Enumerative Aspects of Nullstellensatz Certificates

TL;DR

This paper explores how Nullstellensatz certificates, used to prove infeasibility of polynomial systems modeling combinatorial problems, can be interpreted as enumerating combinatorial structures, linking algebraic certificates to combinatorial enumeration.

Contribution

It generalizes known results and demonstrates that Nullstellensatz certificates can be viewed as enumeration tools for combinatorial structures.

Findings

Nullstellensatz certificates encode combinatorial enumeration.

Gröbner basis algorithms implicitly perform enumeration.

Certificates have meaningful combinatorial interpretations.

Abstract

Using polynomial equations to model combinatorial problems has been a popular tool both in computational combinatorics as well as an approach to proving new theorems. In this paper, we look at several combinatorics problems modeled by systems of polynomial equations satisfying special properties. If the equations are infeasible, Hilbert's Nullstellensatz gives a certificate of this fact. These certificates have been studied and exhibit combinatorial meaning. In this paper, we generalize some known results and show that the Nullstellensatz certificate can be viewed as enumerating combinatorial structures. As such, Gr\"obner basis algorithms for solving these decision problems may implicitly be solving the enumeration problem as well.