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

Abstract

Let $F(x,y)\in \mathbb{C}[x,y]$ be a polynomial of degree $d$ and let $G(x,y)\in \mathbb{C}[x,y]$ 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 $(a,b)\in {\mathbb{C}}^2$ with nonzero coordinates is no greater than $\frac{5}{2}{d}^2{t}^2$. 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.

v1: 18 pages; v2: minor revision, additional remarks and an example, 21 pages, 1 figure