A Pommaret Bases Approach to the Degree of a Polynomial Ideal

TL;DR

This paper explores the relationship between Pommaret bases and Hilbert series, deriving explicit formulas for the degree of polynomial ideals and establishing new bounds for ideal membership problems.

Contribution

It introduces new formulas linking Pommaret bases to Hilbert series and proposes a novel dimension-dependent bound for the degree and ideal membership.

Findings

Derived explicit formulas for Hilbert series and ideal degree from Pommaret bases.

Established a new dimension-dependent Bezout bound for the degree.

Proposed a dimension-dependent bound for the ideal membership problem.

Abstract

In this paper, we study first the relationship between Pommaret bases and Hilbert series. Given a finite Pommaret basis, we derive new explicit formulas for the Hilbert series and for the degree of the ideal generated by it which exhibit more clearly the influence of each generator. Then we establish a new dimension depending Bezout bound for the degree and use it to obtain a dimension depending bound for the ideal membership problem.