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On the limits of depth reduction at depth 3 over small finite fields

2017
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Information and Computation
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In a surprising recent result, Gupta et al. [GKKS13b] have 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 log 3/2 n) 1 . Over fixed-size finite fields, Grigoriev and Karpinski proved that any ΣΠΣ circuit that computes the determinant (or the permanent) polynomial of a n × n matrix must be of size 2 Ω(n) . In this paper, for an explicit polynomial in VP (over fixed-size finite fields), we prove that any ΣΠΣ

doi:10.1016/j.ic.2017.04.007
fatcat:ys6nzr3lxfd3tgkkxwsffvphla