Effective membership problem and systems of polynomial equations with multiple roots

TL;DR

This paper introduces a method to solve the effective membership problem for polynomial systems with specific Newton polytopes, linking it to mixed discriminants and multiple roots of polynomial equations.

Contribution

It provides a novel computational approach for the effective membership problem in polynomial rings with applications to systems with multiple roots.

Findings

Method to compute intersection of Laurent polynomial ideals with polynomial spaces

Connection established between membership problem and mixed discriminants

Insights into solutions of polynomial systems with multiple roots

Abstract

For a tuple of $k+1$ convex polytopes $(A, B,\ldots, B)$ we solve the so-called effective membership problem, i.e. for a tuple $f=(f_1,\ldots, f_k)$ 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.