Subalgebra Analogue to Standard Basis for Ideal

TL;DR

This paper develops the theory of Sasbi bases, a subalgebra analogue to standard bases for ideals, providing algorithms for finite cases and implementing them in the SINGULAR computer algebra system.

Contribution

It introduces algorithms for computing finite Sasbi bases of subalgebras and implements these algorithms in SINGULAR.

Findings

Algorithms successfully compute finite Sasbi bases.

Implementation in SINGULAR enhances computational capabilities.

Theoretical foundation extends standard basis concepts to subalgebras.

Abstract

The theory of "subalgebra basis" analogous to standard basis (the generalization of Gr\"{o}bner bases to monomial ordering which are not necessarily well ordering \cite{GP1}.) for ideals in polynomial rings over a field is developed. We call these bases "SASBI Basis" for "Subalgebra Analogue to Standard Basis for Ideals". The case of global orderings, here they are called "SAGBI Basis" for "Subalgebra Analogue to Gr\"{o}bner Basis for Ideals", is treated in \cite{RS1}. Sasbi bases may be infinite. In this paper we consider subalgebras admitting a finite Sasbi basis and give algorithms to compute them. The algorithms have been implemented as a library for the computer algebra system SINGULAR \cite{GPS1}.