Bivariate systems of polynomial equations with roots of high multiplicity

TL;DR

This paper investigates the possible multiplicities of solutions in bivariate polynomial systems with fixed supports, providing a classification of systems based on the maximum achievable solution multiplicities.

Contribution

It proves the existence of systems with solutions of all multiplicities within a specific range and classifies support pairs that cannot have solutions of multiplicity three.

Findings

Existence of systems with solutions of all multiplicities in a certain range.

Classification of support pairs without solutions of multiplicity 3.

Explicit bounds on solution multiplicities based on support sets.

Abstract

Given a bivariate system of polynomial equations with fixed support sets $A, B$ it is natural to ask which multiplicities its solutions can have. We prove that there exists a system with a solution of multiplicity $i$ for all $i$ in the range $\{0,1,...,|A|-|\mbox{conv}(A)\ominus B|-1\}$, where $A\ominus B$ is the set of all integral vectors that shift B to a subset of $A$. As an application of this result we classify all pairs $(A, B)$ such that the system supported at $(A, B)$ does not have a solution of multiplicity $3$.