THE METHOD OF SHIFTED PARTIAL DERIVATIVES CANNOT SEPARATE THE PERMANENT FROM THE DETERMINANT Academic Article

The method of shifted partial derivatives introduced A. Gupta etal. [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 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, Macaulays 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.