The Method Of Shifted Partial Derivatives Cannot Separate The Permanent From The Determinant

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)perm(m) cannot be realized inside the GL(n)2-orbit closure of the determinant det(n) when 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.