Incidence Results and Bounds of Trilinear and Quadrilinear Exponential Sums

TL;DR

This paper introduces improved bounds on collinear triples in finite fields and applies these results to derive tighter bounds on trilinear and quadrilinear exponential sums, advancing understanding in finite field combinatorics.

Contribution

It provides new bounds on collinear triples and uses them to improve bounds on exponential sums, surpassing previous methods relying on Cauchy inequality.

Findings

Enhanced bounds on collinear triples in finite fields

Tighter bounds on trilinear exponential sums

Tighter bounds on quadrilinear exponential sums

Abstract

We give a new bound on the number of collinear triples for two arbitrary subsets of a finite field. This improves on existing results which rely on the Cauchy inequality. We then us this to provide a new bound on trilinear and quadrilinear exponential sums.