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 (so-called unimodular 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.
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