Intersection multiplicity of a sparse curve and a low-degree curve

Abstract

Let F(x,y)∈C[x,y] be a polynomial of degree d and let G(x,y)∈C[x,y] be a polynomial with t monomials. We want to estimate the maximal multiplicity of a solution of the system F(x,y)=G(x,y)=0. Our main result is that the multiplicity of any isolated solution (a,b)∈C2 with nonzero coordinates is no greater than 52d2t2. We ask whether this intersection multiplicity can be polynomially bounded in the number of monomials of F and G, and we briefly review some connections between sparse polynomials and algebraic complexity theory.