Restricted Gröbner fans and re-embeddings of affine algebras

Abstract

In this paper we continue the study of good re-embeddings of affine K-algebras started in Kreuzer et al. (J Algebra Appl, 2021. https://doi.org/10.1142/S0219498822501882). The idea is to use special linear projections to find isomorphisms between a given affine K-algebra K[X]/I, where $X=(x_1,\dots ,x_n)$, and K-algebras having fewer generators. These projections are induced by particular tuples of indeterminates Z and by term orderings $\sigma$ which realize Z as leading terms of a tuple F of polynomials in I. In order to efficiently find such tuples, we provide two major new tools: an algorithm which reduces the check whether a given tuple F is Z-separating to an LP feasibility problem, and an isomorphism between the part of the Gröbner fan of I consisting of marked reduced Gröbner bases which contain a Z-separating tuple and the Gröbner fan of $I\cap K[X\setminus Z]$. We also indicate a possible generalization to tuples Z which consist of terms. All results are illustrated by explicit examples.