The method of shifted partial derivatives cannot separate the permanent from the determinant

Klim Efremenko, J. M. Landsberg, Hal Schenck, Jerzy Weyman

Research output: Contribution to journalArticlepeer-review

2 Scopus citations

Abstract

The method of shifted partial derivatives introduced A. Gupta et al. [Approaching the chasm at depth four, IEEE Comp. Soc., 2013, pp. 65-73] and N. Kayal [An exponential lower bound for the sum of powers of bounded degree polynomials, ECCC 19, 2010, p. 81], was used to prove a super-polynomial lower bound on the size of depth four circuits needed to compute the permanent. We show that this method alone cannot prove that the padded permanent ℓn-m permm cannot be realized inside the GLn2 -orbit closure of the determinant detn when n > 2m2 +2m. Our proof relies on several simple degenerations of the determinant polynomial, Macaulay's theorem, which gives a lower bound on the growth of an ideal, and a lower bound estimate from [Approaching the chasm at depth four, IEEE Comp. Soc., 2013, pp. 65-73] regarding the shifted partial derivatives of the determinant.

Original languageEnglish
Pages (from-to)2037-2045
Number of pages9
JournalMathematics of Computation
Volume87
Issue number312
DOIs
StatePublished - 1 Jan 2018

Keywords

  • Computational complexity
  • Determinant
  • Free resolution
  • Permanent

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