An analysis of inhomogeneous signature-based Gr\"obner basis computations

TL;DR

This paper investigates how signature-based Gr"obner basis algorithms perform with inhomogeneous inputs, highlighting the trade-offs between reduction safety and efficiency, and analyzing the impact of critical pair handling and degree sorting.

Contribution

It provides an in-depth analysis of the behavior of signature-based Gr"obner basis algorithms on inhomogeneous inputs, revealing performance penalties and efficiency mechanisms.

Findings

Restriction to sig-safe reductions can reduce performance.

Sorting by sugar degree improves critical pair handling.

Inhomogeneous inputs affect the degree-signature relationship.

Abstract

In this paper we give an insight into the behaviour of signature-based Gr\"obner basis algorithms, like F5, G2V or SB, for inhomogeneous input. On the one hand, it seems that the restriction to sig-safe reductions puts a penalty on the performance. The lost connection between polynomial degree and signature degree can disallow lots of reductions and can lead to an overhead in the computations. On the other hand, the way critical pairs are sorted and corresponding s-polynomials are handled in signature-based algorithms is a very efficient one, strongly connected to sorting w.r.t. the well-known sugar degree of polynomials.