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On the Limits of Depth Reduction at Depth 3 Over Small Finite Fields
[article]

2013
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arXiv
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pre-print

Recently, Gupta et.al. [GKKS2013] proved that over Q any n^O(1)-variate and n-degree polynomial in VP can also be computed by a depth three ΣΠΣ circuit of size 2^O(√(n)^3/2n). Over fixed-size finite fields, Grigoriev and Karpinski proved that any ΣΠΣ circuit that computes Det_n (or Perm_n) must be of size 2^Ω(n) [GK1998]. In this paper, we prove that over fixed-size finite fields, any ΣΠΣ circuit for computing the iterated matrix multiplication polynomial of n generic matrices of size n× n,

arXiv:1401.0189v1
fatcat:qeb2ot7mrraqnnoi2tcteslz6e