Bivariate systems of polynomial equations with roots of high multiplicity

2019/10/28 by I. Nikitin, Nikitin, I.
Computer Science · Mathematics · #14H50 #14H55 #14M25 #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry (math.AG) #Coding theory and cryptography #Combinatorics (math.CO) #FOS: Mathematics #Polynomial and algebraic computation

paper · pdf · doi:10.48550/arxiv.1910.12541

openalex publication_date 2019/10/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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|-|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.

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