An Algorithm for Computing a Minimal Comprehensive Gr\"obner\, Basis of a Parametric Polynomial System

TL;DR

This paper introduces an algorithm to compute minimal comprehensive Gr"obner bases for parametric polynomial systems, ensuring minimality through polynomial essentiality checks and demonstrating effectiveness on various examples.

Contribution

The paper presents a novel algorithm for deriving minimal comprehensive Gr"obner bases from existing systems, improving efficiency and minimality in parametric polynomial ideal computations.

Findings

Algorithm successfully computes minimal comprehensive Gr"obner bases

Implementation tested on multiple literature examples

Ensures basis minimality via polynomial essentiality checks

Abstract

An algorithm to generate a minimal comprehensive Gr\"obner\, basis of a parametric polynomial system from an arbitrary faithful comprehensive Gr\"obner\, system is presented. A basis of a parametric polynomial ideal is a comprehensive Gr\"obner\, basis if and only if for every specialization of parameters in a given field, the specialization of the basis is a Gr\"obner\, basis of the associated specialized polynomial ideal. The key idea used in ensuring minimality is that of a polynomial being essential with respect to a comprehensive Gr\"obner\, basis. The essentiality check is performed by determining whether a polynomial can be covered for various specializations by other polynomials in the associated branches in a comprehensive Gr\"obner\, system. The algorithm has been implemented and successfully tried on many examples from the literature.