Bihomogeneous forms in many variables

TL;DR

This paper develops a novel approach using the structure of bihomogeneous equations to count integer points on bihomogeneous varieties more efficiently, allowing for fewer variables and variable-sized regions.

Contribution

It introduces a new method leveraging bihomogeneous structure to improve asymptotic counts in fewer variables and for non-uniform variable regions.

Findings

Achieved asymptotic formulas with fewer variables than traditional methods

Extended counting to regions with variables of different sizes

Demonstrated the effectiveness of the bihomogeneous approach

Abstract

We count integer points on bihomogeneous varieties using the Hardy-Littlewood method. The main novelty lies in using the structure of bihomogeneous equations to obtain asymptotics in generically fewer variables than would be necessary in using the standard approach for homogeneous varieties. Also, we consider counting functions where not all the variables have to lie in intervals of the same size, which arises as a natural question in the setting of bihomogeneous varieties.