Finding binomials in polynomial ideals

TL;DR

This paper presents an algorithm to detect binomials in polynomial ideals, utilizing tropical geometry and computational number theory, addressing the challenge of hidden binomials in complex algebraic structures.

Contribution

It introduces a novel algorithm that determines the existence of binomials in polynomial ideals, combining tropical geometry reduction with number theory techniques.

Findings

Algorithm effectively detects binomials in polynomial ideals.

Decides the existence of binomials in ideals.

Addresses the difficulty of hidden binomials in algebraic structures.

Abstract

We describe an algorithm which finds binomials in a given ideal $I\subset\mathbb{Q}[x_1,\dots,x_n]$ and in particular decides whether binomials exist in $I$ at all. Binomials in polynomial ideals can be well hidden. For example, the lowest degree of a binomial cannot be bounded as a function of the number of indeterminates, the degree of the generators, or the Castelnuovo--Mumford regularity. We approach the detection problem by reduction to the Artinian case using tropical geometry. The Artinian case is solved with algorithms from computational number theory.