移位偏导数法不能将积式与行列式分开
The method of shifted partial derivatives cannot separate the permanent from the determinant
摘要：
利用移位偏导数的方法证明了计算永久式所需深度四个回路大小的一个超多项式下界。我们表明，这种方法本身不能证明填充永久$\ell^{n-m} perm_m$不能实现内的$GL_{n^2}$ -轨道封闭的行列式$ det_n$时，$n>2m^2+2m$。我们的证明依赖于几个简单的退化的行列式多项式，麦考利定理，给出了一个下界增长的理想，和一个下界估计古普塔等。关于行列式的移位的偏导数。
英文摘要：
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.
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