Recently, Gupta et.al. [GKKS2013] proved that over Q any nO(1)-variate
and n-degree polynomial in VP can also be computed by a depth three
ΣΠΣ circuit of size 2O(nlog3/2n). Over fixed-size
finite fields, Grigoriev and Karpinski proved that any ΣΠΣ
circuit that computes Detn (or Permn) must be of size 2Ω(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, must be of size
2Ω(nlogn). 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 nO(1)-variate and n-degree polynomials in VP by depth 3 circuits of
size 2o(nlogn). The result [GK1998] can only rule out such a possibility
for depth 3 circuits of size 2o(n).
We also give an example of an explicit polynomial (NWn,ϵ(X)) in
VNP (not known to be in VP), for which any ΣΠΣ circuit computing
it (over fixed-size fields) must be of size 2Ω(nlogn). The
polynomial we consider is constructed from the combinatorial design. An
interesting feature of this result is that we get the first examples of two
polynomials (one in VP and one in VNP) such that they have provably stronger
circuit size lower bounds than Permanent in a reasonably strong model of
computation.
Next, we prove that any depth 4
ΣΠ[O(n)]ΣΠ[n] circuit computing
NWn,ϵ(X) (over any field) must be of size 2Ω(nlogn). To the best of our knowledge, the polynomial NWn,ϵ(X) is the
first example of an explicit polynomial in VNP such that it requires
2Ω(nlogn) size depth four circuits, but no known matching
upper bound