Solving degree, last fall degree, and related invariants

TL;DR

This paper explores and connects various algebraic invariants related to the solving degree of polynomial systems, offering a framework to estimate the complexity of solving such systems using Groebner bases.

Contribution

It establishes new theoretical links between the solving degree, last fall degree, degree of regularity, and Castelnuovo-Mumford regularity.

Findings

Connected solving degree with last fall degree

Linked degree of regularity with Castelnuovo-Mumford regularity

Provided a framework for complexity estimation of polynomial system solving

Abstract

In this paper we study and relate several invariants connected to the solving degree of a polynomial system. This provides a rigorous framework for estimating the complexity of solving a system of polynomial equations via Groebner bases methods. Our main results include a connection between the solving degree and the last fall degree and one between the degree of regularity and the Castelnuovo-Mumford regularity.