We show that, over Q, if an n-variate polynomial of degree d = n O(1) is computable by an arithmetic circuit of size s (respectively by an arithmetic branching program of size s) then it can also be computed by a depth three circuit (i.e. a ΣΠΣ-circuit) of size expIt also means that if we can prove a lower bound of exp(ω( √ d · log d)) on the size of any ΣΠΣ-circuit computing the d × d permanent Perm d then we get superpolynomial lower bounds for the size of any arithmetic branching program computing Perm d . We then give some further results pertaining to derandomizing polynomial identity testing and circuit lower bounds.The ΣΠΣ circuits that we construct have the property that (some of) the intermediate polynomials have degree much higher than d. Indeed such a counterintuitive construction is unavoidable -it is known that in any ΣΠΣ circuit C computing either Det d or Perm d , if every multiplication gate has fanin at most d (or any constant multiple thereof) then C must have size at least exp(Ω(d)).

Agrawal-Vinay [AV08], Koiran [Koi12] and Tavenas [Tav13] have recently shown that an exp(ω( √ n log n)) lower bound for depth four homogeneous circuits computing the permanent with bottom layer of × gates having fanin bounded by √ n translates to super-polynomial lower bound for general arithmetic circuits computing the permanent. Motivated by this, we examine the complexity of computing the permanent and determinant via such homogeneous depth four circuits with bounded bottom fanin.We show here that any homogeneous depth four arithmetic circuit with bottom fanin bounded by √ n computing the permanent (or the determinant) must be of size exp(Ω( √ n)).

Agrawal-Vinay [AV08] and Koiran [Koi12] have recently shown that an exp(ω( √ n log 2 n)) lower bound for depth four homogeneous circuits computing the permanent with bottom layer of × gates having fanin bounded by √ n translates to super-polynomial lower bound for general arithmetic circuits computing the permanent. Motivated by this, we examine the complexity of computing the permanent and determinant via such homogeneous depth four circuits with bounded bottom fanin. We show here that any homogeneous depth four arithmetic circuit with bottom fanin bounded by √ n computing the permanent (or the determinant) must be of size exp(Ω( √ n)).

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