Hermite reduction

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
on is Hermite reduced if the following recursively defined condition is satisfied.

The form is a Hermite reduced form on

For every positive definite form on, there exists a -module isomorphism and a Hermite reduced form on such that
In matrix notation, for every real positive definite matrix, there exists an integer invertible matrix and an Hermite reduced matrix such that
Then is called a Hermite reduction of.
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.