Counting points on bilinear and trilinear hypersurfaces

TL;DR

This paper establishes upper bounds for the number of integer solutions to bilinear and trilinear forms within boxes, with bounds improving as certain determinants increase, using elementary lattice techniques.

Contribution

It provides the first sharp bounds for integer points on bilinear and trilinear hypersurfaces, linking the bounds to determinants and hyperdeterminants.

Findings

Bound for bilinear forms decreases with larger determinant

Bound for trilinear forms improves with larger Cayley hyperdeterminant D

Methods based on elementary lattice results

Abstract

Consider an irreducible bilinear form $f(x_1,x_2;y_1,y_2)$ with integer coefficients. We derive an upper bound for the number of integer points $(\mathbf{x},\mathbf{y})\in\mathbb{P}^1\times\mathbb{P}^1$ inside a box satisfying the equation $f=0$. Our bound seems to be the best possible bound and the main term decreases with a larger determinant of the form $f$. We further discuss the case when $f(x_1,x_2;y_1,y_2;z_1,z_2)$ is an irreducible non-singular trilinear form defined on $\mathbb{P}^1\times \mathbb{P}^1\times\mathbb{P}^1$, with integer coefficients. In this case, we examine the singularity and reducibility conditions of $f$. To do this, we employ the Cayley hyperdeterminant $D$ associated to $f$. We then derive an upper bound for the number of integer points in boxes on such trilinear forms. The main term of the estimate improves with larger $D$. Our methods are based on elementary lattice results.