Upper bounds for integer solutions to a system of two bilinear forms

TL;DR

This paper establishes upper bounds on the number of integer solutions to a system of two bilinear equations with at least 12 variables, showing these bounds are generally tight unless structural obstructions exist.

Contribution

It provides the first general upper bounds for solutions to such bilinear systems, clarifying when these bounds are nearly optimal.

Findings

Upper bounds match expected values up to logarithmic factors.

Bounds hold unless specific structural reasons prevent them.

Results apply to systems with at least 12 variables.

Abstract

We show that the number of integer solutions for a pair of bilinear equations in at least 2*6 variables has (up to logarithms) the expected upper bound unless there is a structural reason why it is not the case.