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

The method of shifted partial derivatives introduced in [9, 7] 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 perm m cannot be realized inside the GLn2 -orbit closure of the determinant detn when n > 2m 2 + 2m. Our proof relies on several simple degenerations of the determinant polynomial, Macaulay’s theorem that gives a lower bound on the growth of an ideal, and a lower bound estimate from [7] regarding the shifted partial derivatives of the determinant.