Restricted Gr\"obner fans and re-embeddings of affine algebras

TL;DR

This paper explores methods for re-embedding affine algebras using linear projections, introducing algorithms and tools to identify isomorphisms with fewer generators via Gr"obner fan analysis.

Contribution

It introduces new algorithms for identifying Z-separating tuples and establishes a connection between parts of the Gr"obner fan and algebra re-embeddings, advancing the understanding of affine algebra structures.

Findings

Algorithm reduces Z-separating tuple check to LP feasibility

Isomorphism between Gr"obner fan parts and algebra intersections

Illustrative examples demonstrating the methods

Abstract

In this paper we continue the study of good re-embeddings of affine K-algebras started in [KLR]. The idea is to use special linear projections to find isomorphisms between a given affine K-algebra K[X]/I, where X=(x_1,...,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 Gr\"obner fan of I consisting of marked reduced Gr\"obner bases which contain a Z-separating tuple and the Gr\"obner fan of the intersection of I and K[X\Z]. We also indicate a possible generalization to tuples Z which consist of terms. All results are illustrated by explicit examples.