Effective membership problem and systems of polynomial equations with multiple roots

Abstract

For a tuple of $k+1$ convex polytopes $(A,B,\dots ,B)$ we solve the so-called effective membership problem, i.e. for a tuple $f=(f_1,\dots ,f_k)$ of polynomials satisfying some certain properties of generality and having Newton polytope $B$ we provide a method to compute ${\mathbb{C}}^A\cap I$, where $I$ is an ideal in the ring of Laurent polynomials generated by $f$. We connect this topic to the study of mixed discriminants and multiple solutions of systems of polynomial equations.

34 pages, the main result is generalized to arbitrary dimension. More minor mistakes fixed