Multilinear Exponential Sums and Collinear Triples

Abstract: 

We provide a background on some recent results on multilinear exponential sums as well as collinear triples over a finite field. We then focus on the specific cases of weighted trilinear and quadrilinear exponential sums. In studying such sums one needs strong bounds on the number of collinear triples. We will share some existing results and a new bound on the number of collinear triples and use these to give some new bounds on trilinear and quadrilinear exponential sums and then we will discuss some of the issues with trying to extend these results to general multilinear sums.