On the Connection Between Ritt Characteristic Sets and Buchberger-Gr\"obner Bases

TL;DR

This paper explores the deep connection between Ritt characteristic sets and Buchberger-Gr"obner bases, showing how to derive Ritt sets from Buchberger bases and analyzing their properties under variable orderings.

Contribution

It establishes that W-characteristic sets from Buchberger-Gr"obner bases are Ritt characteristic sets when ascending, and provides methods to compute Ritt sets via pseudo-division, enhancing polynomial ideal decomposition.

Findings

W-characteristic set is a Ritt characteristic set if ascending.

Ritt characteristic sets can be derived from W-characteristic sets using pseudo-division.

Explicit pseudo-divisibility relations reveal polynomial structure and enable decomposition.

Abstract

For any polynomial ideal $I$, let the minimal triangular set contained in the reduced Buchberger-Gr\"obner basis of $I$ with respect to the purely lexicographical term order be called the W-characteristic set of $I$. In this paper, we establish a strong connection between Ritt's characteristic sets and Buchberger's Gr\"obner bases of polynomial ideals by showing that the W-characteristic set $C$ of $I$ is a Ritt characteristic set of $I$ whenever $C$ is an ascending set, and a Ritt characteristic set of $I$ can always be computed from $C$ with simple pseudo-division when $C$ is regular. We also prove that under certain variable ordering, either the W-characteristic set of $I$ is normal, or irregularity occurs for the $j$th, but not the $(j+1)$th, elimination ideal of $I$ for some $j$. In the latter case, we provide explicit pseudo-divisibility relations, which lead to nontrivial factorizations of certain polynomials in the Buchberger-Gr\"obner basis and thus reveal the structure of such polynomials. The pseudo-divisibility relations may be used to devise an algorithm to decompose arbitrary polynomial sets into normal triangular sets based on Buchberger-Gr\"obner bases computation.