Computing the binomial part of a polynomial ideal

Abstract

Given an ideal I in a polynomial ring $K[x_1,\dots ,x_n]$ over a field K, we present a complete algorithm to compute the binomial part of I, i.e., the subideal $\mathrm{Bin}\left(I\right)$ of I generated by all monomials and binomials in I. This is achieved step-by-step. First we collect and extend several algorithms for computing exponent lattices in different kinds of fields. Then we generalize them to compute exponent lattices of units in 0-dimensional K-algebras, where we have to generalize the computation of the separable part of an algebra to non-perfect fields in characteristic p. Next we examine the computation of unit lattices in finitely generated K-algebras, as well as their associated characters and lattice ideals. This allows us to calculate $\mathrm{Bin}\left(I\right)$ when I is saturated with respect to the indeterminates by reducing the task to the 0-dimensional case. Finally, we treat the computation of $\mathrm{Bin}\left(I\right)$ for general ideals by computing their cellular decomposition and dealing with finitely many special ideals called $(s,t)$-binomial parts. All algorithms have been implemented in SageMath.