## 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.

Original language | English |
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Pages (from-to) | 70-83 |

Number of pages | 14 |

Journal | Journal of Symbolic Computation |

Volume | 66 |

Issue number | 1 |

DOIs | |

State | Published - Jan 2015 |

## Keywords

- Hilbert's Nullstellensatz
- Linear algebra
- Partition

## All Science Journal Classification (ASJC) codes

- Algebra and Number Theory
- Computational Mathematics