On the complexity of Hilbert refutations for partition

S. Margulies, S. Onn, D. V. Pasechnik

Research output: Contribution to journalArticlepeer-review


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.

Original languageEnglish
Pages (from-to)70-83
Number of pages14
JournalJournal of Symbolic Computation
Issue number1
StatePublished - Jan 2015


  • Hilbert's Nullstellensatz
  • Linear algebra
  • Partition

All Science Journal Classification (ASJC) codes

  • Algebra and Number Theory
  • Computational Mathematics


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