A multi-linear geometric estimate

Abstract

We give a generalization of the geometric estimate used by Hart and the second author in their 2008 work on sums and products in finite fields. Their result concerned level sets of non-degenerate bilinear forms over finite fields, while in this work we prove that if E ⊂ F d q is sufficiently large and ̟ is a non-degenerate multi-linear form then ̟ will attain all possible nonzero values as its arguments vary over E, under a certain quantitative assumption on the extent to which E is projective. We show that our bound is nontrivial in the case that n = 3 and d = 2 and construct examples of sets to which this applies. In particular, we give conditions under which every member of F * q