On the complexity of Hilbert refutations for partition. (January 2015) - Margulies, S.; Onn, S.; Pasechnik, D.V.
Abstract: Given a set of integers W, thePartition problem determines whether W can be divided into two disjoint subsets with equal sums. We model thePartition 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 thePartition 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 .