2015/02/26 by Thomas Reuss, Reuss, Thomas
Engineering · Mathematics · #Advanced Algebra and Geometry #Advanced Numerical Analysis Techniques #FOS: Mathematics #Fixed Point Theorems Analysis #Number Theory (math.NT) #math.NT
paper · pdf · doi:10.48550/arxiv.1502.07594
arxiv created 2015/02/26 · openalex publication_date 2015/02/26 · arxiv updated 2015/02/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Consider an irreducible bilinear form f(x1,x2;y1,y2) with integer coefficients. We derive an upper bound for the number of integer points (x,y)∈ℙ1×ℙ1 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(x1,x2;y1,y2;z1,z2) is an irreducible non-singular trilinear form defined on ℙ1× ℙ1×ℙ1, 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.