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

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 ℓ n − m p e r m m \ell ^{n-m}\mathrm {perm}_m cannot be realized inside the G L n 2 GL_{n^2} -orbit closure of the determinant d e t n \mathrm {det}_n when n > 2 m 2 + 2 m n>2m^2+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.