Computing the Binomial Part of a Polynomial Ideal

TL;DR

This paper introduces a comprehensive algorithm for computing the binomial part of polynomial ideals, integrating lattice computations, algebraic extensions, and cellular decompositions, with implementations in SageMath.

Contribution

It generalizes existing algorithms to compute the binomial part of any ideal over various fields, including non-perfect fields, and provides a complete step-by-step method.

Findings

Algorithms successfully compute binomial parts for various ideals.

Implementation in SageMath demonstrates practical applicability.

Method handles non-perfect fields and complex ideal decompositions.

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 ${\rm Bin}(I)$ 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 affine $K$-algebras, as well as their associated characters and lattice ideals. This allows us to calculate ${\rm Bin}(I)$ when $I$ is saturated with respect to the indeterminates by reducing the task to the 0-dimensional case. Finally, we treat the computation of ${\rm Bin}(I)$ 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.