Dimension-Dependent Upper Bounds for Grobner Bases

TL;DR

This paper establishes new upper bounds for the degrees of Grobner bases and related invariants of polynomial ideals in specific generic positions, improving understanding of their algebraic complexity.

Contribution

It introduces dimension-dependent bounds for Grobner bases, Castelnuovo-Mumford regularity, and degrees of basis elements in strongly stable and quasi stable positions.

Findings

Derived new degree bounds for Grobner bases in generic positions.

Established dimension- and depth-dependent bounds for regularity.

Focused on strongly stable and quasi stable positions for polynomial ideals.

Abstract

We improve certain degree bounds for Grobner bases of polynomial ideals in generic position. We work exclusively in deterministically verifiable and achievable generic positions of a combinatorial nature, namely either strongly stable position or quasi stable position. Furthermore, we exhibit new dimension- (and depth-)dependent upper bounds for the Castelnuovo-Mumford regularity and the degrees of the elements of the reduced Grobner basis (w.r.t. the degree reverse lexicographical ordering) of a homogeneous ideal in these positions.