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Enumeration of Nondegenerate $2 \times (k+1) \times k$ Hypermatrices

Brandon Koprowski, Joel Brewster Lewis

Abstract

We consider the problem of enumerating hypermatrices of format $2 \times (k + 1) \times k$ over a finite field that have nonzero hyperdeterminant and whose nonzero entries are restricted to a plane partition. We conjecture an attractive product formula for the enumeration, and prove it in many cases. In general, we show that the enumeration is given (up to a power of $q - 1$) by a polynomial in $q$ with nonnegative integer coefficients, whose value at $q = 1$ enumerates a natural family of three-dimensional rook placements.

Enumeration of Nondegenerate $2 \times (k+1) \times k$ Hypermatrices

Abstract

We consider the problem of enumerating hypermatrices of format over a finite field that have nonzero hyperdeterminant and whose nonzero entries are restricted to a plane partition. We conjecture an attractive product formula for the enumeration, and prove it in many cases. In general, we show that the enumeration is given (up to a power of ) by a polynomial in with nonnegative integer coefficients, whose value at enumerates a natural family of three-dimensional rook placements.
Paper Structure (14 sections, 31 theorems, 139 equations, 1 figure)

This paper contains 14 sections, 31 theorems, 139 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Hypermatrices and Nondegeneracy
  3. Main Conjecture
  4. The case mu is empty
  5. The case P equals Delta
  6. The case k = 2
  7. Hyperrook placements and a weak form of the main conjecture
  8. Hyperrook placements
  9. Decomposing the space of nondegenerate hypermatrices
  10. Further Research
  11. Proving the main conjecture
  12. Other formats
  13. Intrinsic definition of hyperrook placements
  14. Lower rank

Key Result

Theorem 2.2

The hypdeterminant of format $(k_1 + 1) \times \cdots \times (k_r +1)$ is non-trivial if and only if for all $j = 1, \ldots, r$.

Figures (1)

  • Figure 1: Left: the situation described in Proposition \ref{['prop:potentially bad entries in front face']}, with a potential $\lambda$ in gray. Right: the situation described in Proposition \ref{['prop:potentially bad entries in back face']}, with a potential $\mu$ in gray.

Theorems & Definitions (75)

  • Definition 2.1
  • Theorem 2.2: gkz
  • Proposition 2.3: gkz
  • Corollary 2.4: gkz
  • Lemma 2.5: aitken
  • Definition 2.6
  • Definition 2.7
  • Definition 2.8
  • Definition 2.9
  • Theorem 2.10
  • ...and 65 more