Finding binomials in polynomial ideals

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.

11 pages, v2: final version, to appear in Res. Math. Sci