Small solutions of quadratic forms with congruence conditions

TL;DR

This paper proves the existence of small-height solutions to systems of quadratic forms with congruence conditions in three or more variables, including solutions avoiding certain hyperplanes.

Contribution

It establishes new bounds for integral solutions of quadratic systems with congruence constraints and hyperplane avoidance in higher dimensions.

Findings

Existence of two linearly independent integral solutions of bounded height.

Existence of small-height solutions avoiding specified hyperplanes.

Results applicable for systems with at least three variables.

Abstract

We consider a system of homogeneous quadratic forms with congruence conditions in $n\geq 3$ variables and prove the existence of two linearly independent integral solutions of bounded height. We also show the existence of small height integral zeros of this system avoiding a given set of hyperplanes.