Editing Quadratic form (section)

== Integral quadratic forms ==
Quadratic forms over the ring of integers are called '''integral quadratic forms''', whereas the corresponding modules are '''quadratic lattices''' (sometimes, simply [[lattice (group)|lattice]]s). They play an important role in [[number theory]] and [[topology]].

An integral quadratic form has integer coefficients, such as {{math|''x''<sup>2</sup> + ''xy'' + ''y''<sup>2</sup>}}; equivalently, given a lattice {{math|Λ}} in a vector space {{math|''V''}} (over a field with characteristic 0, such as {{math|'''Q'''}} or {{math|'''R'''}}), a quadratic form {{math|''Q''}} is integral ''with respect to'' {{math|Λ}} if and only if it is integer-valued on {{math|Λ}}, meaning {{math|''Q''(''x'', ''y'') ∈ '''Z'''}} if {{math|''x'', ''y'' ∈ Λ}}.

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 {{math|''ax''<sup>2</sup> + 2''bxy'' + ''cy''<sup>2</sup>}}, represented by the symmetric matrix
<math display="block">\begin{pmatrix}a & b\\ b&c\end{pmatrix}</math>
This is the convention [[Gauss]] uses in ''[[Disquisitiones Arithmeticae]]''.

In "twos out", binary quadratic forms are of the form {{math|''ax''<sup>2</sup> + ''bxy'' + ''cy''<sup>2</sup>}}, represented by the symmetric matrix
<math display="block">\begin{pmatrix}a & b/2\\ b/2&c\end{pmatrix}.</math>

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 (group)|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]] for [[intersection theory]];
* the [[Lie group]] and [[algebraic group]] aspects.

=== Universal quadratic forms ===
An integral quadratic form whose image consists of all the positive integers is sometimes called ''universal''.  [[Lagrange's four-square theorem]] shows that {{math|''w''<sup>2</sup> + ''x''<sup>2</sup> + ''y''<sup>2</sup> + ''z''<sup>2</sup>}} is universal. [[Ramanujan]] generalized this {{math|''aw''<sup>2</sup> + ''bx''<sup>2</sup> + ''cy''<sup>2</sup> + ''dz''<sup>2</sup>}} and found 54 multisets {{math|{{mset|''a'', ''b'', ''c'', ''d''}}}} that can each generate all positive integers, namely,
{{plainlist | indent = 1 |
* {{math|{{mset|1, 1, 1, ''d''}}, 1 ≤ ''d'' ≤ 7}}
* {{math|{{mset|1, 1, 2, ''d''}}, 2 ≤ ''d'' ≤ 14}}
* {{math|{{mset|1, 1, 3, ''d''}}, 3 ≤ ''d'' ≤ 6}}
* {{math|{{mset|1, 2, 2, ''d''}}, 2 ≤ ''d'' ≤ 7}}
* {{math|{{mset|1, 2, 3, ''d''}}, 3 ≤ ''d'' ≤ 10}}
* {{math|{{mset|1, 2, 4, ''d''}}, 4 ≤ ''d'' ≤ 14}}
* {{math|{{mset|1, 2, 5, ''d''}}, 6 ≤ ''d'' ≤ 10}}
}}
There are also forms whose image consists of all but one of the positive integers.  For example, {{math|{{mset|1, 2, 5, 5}}}} 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.