Hermite reduction

Not to be confused with Hermite normal 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.

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

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

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

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

• "Quadratic forms, reduction of", Encyclopedia of Mathematics, EMS Press, 2001 [1994]