$S$-integral quadratic forms and homogeneous dynamics

TL;DR

This paper uses homogeneous dynamics to derive explicit, polynomial-bounded criteria for $S$-integral equivalence of quadratic forms and to find finite generating sets for $S$-integral orthogonal groups, extending prior results.

Contribution

It provides the first explicit polynomial bounds for $S$-integral quadratic forms and their orthogonal groups, generalizing classical results to the $S$-integral setting.

Findings

Established a criterion for $S$-integral equivalence with polynomial bounds.

Determined finite generating sets for $S$-integral orthogonal groups.

Extended results of Li and Margulis to the $S$-integral case.

Abstract

Let $S = \{ \infty \} \cup S_f$ be a finite set of places of $\mathbb{Q}$. Using homogeneous dynamics, we establish two new quantitative and explicit results about integral quadratic forms in three or more variables: The first is a criterion of $S$-integral equivalence. The second determines a finite generating set of any $S$-integral orthogonal group. Both theorems--which extend results of H. Li and G. Margulis for $S = \{ \infty\}$--are given by polynomial bounds on the size of the coefficients of the quadratic forms.