A new conception for computing gr\"{o}bner basis and its applications

TL;DR

This paper introduces a novel conceptual framework for computing Gröbner bases, transforming existing algorithms into a unified type, analyzing their termination conditions, and proposing an improved criterion for their efficiency.

Contribution

It presents a new conception that unifies and analyzes Gröbner basis algorithms, providing conditions for their finite termination and an improved criterion for algorithmic efficiency.

Findings

Unified framework for Gröbner basis algorithms.

Finite termination conditions for transformed algorithms.

Enhanced criterion improving algorithm performance.

Abstract

This paper presents a conception for computing gr\"{o}bner basis. We convert some of gr\"{o}bner-computing algorithms, e.g., F5, extended F5 and GWV algorithms into a special type of algorithm. The new algorithm's finite termination problem can be described by equivalent conditions, so all the above algorithms can be determined when they terminate finitely. At last, a new criterion is presented. It is an improvement for the Rewritten and Signature Criterion.