Some Complexity Results for Prefix Gröbner Bases in Free Monoid Rings

Abstract

We establish the following complexity results for prefix Gröbner bases in free monoid rings: 1. $|{\cal R}| \cdot size(p)$ reduction steps are sufficient to normalize a given polynomial p w.r.t. a given right-normalized system ${\cal R}$ of prefix rules compatible with some total admissible wellfounded ordering >. 2. $O(|{\cal R}|^2 \cdot size({\cal R}))$ basic steps are sufficient to transform a given terminating system ${\cal R}$ of prefix rules into an equivalent right-normalized system. 3. $O(|{\cal R}|^3 \cdot size({\cal R}))$ basic steps are sufficient to decide whether or not a given terminating system ${\cal R}$ of prefix rules is a prefix Gröbner basis. The latter result answers an open question posed by Zeckzer in [10].