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

TL;DR

The paper demonstrates that the shifted partial derivatives method cannot distinguish the permanent from the determinant in certain algebraic complexity settings, highlighting limitations of this approach.

Contribution

It shows the fundamental limitations of the shifted partial derivatives method in separating the permanent from the determinant in algebraic complexity theory.

Findings

The method cannot prove the permanent is outside the determinant's orbit closure for certain parameters.

The proof uses degenerations of the determinant polynomial and Macaulay's theorem.

It relies on a lower bound estimate for shifted partial derivatives of the determinant.

Abstract

The method of shifted partial derivatives 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 $\ell^{n-m} perm_m$ 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 that gives a lower bound on the growth of an ideal, and a lower bound estimate from Gupta et. al. regarding the shifted partial derivatives of the determinant.