We consider arithmetic formulas consisting of alternating layers of addition (+) and multiplication (×) gates such that the fanin of all the gates in any fixed layer is the same. Such a formula Φ which additionally has the property that its formal/syntactic degree is at most twice the (total) degree of its output polynomial, we refer to as a regular formula. As usual, we allow arbitrary constants from the underlying field F on the incoming edges to a + gate so that a + gate can in fact compute an arbitrary F-linear combination of its inputs. We show that there is an (n 2 + 1)-variate polynomial of degree 2n in VNP such that any regular formula computing it must be of size at least n Ω(log n) .Along the way, we examine depth four (ΣΠΣΠ) regular formulas wherein all multiplication gates in the layer adjacent to the inputs have fanin a and all multiplication gates in the layer adjacent to the output node have fanin b. We refer to such formulas as ΣΠ [b] ΣΠ [a] -formulas. We show that there exists an n 2

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)).

We give deterministic black-box polynomial identity testing algorithms for multilinear read-once oblivious algebraic branching programs (ROABPs), in n O(lg 2 n) time.Further, our algorithm is oblivious to the order of the variables. This is the first sub-exponential time algorithm for this model. Furthermore, our result has no known analogue in the model of read-once oblivious boolean branching programs with unknown order.We obtain our results by recasting, and improving upon, the ideas of Agrawal, Saha and Saxena [ASS13]. We phrase the ideas in terms of rank condensers and Wronskians, and show that our results improve upon the classical multivariate Wronskian, which may be of independent interest.In addition, we give the first n O(lg lg n) black-box polynomial identity testing algorithm for the so called model of diagonal circuits. This result improves upon the n Θ(lg n) -time algorithms given by Agrawal, Saha and Saxena [ASS13], and Forbes and Shpilka [FS13b] for this class. More generally, our result holds for any model computing polynomials whose partial derivatives (of all orders) span a low dimensional linear space.

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)).

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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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