A quadratic lower bound for the permanent and determinant problem over any characteristic ≠ 2

  • Authors:

  • Jin-Yi Cai;Xi Chen;Dong Li

  • Affiliations:

  • University of Wisconsin, Madison, Madison, WI, USA;Institute for Advanced Study, Princeton, NJ, USA;Institute for Advanced Study, Princeton, NJ, USA

  • Venue:
  • STOC '08 Proceedings of the fortieth annual ACM symposium on Theory of computing
  • Year:
  • 2008

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Abstract

In Valiant's theory of arithmetic complexity, the classes VP and VNP are analogs of P and NP. A fundamental problem concerning these classes is the Permanent and Determinant Problem: Given a field F of characteristic ≠2, and an integer n, what is the minimum m such that the permanent of an n x n matrix X=(xij) can be expressed as a determinant of an m x m matrix, where the entries of the determinant matrix are affine linear functions of xij's, and the equality is in F [X]. Mignon and Ressayre (2004) [11] proved a quadratic lower bound m=Ω(n2) for fields of characteristic 0. We extend the Mignon-Ressayre quadratic lower bound to all fields of characteristic ≠2.