On the Complexity of Hilbert Refutations for Partition

TL;DR

This paper explores the complexity of Hilbert Nullstellensatz refutations for the Partition problem by linking it to the determinant of a specially constructed matrix, providing explicit certificates and analyzing their properties.

Contribution

It introduces an explicit construction of minimum-degree certificates and relates the Partition problem to the determinant of a novel partition matrix.

Findings

Constructed a minimum-degree Hilbert Nullstellensatz certificate.

Established the equivalence between the Partition problem and the determinant of the partition matrix.

Demonstrated the determinant factors over all possible partitions.

Abstract

Given a set of integers W, the Partition problem determines whether W can be divided into two disjoint subsets with equal sums. We model the Partition problem as a system of polynomial equations, and then investigate the complexity of a Hilbert's Nullstellensatz refutation, or certificate, that a given set of integers is not partitionable. We provide an explicit construction of a minimum-degree certificate, and then demonstrate that the Partition problem is equivalent to the determinant of a carefully constructed matrix called the partition matrix. In particular, we show that the determinant of the partition matrix is a polynomial that factors into an iteration over all possible partitions of W.