The method of shifted partial derivatives cannot separate the permanent from the determinant Academic Article uri icon

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 $\ell^{nm}\mathrm {perm} _m $ cannot be realized inside the $ GL_ {n^ 2} $-orbit closure of the determinant $\mathrm {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 …

publication date

  • January 1, 2017