then the form is called ; this includes positive definite, negative definite, and indefinite (a mix of 1 and −1); equivalently, a nondegenerate quadratic form is one whose associated symmetric form is a nondegenerate ''bilinear'' form. A real vector space with an indefinite nondegenerate quadratic form of index (denoting ''p'' 1s and ''q'' −1s) is often denoted as R''p'',''q'' particularly in the physical theory of
spacetime In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differ ...
. The discriminant of a quadratic form, concretely the class of the determinant of a representing matrix in ''K''/(''K''×)2 (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, $\left(-1\right)^.$ These results are reformulated in a different way below. Let ''q'' be a quadratic form defined on an ''n''-dimensional real vector space. Let ''A'' be the matrix of the quadratic form ''q'' in a given basis. This means that ''A'' is a symmetric matrix such that : $q\left(v\right)=x^\mathrm Ax,$ where ''x'' is the column vector of coordinates of ''v'' in the chosen basis. Under a change of basis, the column ''x'' is multiplied on the left by an invertible matrix ''S'', and the symmetric square matrix ''A'' is transformed into another symmetric square matrix ''B'' of the same size according to the formula : $A\to B=S^\mathrmAS.$ Any symmetric matrix ''A'' can be transformed into a diagonal matrix : $B=\begin \lambda_1 & 0 & \cdots & 0\\ 0 & \lambda_2 & \cdots & 0\\ \vdots & \vdots & \ddots & 0\\ 0 & 0 & \cdots & \lambda_n \end$ by a suitable choice of an orthogonal matrix ''S'', and the diagonal entries of ''B'' are uniquely determined – this is Jacobi's theorem. If ''S'' is allowed to be any invertible matrix then ''B'' can be made to have only 0,1, and −1 on the diagonal, and the number of the entries of each type (''n''0 for 0, ''n''+ for 1, and ''n'' for −1) depends only on ''A''. This is one of the formulations of Sylvester's law of inertia and the numbers ''n''+ and ''n'' are called the positive and negative indices of inertia. Although their definition involved a choice of basis and consideration of the corresponding real symmetric matrix ''A'', Sylvester's law of inertia means that they are invariants of the quadratic form ''q''. The quadratic form ''q'' is positive definite (resp., negative definite) if (resp., ) for every nonzero vector ''v''. When ''q''(''v'') assumes both positive and negative values, ''q'' is an indefinite quadratic form. The theorems of Jacobi and Sylvester show that any positive definite quadratic form in ''n'' variables can be brought to the sum of ''n'' squares by a suitable invertible linear transformation: geometrically, there is only ''one'' positive definite real quadratic form of every dimension. Its isometry group is a ''
compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in Britis ...
'' orthogonal group O(''n''). This stands in contrast with the case of indefinite forms, when the corresponding group, the indefinite orthogonal group O(''p'', ''q''), is non-compact. Further, the isometry groups of ''Q'' and −''Q'' are the same (, but the associated Clifford algebras (and hence pin groups) are different.

# Definitions

A quadratic form over a field ''K'' is a map $q: V \to K$ from a finite-dimensional ''K''-vector space to ''K'' such that $q\left(av\right) = a^2q\left(v\right)$ for all $a \in K, v \in V$ and the function $q\left(u+v\right) - q\left(u\right) - q\left(v\right)$ is bilinear. More concretely, an ''n''-ary quadratic form over a field ''K'' is a
homogeneous polynomial In mathematics, a homogeneous polynomial, sometimes called quantic in older texts, is a polynomial whose nonzero terms all have the same degree. For example, x^5 + 2 x^3 y^2 + 9 x y^4 is a homogeneous polynomial of degree 5, in two variables; ...
of degree 2 in ''n'' variables with coefficients in ''K'': : $q\left(x_1,\ldots,x_n\right) = \sum_^\sum_^a_, \quad a_\in K.$ This formula may be rewritten using matrices: let ''x'' be the column vector with components ''x''1, ..., ''x''''n'' and be the ''n''×''n'' matrix over ''K'' whose entries are the coefficients of ''q''. Then : $q\left(x\right)=x^\mathrmAx.$ A vector $v = \left(x_1,\ldots,x_n\right)$ is a null vector if ''q''(''v'') = 0. Two ''n''-ary quadratic forms ''φ'' and ''ψ'' over ''K'' are equivalent if there exists a nonsingular linear transformation such that : $\psi\left(x\right)=\varphi\left(Cx\right).$ Let the characteristic of ''K'' be different from 2. The coefficient matrix ''A'' of ''q'' may be replaced by the symmetric matrix with the same quadratic form, so it may be assumed from the outset that ''A'' is symmetric. Moreover, a symmetric matrix ''A'' is uniquely determined by the corresponding quadratic form. Under an equivalence ''C'', the symmetric matrix ''A'' of ''φ'' and the symmetric matrix ''B'' of ''ψ'' are related as follows: : $B=C^\mathrmAC.$ The associated bilinear form of a quadratic form ''q'' is defined by : $b_q\left(x,y\right)=\tfrac\left(q\left(x+y\right)-q\left(x\right)-q\left(y\right)\right) = x^\mathrmAy = y^\mathrmAx.$ Thus, ''b''''q'' is a symmetric bilinear form over ''K'' with matrix ''A''. Conversely, any symmetric bilinear form ''b'' defines a quadratic form : $q\left(x\right)=b\left(x,x\right),$ 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 ''n'' variables are essentially the same.

Given an ''n''-dimensional
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
''V'' over a field ''K'', a ''quadratic form'' on ''V'' is a function $Q:V\to K$ that has the following property: for some basis, the function ''q'' that maps the coordinates of $v\in V$ to $Q\left(v\right)$ is a quadratic form. In particular, if $V=K^n$ with its
standard basis In mathematics, the standard basis (also called natural basis or canonical basis) of a coordinate vector space (such as \mathbb^n or \mathbb^n) is the set of vectors whose components are all zero, except one that equals 1. For example, in the ...
, one has : The change of basis formulas show that the property of being a quadratic form does not depend on the choice of a specific basis in ''V'', although the quadratic form ''q'' depends on the choice of the basis. A finite-dimensional vector space with a quadratic form is called a quadratic space. The map ''Q'' is a
homogeneous function In mathematics, a homogeneous function is a function of several variables such that, if all its arguments are multiplied by a scalar, then its value is multiplied by some power of this scalar, called the degree of homogeneity, or simply the ''d ...
of degree 2, which means that it has the property that, for all ''a'' in ''K'' and ''v'' in ''V'': :$Q\left(av\right) = a^2 Q\left(v\right).$ When the characteristic of ''K'' is not 2, the bilinear map over ''K'' is defined: :$B\left(v,w\right)= \tfrac\left(Q\left(v+w\right)-Q\left(v\right)-Q\left(w\right)\right).$ This bilinear form ''B'' is symmetric. That is, for all ''x'', ''y'' in ''V'', and it determines ''Q'': for all ''x'' in ''V''. When the characteristic of ''K'' is 2, so that 2 is not a unit, it is still possible to use a quadratic form to define a symmetric bilinear form . However, ''Q''(''x'') can no longer be recovered from this ''B''′ in the same way, since for all ''x'' (and is thus alternating).This alternating form associated with a quadratic form in characteristic 2 is of interest related to the Arf invariant – . Alternatively, there always exists a bilinear form ''B''″ (not in general either unique or symmetric) such that . The pair consisting of a finite-dimensional vector space ''V'' over ''K'' and a quadratic map ''Q'' from ''V'' to ''K'' is called a quadratic space, and ''B'' as defined here is the associated symmetric bilinear form of ''Q''. The notion of a quadratic space is a coordinate-free version of the notion of quadratic form. Sometimes, ''Q'' is also called a quadratic form. Two ''n''-dimensional quadratic spaces and are isometric if there exists an invertible linear transformation (isometry) such that : $Q\left(v\right) = Q\text{'}\left(Tv\right) \text v\in V.$ The isometry classes of ''n''-dimensional quadratic spaces over ''K'' correspond to the equivalence classes of ''n''-ary quadratic forms over ''K''.

## Generalization

Let ''R'' be a commutative ring, ''M'' be an ''R''-
module Module, modular and modularity may refer to the concept of modularity. They may also refer to: Computing and engineering * Modular design, the engineering discipline of designing complex devices using separately designed sub-components * Modul ...
, and be an ''R''-bilinear form. A mapping is the ''associated quadratic form'' of ''b'', and is the ''polar form'' of ''q''. A quadratic form may be characterized in the following equivalent ways: *There exists an ''R''-bilinear form such that ''q''(''v'') is the associated quadratic form. * for all and , and the polar form of ''q'' is ''R''-bilinear.

## Related concepts

Two elements ''v'' and ''w'' of ''V'' are called orthogonal if . The kernel of a bilinear form ''B'' consists of the elements that are orthogonal to every element of ''V''. ''Q'' is non-singular if the kernel of its associated bilinear form is . If there exists a non-zero ''v'' in ''V'' such that , the quadratic form ''Q'' is
isotropic Isotropy is uniformity in all orientations; it is derived . Precise definitions depend on the subject area. Exceptions, or inequalities, are frequently indicated by the prefix ' or ', hence '' anisotropy''. ''Anisotropy'' is also used to describ ...
, otherwise it is anisotropic. This terminology also applies to vectors and subspaces of a quadratic space. If the restriction of ''Q'' to a subspace ''U'' of ''V'' is identically zero, then ''U'' is totally singular. The orthogonal group of a non-singular quadratic form ''Q'' is the group of the linear automorphisms of ''V'' that preserve ''Q'': that is, the group of isometries of into itself. If a quadratic space has a product so that ''A'' is an algebra over a field, and satisfies :$\forall x, y \isin A \quad Q\left(x y\right) = Q\left(x\right) Q\left(y\right) ,$ then it is a composition algebra.

# Equivalence of forms

Every quadratic form ''q'' in ''n'' variables over a field of characteristic not equal to 2 is equivalent to a diagonal form : $q\left(x\right)=a_1 x_1^2 + a_2 x_2^2+ \cdots +a_n x_n^2.$ Such a diagonal form is often denoted by $\langle a_1,\ldots,a_n\rangle.$ Classification of all quadratic forms up to equivalence can thus be reduced to the case of diagonal forms.

# Geometric meaning

Using
Cartesian coordinates A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in ...
in three dimensions, let $\mathbf = \left(x,y,z\right)^\text$, and let $A$ be a symmetric 3-by-3 matrix. Then the geometric nature of the
solution set In mathematics, a solution set is the set of values that satisfy a given set of equations or inequalities. For example, for a set of polynomials over a ring , the solution set is the subset of on which the polynomials all vanish (evaluate t ...
of the equation $\mathbf^\textA\mathbf+\mathbf^\text\mathbf=1$ depends on the eigenvalues of the matrix $A$. If all
eigenvalue In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denote ...
s of $A$ are non-zero, then the solution set is an
ellipsoid An ellipsoid is a surface that may be obtained from a sphere by deforming it by means of directional scalings, or more generally, of an affine transformation. An ellipsoid is a quadric surface;  that is, a surface that may be defined as th ...
or a
hyperboloid In geometry, a hyperboloid of revolution, sometimes called a circular hyperboloid, is the surface generated by rotating a hyperbola around one of its principal axes. A hyperboloid is the surface obtained from a hyperboloid of revolution by def ...
. If all the eigenvalues are positive, then it is an ellipsoid; if all the eigenvalues are negative, then it is an ''imaginary ellipsoid'' (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. If there exist one or more eigenvalues $\lambda_i = 0$, then the shape depends on the corresponding $b_i$. If the corresponding $b_i \neq 0$, then the solution set is a paraboloid (either elliptic or hyperbolic); if the corresponding $b_i = 0$, then the dimension $i$ degenerates and does not come into play, and the geometric meaning will be determined by other eigenvalues and other components of $\mathbf$. 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.

Quadratic forms over the ring of integers are called integral quadratic forms, whereas the corresponding modules are quadratic lattices (sometimes, simply lattices). They play an important role in
number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Ma ...
and
topology In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
. An integral quadratic form has integer coefficients, such as ; equivalently, given a lattice Λ in a vector space ''V'' (over a field with characteristic 0, such as Q or R), a quadratic form ''Q'' is integral ''with respect to'' Λ if and only if it is integer-valued on Λ, meaning if . This is the current use of the term; in the past it was sometimes used differently, as detailed below.

## Historical use

Historically there was some confusion and controversy over whether the notion of integral quadratic form should mean: ;''twos in'': the quadratic form associated to a symmetric matrix with integer coefficients ;''twos out'': a polynomial with integer coefficients (so the associated symmetric matrix may have half-integer coefficients off the diagonal) 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). In "twos in", binary quadratic forms are of the form $ax^2+2bxy+cy^2$, represented by the symmetric matrix :$\begina & b\\ b&c\end$ this is the convention Gauss uses in '' Disquisitiones Arithmeticae''. In "twos out", binary quadratic forms are of the form $ax^2+bxy+cy^2$, represented by the symmetric matrix :$\begina & b/2\\ b/2&c\end.$ Several points of view mean that ''twos out'' has been adopted as the standard convention. Those include: * better understanding of the 2-adic theory of quadratic forms, the 'local' source of the difficulty; * the lattice point of view, which was generally adopted by the experts in the arithmetic of quadratic forms during the 1950s; * the actual needs for integral quadratic form theory in
topology In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
for intersection theory; * the
Lie group In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the addi ...
and algebraic group aspects.

An integral quadratic form whose image consists of all the positive integers is sometimes called ''universal''. Lagrange's four-square theorem shows that $w^2+x^2+y^2+z^2$ is universal. Ramanujan generalized this to $aw^2+bx^2+cy^2+dz^2$ and found 54 multisets that can each generate all positive integers, namely, :, 1 ≤ ''d'' ≤ 7 :, 2 ≤ ''d'' ≤ 14 :, 3 ≤ ''d'' ≤ 6 :, 2 ≤ ''d'' ≤ 7 :, 3 ≤ ''d'' ≤ 10 :, 4 ≤ ''d'' ≤ 14 :, 6 ≤ ''d'' ≤ 10 There are also forms whose image consists of all but one of the positive integers. For example, has 15 as the exception. Recently, the 15 and 290 theorems 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 integers up through 15.

Cubic form In mathematics, a cubic form is a homogeneous polynomial of degree 3, and a cubic hypersurface is the zero set of a cubic form. In the case of a cubic form in three variables, the zero set is a cubic plane curve. In , Boris Delone and Dmitry ...
* Discriminant of a quadratic form * Hasse–Minkowski theorem * Quadric * Ramanujan's ternary quadratic form * Square class * Witt group * Witt's theorem

* * *