Hermite reduction

Not to be confused with Hermitian form.

In the theory of quadratic forms, a Hermite reduction of a real positive definite form is another real positive definite form integrally equivalent to it whose coefficients are reasonably small in the sense defined below.

Definition

A positive definite form

$Q(x)=\sum _{i=1}^n\sum _{j=1}^n{Q}_{ij}{x}_i{x}_j$

on ${\mathbb{ℝ}}^n$ is Hermite reduced if the following recursively defined condition is satisfied.

• $0<|Q_{11}|\le (4/3{)}^{(n-1)/2}\sqrt[n]{\det Q}$
• $2|Q_{1i}|\le |Q_{11}|(i=2,\dots ,n)$
• The form $Q^{\prime }(x_2,\dots ,x_n)=Q_{11}Q(x)-(Q_{11}{x}_1+\cdots +Q_{1n}{x}_n{)}^2$ is a Hermite reduced form on ${\mathbb{ℝ}}^{n-1}$

For every positive definite form $Q$ on ${\mathbb{ℝ}}^n$, there exists a $\mathbb{\mathbb{Z}}$-module isomorphism $U:{\mathbb{\mathbb{Z}}}^n\to {\mathbb{\mathbb{Z}}}^n$ and a Hermite reduced form $\tilde{Q}$ on ${\mathbb{ℝ}}^n$ such that

$Q\circ (U{\otimes }_{\mathbb{\mathbb{Z}}}\mathbb{ℝ})=\tilde{Q}.$

In matrix notation, for every real $n\times n$ positive definite matrix $Q$, there exists an integer $n\times n$ invertible matrix $U$ (so-called unimodular matrix) and an $n\times n$ Hermite reduced matrix $\tilde{Q}$ such that

$U^{T}QU=\tilde{Q}.$

Then $\tilde{Q}$ is called a Hermite reduction of $Q$.

Each real positive definite form has only a finite number of Hermite reductions; they are not unique in general.

Application

The Hermite reduction of a binary or ternary positive definite form with integer coefficients with determinant 1 is simply the sum of squares. This is used in a proof of Legendre's three-square theorem: to show that an integer is a sum of squares of three integers it is sufficient to show that it can be represented by a ternary positive definite form with determinant 1.

Historical note

The Hermite reduction is named after Charles Hermite.

External links

• Hazewinkel, Michiel, ed. (2001), "Quadratic forms, reduction of", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=Main_Page