View source for Quadratic form

''over a commutative ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q0760801.png" /> with an identity''

A homogeneous polynomial

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in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q0760803.png" /> variables with coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q0760804.png" />. Usually <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q0760805.png" /> is the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q0760806.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q0760807.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q0760808.png" />, or else the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q0760809.png" />, the ring of integer elements of an algebraic number field or the ring of integers of the completion of an algebraic number field with respect to a non-Archimedian norm.

The symmetric square matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q07608010.png" /> of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q07608011.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q07608012.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q07608013.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q07608014.png" />, is called the Kronecker matrix of the quadratic form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q07608015.png" />; in Siegel's notation: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q07608016.png" />. If the [[Discriminant|discriminant]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q07608017.png" /> of the quadratic form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q07608018.png" /> is non-zero, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q07608019.png" /> is said to be a non-degenerate quadratic form, while if it is zero, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q07608020.png" /> is called degenerate.

A quadratic form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q07608021.png" /> is called Gaussian if it can be expressed in symmetric notation:

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that is, there exist <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q07608023.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q07608024.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q07608025.png" />. The symmetric square matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q07608026.png" /> is called the matrix (or Gaussian matrix) of the quadratic form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q07608027.png" />. The quantity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q07608028.png" /> is called the determinant of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q07608029.png" />; in this connection,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q07608030.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q07608031.png" /></td> </tr></table>

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q07608032.png" /> is a field of characteristic distinct from 2, then every quadratic form over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q07608033.png" /> is Gaussian. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q07608034.png" /> is imbeddable in a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q07608035.png" /> of characteristic distinct from 2, then a quadratic form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q07608036.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q07608037.png" /> can be regarded as Gaussian, but with matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q07608038.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q07608039.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q07608040.png" />.

Two quadratic forms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q07608041.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q07608042.png" /> are equivalent over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q07608043.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q07608044.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q07608045.png" />) if one can be obtained from the other by an invertible (with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q07608046.png" />) linear homogeneous change of variables, that is, if there exists an invertible square matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q07608047.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q07608048.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q07608049.png" />. The collection of quadratic forms over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q07608050.png" /> equivalent over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q07608051.png" /> to a given one is called the class of that quadratic form. The discriminant of the quadratic form is, up to the square of an invertible element in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q07608052.png" />, an invariant of the class.

Another way of looking at quadratic forms is the following. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q07608053.png" /> be a unital <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q07608054.png" />-module; a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q07608055.png" /> is called a quadratic mapping (or a quadratic form) on the module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q07608056.png" /> if 1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q07608057.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q07608058.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q07608059.png" />; and 2) the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q07608060.png" /> given by

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is a [[Bilinear form|bilinear form]] on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q07608062.png" />. The pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q07608063.png" /> is called a quadratic module. The form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q07608064.png" /> is always symmetric.

To each bilinear form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q07608065.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q07608066.png" /> corresponds the quadratic form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q07608067.png" />; here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q07608068.png" />.

If in the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q07608069.png" /> the element 2 has an inverse, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q07608070.png" /> is a one-to-one correspondence between the quadratic and symmetric forms on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q07608071.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q07608072.png" /> is a free <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q07608073.png" />-module of rank <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q07608074.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q07608075.png" /> is a quadratic form on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q07608076.png" />, then to each basis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q07608077.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q07608078.png" /> corresponds a quadratic form in the classical sense,

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where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q07608080.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q07608081.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q07608082.png" />. Every quadratic form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q07608083.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q07608084.png" /> is obtained in this way from some quadratic module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q07608085.png" />, and conversely. Under a change of basis the quadratic form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q07608086.png" /> is converted to an equivalent one.

An element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q07608087.png" /> is said to be representable by the quadratic form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q07608088.png" /> (or one says that the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q07608089.png" /> represents <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q07608090.png" />) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q07608091.png" /> is the value of this form for certain values of the variables. Equivalent quadratic forms represent the same elements. A quadratic form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q07608092.png" /> over an ordered field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q07608093.png" /> is called indefinite if it represents both positive and negative elements, and positive (negative) definite if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q07608094.png" /> (respectively <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q07608095.png" />) for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q07608096.png" />. A non-degenerate quadratic form which represents 0 non-trivially is called isotropic, otherwise it is called anisotropic. Similarly, a quadratic form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q07608097.png" /> is called representable by a quadratic form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q07608098.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q07608099.png" /> can be converted to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080100.png" /> by a substitution of certain linear forms in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080101.png" /> for the variables in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080102.png" />; that is, if there is a rectangular <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080103.png" /> matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080104.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080105.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080106.png" /> (there <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080107.png" /> denotes transposition).

==Algebraic theory of quadratic forms.==
This is the theory of quadratic forms over fields.

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080108.png" /> be an arbitrary field of characteristic distinct from 2. The problem of representing a form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080109.png" /> by a form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080110.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080111.png" /> reduces to the problem of equivalence of forms, because (Pall's theorem) in order that a non-degenerate quadratic form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080112.png" /> be representable by a non-degenerate quadratic form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080113.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080114.png" />, it is necessary and sufficient that there exist a form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080115.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080116.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080117.png" /> are equivalent over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080118.png" />. Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080119.png" /> is the orthogonal direct sum of forms, that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080120.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080121.png" /> have no variables in common.

Witt's cancellation theorem. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080122.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080123.png" />.

Every quadratic form over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080124.png" /> is equivalent to a diagonal one:

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It will be assumed that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080126.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080127.png" />. The number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080128.png" /> is called the rank of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080129.png" />, which is the same as the [[Rank|rank]] of the matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080130.png" />. If every element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080131.png" /> has a square root, then a quadratic form over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080132.png" /> is equivalent to the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080133.png" /> (the normal form of a quadratic form).

Every non-degenerate quadratic form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080134.png" /> is equivalent to a form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080135.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080136.png" /> is anisotropic; here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080137.png" /> uniquely determines <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080138.png" /> and the class of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080139.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080140.png" />, called the anisotropic kernel of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080141.png" /> (see also [[Witt decomposition|Witt decomposition]]). Two forms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080142.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080143.png" /> having the same anisotropic kernel are called similar in the sense of Witt. A ring structure can be defined on the classes of similar forms (see [[Witt ring|Witt ring]]).

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080144.png" /> be an ordered field (in particular the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080145.png" />) and suppose that every positive element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080146.png" /> is a square. Then every quadratic form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080147.png" /> is reducible to the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080148.png" /></td> </tr></table>

Here the numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080149.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080150.png" /> (the positive and negative indices of inertia) are uniquely determined by the form (see [[Law of inertia|Law of inertia]]). Thus, for these fields the problem of equivalence of quadratic forms is solved.

The problem of equivalence over the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080151.png" /> reduces to the analogous problem for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080152.png" />-adic number fields: In order that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080153.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080154.png" /> be equivalent over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080155.png" />, it is necessary and sufficient that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080156.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080157.png" /> be equivalent over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080158.png" /> for all primes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080159.png" /> and over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080160.png" /> (the Minkowski–Hasse theorem). A similar assertion holds for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080162.png" />-fields — algebraic number fields and fields of algebraic functions in one variable over a finite field of constants. This is a particular case of the [[Hasse principle|Hasse principle]]. In the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080163.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080164.png" />) the problem of equivalence is solved by means of the [[Hasse invariant|Hasse invariant]].

A quadratic form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080165.png" /> over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080166.png" /> is called multiplicative over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080167.png" /> if

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080168.png" /></td> </tr></table>

where the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080169.png" /> are rational functions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080170.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080171.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080172.png" />. If, in addition, the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080173.png" /> are bilinear functions, then the form is said to have a composition. Composition is possible only for the cases <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080174.png" /> (Hurwitz's theorem). There is a simple description of multiplicative forms [[#References|[16]]].

The algebraic theory of quadratic forms has been generalized [[#References|[7]]] to the case of a field of characteristic 2.

==Arithmetic theory of quadratic forms.==
This is the theory of quadratic forms over rings.

This theory arose in connection with problems of solving Diophantine equations of the second degree. The question of solving such equations reduces to the problem of representing integers by an integral quadratic form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080175.png" />, that is, the problem of solving the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080176.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080177.png" />. Algorithms are known which reduce the determination (description) of all solutions of this equation to the problem of equivalence of quadratic forms over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080178.png" />, that is, the problem of finding for given quadratic forms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080179.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080180.png" /> invertible matrices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080181.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080182.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080183.png" />. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080184.png" /> such algorithms were constructed by J.L. Lagrange and C.F. Gauss, who created the general theory of binary quadratic forms (cf. [[Binary quadratic form|Binary quadratic form]]). They were generalized to arbitrary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080185.png" /> by H. Smith and H. Minkowski.

One of the central problems of the arithmetic theory is that of finding simple criteria for the existence of representations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080186.png" /> of a form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080187.png" /> by a form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080188.png" />, that is, criteria for the solutions of the matrix equation

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080189.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>

and also the problem of constructing a formula for the number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080190.png" /> of such representations. In this connection, if the number of representations is infinite, then it becomes a question of the number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080191.png" /> of "essentially different" representations. (Two representations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080192.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080193.png" /> are identified if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080194.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080195.png" /> is an integral automorphism of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080196.png" />, that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080197.png" />.) A necessary condition for the existence of representations is the solvability of (1) over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080198.png" /> and the solvability over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080199.png" /> of the matrix congruence

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080200.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>

for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080201.png" />. (For the solvability of all congruences (2) it is sufficient that (2) be solvable for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080202.png" />.) These necessary conditions, called "generic conditions for the solvability of a matrix equationgeneric" , are equivalent to the solvability of (1) over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080203.png" /> for every prime <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080204.png" /> and over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080205.png" />. They are also equivalent to the solvability of (1) over the field of rationals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080206.png" /> "without an essential denominator" , that is, the existence of a rational solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080207.png" /> with common denominator coprime to any pre-assigned integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080208.png" /> (it being sufficient to confine oneself to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080209.png" />). The solvability conditions of (2) can be expressed in terms of generic invariants of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080210.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080211.png" />. The number of solutions of (2) is found by means of Gauss sums.

The genus of a quadratic form over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080212.png" /> is the set of quadratic forms over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080213.png" /> equivalent to one another over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080214.png" /> for all primes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080215.png" />, including <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080216.png" />. The genus of a quadratic form consists of a finite number of classes with the same discriminant. The genus of a quadratic form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080217.png" /> can be given by a finite number of generic invariants — order invariants expressed in terms of the elementary divisors of A — and characters of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080218.png" />. The genus can also be given by the values of Gauss sums. An important role in the theory of quadratic forms is also played by the notion of a spinor genus, a more delicate notion than that of a genus.

The number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080219.png" /> of essentially different representations of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080220.png" /> by the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080221.png" /> is in a simple way related to the number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080222.png" /> of essentially different primitive representations, that is, representations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080223.png" /> such that the greatest common divisor of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080224.png" />-th order minors (cf. [[Minor|Minor]]) of the matrix is 1. For the quantity

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080225.png" /></td> </tr></table>

(the averaging function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080226.png" /> over the genus of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080227.png" />), where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080228.png" /> are representatives of all classes of the genus of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080229.png" /> (one from each class), there are formulas (see [[#References|[11]]], [[#References|[15]]]) expressing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080230.png" /> in terms of the number of solutions of certain congruences. In case the genus of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080231.png" /> consists of a single class, these formulas completely solve the question of the number of representations. In the case of genera having several classes, only asymptotic formulas are known for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080232.png" />, as well as "precise" formulas for certain concrete quadratic forms.

==Analytic theory of quadratic forms.==
Analytic methods were brought into the theory of quadratic forms by P.G.L. Dirichlet. Developing these methods, C.L. Siegel arrived at general formulas for the number of representations of a form by genera of forms.

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080233.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080234.png" /> be positive-definite quadratic forms over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080235.png" />.

The number

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080236.png" /></td> </tr></table>

is called the Siegel mean with respect to the genus for the number of representations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080237.png" /> of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080238.png" /> by the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080239.png" />. Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080240.png" /> is the number of automorphisms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080241.png" /> and

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080242.png" /></td> </tr></table>

is the weight of the genus of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080243.png" />. Let

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080244.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080245.png" /> is a neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080246.png" /> in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080247.png" />-dimensional space of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080248.png" />-ary quadratic forms over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080249.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080250.png" /> is the corresponding domain of solutions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080251.png" /> of the matrix equation (1) over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080252.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080253.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080254.png" /> are their volumes.

Siegel's formula for the quadratic forms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080255.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080256.png" /> is

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080257.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080258.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080259.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080260.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080261.png" /> otherwise. Here

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080262.png" /></td> </tr></table>

where the limit is taken over those sequences of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080263.png" />'s for which any natural number is a divisor of almost-all terms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080264.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080265.png" /> is the number of distinct prime divisors of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080266.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080267.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080268.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080269.png" /> is the number of representations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080270.png" /> by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080271.png" />, that is, the number of solutions of the matrix congruence

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080272.png" /></td> </tr></table>

The following formula holds:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080273.png" /></td> </tr></table>

There are a number of equivalent definitions for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080274.png" /> and an expression (see [[#References|[17]]]) in terms of generalized Gauss sums. Formula (3) includes as a special case Minkowski's formula for the weight of the genus:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080275.png" /></td> </tr></table>

For the case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080276.png" /> the latter gives the Dirichlet formula for the number of classes.

Formulas analogous to (3) hold also for indefinite forms and forms with integral algebraic coefficients (see [[#References|[17]]], [[#References|[18]]]).

An application of the theory of the number of representations of numbers by positive quadratic forms in an even number of variables has been given by E. Hecke [[#References|[10]]]. The theory of modular forms (cf. [[Modular form|Modular form]]) enables one to obtain formulas for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080277.png" /> (see the survey [[#References|[5]]]).

The [[Circle method|circle method]] (see [[#References|[4]]]) has been applied to the question of the representation of numbers by quadratic forms in four or more variables. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080278.png" /> is a positive-definite quadratic form over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080279.png" />, then for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080280.png" /> an application of the circle method leads to the asymptotic formula

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080281.png" /></td> </tr></table>

Similar asymptotic formulas can also be obtained by the circle method for indefinite quadratic forms with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080282.png" />.

For the investigation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080283.png" /> in the case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080284.png" />, Linnik's discrete ergodic method (cf. [[Linnik discrete ergodic method|Linnik discrete ergodic method]]) has been applied (see [[#References|[3]]], [[#References|[4]]]). It is based on the fact that on some set of representations of numbers by ternary quadratic forms, an ergodic flow of the representations can be constructed which is regulated by an operator connected with the problem of the representation of numbers by quaternary quadratic forms. The ergodic method leads (under certain necessary conditions) to an estimate of the type

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080285.png" /></td> </tr></table>

and in a number of cases asymptotic formulas have also been obtained.

Recent results of H. Iwaniec [[#References|[22]]] concerning means of Kloosterman sums lead to the asymptotic formula [[#References|[23]]]

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080286.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080287.png" /> is square-free (or where the square-free part of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080288.png" /> is bounded). This formula is non-trivial already for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080289.png" />. This formula provides an approach to a conjecture of H. Petersson. (Petersson's conjecture has already been proved by P. Deligne for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080290.png" /> even, see [[#References|[24]]].)

==Geometric theory of quadratic forms.==
For the study of such questions in the theory of quadratic forms as reduction theory, automorphisms and arithmetic minima of quadratic forms, the method of continuous parameters was developed by Ch. Hermite, which thereupon became an extensive branch of the theory of quadratic forms, namely the geometric theory of quadratic forms, or the geometry of quadratic forms (which can also be regarded as part of the [[Geometry of numbers|geometry of numbers]]). The idea of the method consists in the following. With a given <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080291.png" />-dimensional point lattice <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080292.png" /> some kind of arithmetical quantity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080293.png" /> is associated and the behaviour of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080294.png" /> is considered under small changes of the parameters of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080295.png" />. A characteristic feature of the geometry of quadratic forms is the systematic use of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080296.png" />-dimensional coefficient (parameter) space, in which the lattice <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080297.png" /> is represented by a point. Let

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080298.png" /></td> </tr></table>

be a quadratic form with real coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080299.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080300.png" />). Associated with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080301.png" /> is the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080302.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080303.png" />-dimensional Euclidean space (where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080304.png" />), called the coefficient space. Then corresponding to the positive-definite form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080305.png" /> is an open convex cone <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080306.png" /> with vertex at the origin, called the cone of positivity. The lattice <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080307.png" /> is related to the class of equivalent <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080308.png" />-ary positive-definite quadratic forms; by means of a basis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080309.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080310.png" />, the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080311.png" /></td> </tr></table>

is associated with it. Thus, an infinite discrete set of points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080312.png" /> has been associated with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080313.png" />. If one chooses the correct domain of reduction <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080314.png" /> of positive-definite quadratic forms, then each lattice is uniquely associated with a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080315.png" /> of the coefficient space. Small changes in the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080316.png" /> correspond to small changes in the parameters of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080317.png" />.

The geometric theory of quadratic forms divides up into a number of fairly independent theories related by a single method of investigation. At its foundation is the reduction theory of positive quadratic forms, which, by studying the domains of reduction <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080318.png" />, solves the problem of the equivalence of positive quadratic forms, one of the central problems in the arithmetical theory of quadratic forms (see [[Quadratic forms, reduction of|Quadratic forms, reduction of]]).

An essential role is played by the theory of [[Voronoi lattice types|Voronoi lattice types]]. It has important applications in the theory of parallelohedra. The theory of types has found application in the solution of problems on the most economic lattice covering of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080319.png" />-dimensional space by balls.

Another traditional branch of the geometric theory of quadratic forms is the theory of perfect forms, also created by G.F. Voronoi. This theory allows one to solve the [[Hermite problem|Hermite problem]] on the arithmetical minima of positive quadratic forms; this is equivalent to the problem on the densest lattice [[Packing|packing]] of balls in an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080320.png" />-dimensional space. The problem on the densest lattice packing of balls and that of the most economic lattice covering by balls are the best known examples of the extremal problems which constitute a significant part of the geometry of quadratic forms.

Also related to the geometric theory of quadratic forms are certain generalizations of the continued fractions algorithm; for example, the algorithm of Voronoi for calculating the units of a cubic field, and the theory of fundamental domains of automorphisms of indefinite quadratic forms.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> Z.I. Borevich, I.R. Shafarevich, "Number theory" , Acad. Press (1966) (Translated from Russian) (German translation: Birkhäuser, 1966) {{MR|0195803}} {{ZBL|0145.04902}} </TD></TR><TR><TD valign="top">[2a]</TD> <TD valign="top"> B.N. Delone, "The geometry of positive definite quadratic forms" ''Uspekhi Mat. Nauk'' , '''3''' (1937) pp. 16–62 (In Russian)</TD></TR><TR><TD valign="top">[2b]</TD> <TD valign="top"> B.N. Delone, "The geometry of positive definite quadratic forms" ''Uspekhi Mat. Nauk'' , '''4''' (1938) pp. 102–164 (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> Yu.V. Linnik, "Ergodic properties of algebraic fields" , Springer (1968) (Translated from Russian) {{MR|0238801}} {{ZBL|0162.06801}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> A.V. Malyshev, "On the representation of integers by positive quadratic forms" , Moscow-Leningrad (1962) (In Russian) {{MR|}} {{ZBL|0425.10024}} {{ZBL|0163.04604}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> A.V. Malyshev, "On formulas for the representation of numbers by positive quadratic forms (problems)" , ''Current problems in analytic number theory'' , Minsk (1974) pp. 119–137 (In Russian)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> J.-P. Serre, "A course in arithmetic" , Springer (1973) (Translated from French) {{MR|0344216}} {{ZBL|0256.12001}} </TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top"> C. Arf, "Untersuchungen über quadratischen Formen in Körpern der Characteristik 2, I" ''J. Reine Angew. Math.'' , '''183''' (1941) pp. 148–167</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top"> M. Eichler, "Quadratische Formen und orthogonale Gruppen" , Springer (1952) {{MR|0051875}} {{ZBL|0049.31106}} </TD></TR><TR><TD valign="top">[9a]</TD> <TD valign="top"> H. Hasse, "Ueber die Darstellbarkeit von Zahlen durch quadratische Formen im Körper der rationalen Zahlen" ''J. Reine Angew. Math.'' , '''152''' (1923) pp. 129–148 {{MR|}} {{ZBL|49.0102.01}} </TD></TR><TR><TD valign="top">[9b]</TD> <TD valign="top"> H. Hasse, "Ueber die Aequivalenz quadratischer Formen im Körper der rationalen Zahlen" ''J. Reine Angew. Math.'' , '''152''' (1923) pp. 205–224 {{MR|}} {{ZBL|49.0102.02}} </TD></TR><TR><TD valign="top">[9c]</TD> <TD valign="top"> H. Hasse, "Symmetrische Matrizen im Körper der rationalen Zahlen" ''J. Reine Angew. Math.'' , '''153''' (1924) pp. 12–43 {{MR|}} {{ZBL|49.0104.01}} </TD></TR><TR><TD valign="top">[9d]</TD> <TD valign="top"> H. Hasse, "Zur Theorie des quadratischen Hilbertschen Normenrestsymbols in algebraischen Körper" ''J. Reine Angew. Math.'' , '''153''' (1924) pp. 76–93</TD></TR><TR><TD valign="top">[9e]</TD> <TD valign="top"> H. Hasse, "Darstellbarkeit von Zahlen durch quadratische Formen in einem beliebigen algebraischen Zahlkörper" ''J. Reine Angew. Math.'' , '''153''' (1924) pp. 113–130 {{MR|}} {{ZBL|49.0114.01}} </TD></TR><TR><TD valign="top">[9f]</TD> <TD valign="top"> H. Hasse, "Aequivalenz quadratischer Formen in einem beliebigen algebraischen Zahlkörper" ''J. Reine Angew. Math.'' , '''153''' (1924) pp. 158–162 {{MR|}} {{ZBL|50.0104.03}} </TD></TR><TR><TD valign="top">[9g]</TD> <TD valign="top"> H. Hasse, "Zur Theorie des Hilbertschen Normenrestsymbols in algebraischen Zahlkörpern" ''J. Reine Angew. Math.'' , '''153''' (1924) pp. 184–191 {{MR|}} {{ZBL|50.0105.01}} </TD></TR><TR><TD valign="top">[9h]</TD> <TD valign="top"> H. Hasse, "Das allgemeine Reziprocitätsgesetz und seine Ergänzungssätze in beliebigen Zahlkörpern für gewisse nicht-primäre Zahlen" ''J. Reine Angew. Math.'' , '''153''' (1924) pp. 192–207</TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top"> E. Hecke, "Mathematische Werke" , Vandenhoeck &amp; Ruprecht (1959) {{MR|0104550}} {{ZBL|0092.00102}} </TD></TR><TR><TD valign="top">[11]</TD> <TD valign="top"> B.W. Jones, "The arithmetic theory of quadratic forms" , Math. Assoc. Amer. (1950) {{MR|0037321}} {{ZBL|0041.17505}} </TD></TR><TR><TD valign="top">[12]</TD> <TD valign="top"> T.Y. Lam, "The algebraic theory of quadratic forms" , Benjamin (1973) {{MR|0396410}} {{ZBL|0259.10019}} </TD></TR><TR><TD valign="top">[13]</TD> <TD valign="top"> H. Minkowski, "Gesammelte Abhandlungen" , '''1–2''' , Teubner (1911) {{MR|}} {{ZBL|42.0023.03}} </TD></TR><TR><TD valign="top">[14]</TD> <TD valign="top"> O.T. O'Meara, "Introduction to quadratic forms" , Springer (1963) {{MR|}} {{ZBL|0107.03301}} </TD></TR><TR><TD valign="top">[15]</TD> <TD valign="top"> G. Pall, "Representation by quadratic forms" ''Canad. J. Math.'' , '''1''' (1949) pp. 344–364 {{MR|0034795}} {{ZBL|0034.02201}} </TD></TR><TR><TD valign="top">[16]</TD> <TD valign="top"> A. Pfister, "Multiplikative quadratische Formen" ''Arch. Math.'' , '''16''' (1965) pp. 363–370 {{MR|0184937}} {{ZBL|0146.26001}} </TD></TR><TR><TD valign="top">[17]</TD> <TD valign="top"> C.L. Siegel, "Lectures on quadratic forms" , Tata Inst. (1963) {{MR|0271028}} {{ZBL|0115.04401}} </TD></TR><TR><TD valign="top">[18]</TD> <TD valign="top"> C.L. Siegel, "Gesammelte Abhandlungen" , '''1–4''' , Springer (1966–1979) {{MR|0543842}} {{MR|0197270}} {{ZBL|0402.01011}} {{ZBL|0143.00101}} </TD></TR><TR><TD valign="top">[19]</TD> <TD valign="top"> H.J.S. Smith, "The collected mathematical papers" , '''1–2''' , Chelsea, reprint (1965–1979) {{MR|}} {{ZBL|25.0029.02}} </TD></TR><TR><TD valign="top">[20]</TD> <TD valign="top"> G.L. Watson, "Integral quadratic forms" , Cambridge Univ. Press (1960) {{MR|0118704}} {{ZBL|0090.03103}} </TD></TR><TR><TD valign="top">[21]</TD> <TD valign="top"> O.M. Fomenko, "Applications of the theory of modular forms to number theory" ''J. Soviet Math.'' , '''14''' : 4 (1980) pp. 1307–1362 ''Itogi Nauk. i Tekhn. Algebra Topol. Geom.'' , '''15''' (1977) pp. 5–91 {{MR|0491505}} {{ZBL|0446.10021}} </TD></TR><TR><TD valign="top">[22]</TD> <TD valign="top"> H. Iwaniec, "Fourier coefficients of modular forms of half-integral weight" ''Invent. Math.'' , '''87''' (1987) pp. 385–401 {{MR|0870736}} {{ZBL|0606.10017}} </TD></TR><TR><TD valign="top">[23]</TD> <TD valign="top"> W. Duke, ''Sém. Théorie des Nombres de Bordeaux'' , '''37''' (1987–1988) pp. 1–7</TD></TR><TR><TD valign="top">[24]</TD> <TD valign="top"> E. Freitag, R. Kiehl, "Etale cohomology and the Weil conjecture" , Springer (1988) (Translated from German) {{MR|0926276}} {{ZBL|0643.14012}} </TD></TR></table>



====Comments====
Recent accounts of the geometric theory of quadratic forms are contained in [[#References|[a4]]]. There Dirichlet–Voronoi cells, parallelohedra, reduction theory, and relations to the problem of densest lattice packing and of thinnest lattice covering with Euclidean balls are discussed in detail.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> B.N. Delone, S.S. Ryshkov, "Extremal problems in the theory of positive quadratic forms" ''Proc. Steklov Inst. Math.'' , '''112''' (1971) pp. 211–231 ''Trudy Mat. Inst. Steklov.'' , '''112''' (1971) pp. 203–223 {{MR|340183}} {{ZBL|0261.10020}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> P. Erdös, P.M. Gruber, J. Hammer, "Lattice points" , Longman (1989) {{MR|1003606}} {{ZBL|0683.10025}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> P.M. Gruber, C.G. Lekkerkerker, "Geometry of numbers" , North-Holland (1987) pp. Sect. (iv) (Updated reprint) {{MR|0893813}} {{ZBL|0611.10017}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> C.L. Siegel, "Lectures on the geometry of numbers" , Springer (1989) {{MR|1020761}} {{ZBL|0691.10021}} </TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> S.S. Ryshkov, E.P. Baranovskii, "<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080321.png" />-types of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076080/q076080322.png" />-dimensional lattices and 5-dimensional primitive parallelohedra" , Amer. Math. Soc. (1978) (Translated from Russian)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top"> J.W.S. Cassels, "Rational quadratic forms" , Acad. Press (1978) {{MR|0522835}} {{ZBL|0395.10029}} </TD></TR></table>