Converting Subalgebra Bases with the Sagbi Walk

TL;DR

This paper introduces a Sagbi walk algorithm that efficiently converts Sagbi bases between different term orderings for polynomial subalgebras, with implementation in SINGULAR.

Contribution

It presents a novel algorithm for converting Sagbi bases across term orderings using a path-following approach in the Sagbi Fan.

Findings

Algorithm successfully converts Sagbi bases between orderings.

Implementation available in SINGULAR enhances practical usability.

Method applicable to subalgebras with finite Sagbi bases.

Abstract

We present an algorithm which converts a given Sagbi basis of a polynomial $K$-subalgebra $\mathcal{A}$ to a Sagbi basis of $\mathcal{A}$ in a polynomial ring with respect to another term ordering, under the assumption that subalgebra $\mathcal{A}$ admits a finite Sagbi basis with respect to all term ordering. The Sagbi walk method converts a Sagbi basis by partitioning the computations following a path in the Sagbi Fan. The algorithms have been implemented as a library for the computer algebra system SINGULAR \cite{GPS1}.