Gr\"obner bases and final polynomials

TL;DR

This paper challenges a previous claim about the appearance of final polynomials in lexicographic Gr"obner bases, providing a counterexample and proposing an extended concept with an extra variable to restore the claim.

Contribution

It provides a counterexample to a known claim and introduces extended final polynomials with an extra variable to address the issue.

Findings

Counterexample disproves the claim about final polynomials always appearing in lexicographic Gr"obner bases

Introducing an extra variable restores the claim in a deformed setup

Extended final polynomials generalize the concept to include the counterexample case

Abstract

In [4] Sturmfels linked the Hilbert Nullstellensatz to Gr\"obner bases through final polynomials. In (loc. cit.) it was claimed that final polynomials always appear in a lexicographic Gr\"obner basis of a certain ideal. In this paper, we give a counterexample to this claim. We also show how the introduction of an extra variable restores the claim in a deformed setup, which we call extended final polynomials.