Counting points on bilinear and trilinear hypersurfaces

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.