Reduced Groebner Bases With Double Exponential Cardinality

TL;DR

This paper demonstrates that for certain polynomial families and monomial orderings, the reduced Groebner bases can have double exponential size, highlighting potential computational complexity issues.

Contribution

It identifies a family of polynomials and a criterion under which the reduced Groebner basis size becomes double exponential for common monomial orderings.

Findings

Reduced Groebner bases can have double exponential size.

The criterion applies to lexicographic, degree lexicographic, and weighted orderings.

Highlights complexity challenges in Groebner basis computations.

Abstract

In this article, we investigate the cardinality of Groebner bases under various monomial orderings. We identify a family of polynomials F and a criterion such that the reduced Groebner basis of F is double exponential in cardinality with respect to any monomial ordering which satisfies this criterion. We also show that the said criterion is satisfied by orderings such as the lexicographic, degree lexicographic and weighted orderings.