Distinct Values Of Bilinear Forms On Algebraic Curves

Let B-M : C x C -> C be a bilinear form B-M(p, q) - p(T)Mq, with an invertible matrix M is an element of C-2x2. We prove that any finite set S contained in an irreducible algebraic curve C of degree d in C determines Omega(d)(vertical bar S vertical bar(4/3)) distinct values of B-M, unless C has an exceptional form. This strengthens a result of Charalambides [1] in several ways. The proof is based on that of Pach and De Zeeuw [9], who proved a similar statement for the Euclidean distance function in R. Our main motivation for this paper is that for bilinear forms, this approach becomes more natural, and should better lend itself to understanding and generalization.