Multidimensional permanents of polystochastic matrices

Billy Child, Ian M. Wanless

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Abstract

A d-dimensional matrix is called 1-polystochastic if it is non-negative and the sum over each line equals 1. Such a matrix that has a single 1 in each line and zeros elsewhere is called a 1-permutation matrix. A diagonal of a d-dimensional matrix of order n is a choice of n elements, no two in the same hyperplane. The permanent of a d-dimensional matrix is the sum over the diagonals of the product of the elements within the diagonal. For a given order n and dimension d, the set of 1-polystochastic matrices forms a convex polytope that includes the 1-permutation matrices within its set of vertices. For even n and odd d, we give a construction for a class of 1-permutation matrices with zero permanent. Consequently, we show that the set of 1-polystochastic matrices with zero permanent contains at least nn3/2(1/2−o(1)) 1-permutation matrices and contains a polytope of dimension at least cn3/2 for fixed c,d and even n→∞. We also provide counterexamples to a conjecture by Taranenko [11] about the location of local extrema of the permanent. For odd d, we give a construction of 1-permutation matrices that decompose into a convex linear sum of positive diagonals. These combine with a theorem of Taranenko [11] to provide counterexamples to a conjecture by Dow and Gibson [4] generalising van der Waerden's conjecture to higher dimensions.

Original languageEnglish
Pages (from-to)89-102
Number of pages14
JournalLinear Algebra and Its Applications
Volume586
DOIs
Publication statusPublished - 1 Feb 2020

Keywords

  • Birkhoff polytope
  • Hypercube
  • Permanent
  • Polystochastic
  • Transversal

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