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

International audienceLet $F(x, y)∈C[x, y]$ be a polynomial of degreed 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 boundedin the number of monomials of F and G, and we briefly review someconnections between sparse polynomials and algebraic complexity the-ory