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Let $B$ be a bilinear form on pairs of points in the complex plane, of theform $B(p,q) = p^TMq$, for an invertible $2\times2$ complex matrix $M$. Weprove that any finite set $S$ contained in an irreducible algebraic curve $C$of degree $d$ in $\mathbb{C}^2$ determines at least $c_d|S|^{4/3}$ distinctvalues of $B$, unless the curve $C$ has an exceptional form. This strengthens aresult of Charalambides in several ways. The proof is based on that of Pach andDe Zeeuw, who proved a similar statement for the Euclidean distance function inthe real plane. Our main motivation for this paper is that for bilinear forms,this approach becomes more natural, and should better lend itself tounderstanding and generalization.

Distinct values of bilinear forms on algebraic curves