The method of shifted partial derivatives cannot separate the permanent from the determinant
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© 2017 American Mathematical Society. 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.
author list (cited authors)
Efremenko, K., Landsberg, J. M., Schenck, H., & Weyman, J.