Effective membership problem and systems of polynomial equations with multiple roots

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.