On the Limits of Depth Reduction at Depth-3 Over Small Finite Fields

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Suryajith Chillara and Partha Mukhopadhyay

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In a surprising recent result, Gupta-Kamath-Kayal-Saptharishi have proved that over $\mathbb{Q}$ any $n^{O(1)}$-variate and $n$-degree polynomial in $VP$ can also be computed by a depth three $\Sigma\Pi\Sigma$ circuit of size $2^{O(\sqrt{n} \log^{3/2}n)}$. Over fixed-size finite fields, Grigoriev and Karpinski proved that any $\Sigma\Pi\Sigma$ circuit that computes the determinant (or the permanent) polynomial of a $n\times n$ matrix must be of size $2^{\Omega(n)}$. In this paper, for an explicit polynomial in $VP$ (over fixed-size finite fields), we prove that any $\Sigma\Pi\Sigma$ circuit computing it must be of size $2^{\Omega(n\log n)}$. The explicit polynomial that we consider is the iterated matrix multiplication polynomial of $n$ generic matrices of size $n\times n$. The importance of this result is that over fixed-size fields there is no depth reduction technique that can be used to compute all the $n^{O(1)}$-variate and $n$-degree polynomials in $VP$ by depth 3 circuits of size $2^{o(n\log n)}$. The result of Grigoriev and Karpinski can only rule out such a possibility for $\Sigma\Pi\Sigma$ circuits of size $2^{o(n)}$.

We also give an example of an explicit polynomial ($NW_{n,\epsilon}(X)$) in $VNP$ (which is not known to be in $VP$), for which any $\Sigma\Pi\Sigma$ circuit computing it (over fixed-size fields) must be of size $2^{\Omega(n\log n)}$. The polynomial we consider is constructed from the combinatorial design of Nisan and Wigderson, and is closely related to the polynomials considered in many recent papers (by Kayal-Saha-Saptharishi, Kayal-Limaye-Saha-Srinivasan, and Kumar-Saraf), where strong depth 4 circuit size lower bounds are shown.