Signature-based standard basis algorithm under the framework of GVW algorithm

Abstract

The GVW algorithm, one of the most important so-called signature-based algorithms, is designed to eliminate a large number of useless polynomial reductions from Buchberger's algorithm. The cover theorem serves as the theoretical foundation of the GVW algorithm, and up to now, it applies only to a certain class of monomial orders, namely global orders and a special class of local orders. In this paper we extend this theorem to any semigroup order, which can be either global, local or even mixed. Building upon the pioneering idea of the Mora normal form algorithm, we propose a more comprehensive and general proof for the cover theorem while bypassing the need to choose a minimal element from an infinite set of monomials in all the existing proofs. Therefore, the algorithm for signature-based standard bases is presented for any semigroup order under the framework of the GVW algorithm, and an example is given to provide an illustration of the algorithm.