Generalised Buchberger and Schreyer algorithms for strongly discrete coherent rings

TL;DR

This paper extends Buchberger and Schreyer algorithms to strongly discrete coherent rings, enabling explicit computation of leading term modules and providing a constructive approach to Hilbert's syzygy theorem in this setting.

Contribution

It introduces generalized algorithms for strongly discrete coherent rings, including a Buchberger-like algorithm and a constructive Hilbert's syzygy theorem.

Findings

The module of leading terms is countably generated.

The Buchberger-like algorithm converges iff the leading term module is finitely generated.

A constructive version of Hilbert's syzygy theorem is provided.

Abstract

Let M be a finitely generated submodule of a free module over a multivariate polynomial ring with coefficients in a discrete coherent ring. We prove that its module MLT(M ) of leading terms is countably generated and provide an algorithm for computing explicitly a generating set. This result is also useful when MLT(M ) is not finitely generated. Suppose that the base ring is strongly discrete coherent. We provide a Buchberger-like algorithm and prove that it converges if, and only if, the module of leading terms is finitely generated. We also provide a constructive version of Hilbert's syzygy theorem by following Schreyer's method.