Abstract. It is shown that any weakly-skew circuit can be converted into a skew circuit with constant factor overhead, while preserving either syntactic or semantic multilinearity. This leads to considering syntactically multilinear algebraic branching programs (ABPs), which are defined by a natural read-once property. A 2 n/4 size lower bound is proven for ordered syntactically multilinear ABPs computing an explicitly constructed multilinear polynomial in 2n variables. Without the ordering restriction a lower bound of level Ω(n 3/2 / log n) is observed, by considering a generalization of a hypercube covering problem by Galvin [1].

We show that over fields of characteristic zero there does not exist a polynomial p(n) and a constant-free succinct arithmetic circuit family {Φ n } using division by constants, 3 where Φ n has size at most p(n) and depth O (1), such that Φ n computes the n × n permanent. A circuit family {Φ n } is succinct if there exists a nonuniform Boolean circuit family {C n } with O (log n) many inputs and size n o(1) such that C n can correctly answer direct connection language queries about Φ n -succinctness is a relaxation of uniformity.To obtain this result we develop a novel technique that further strengthens the connection between black-box derandomization of polynomial identity testing and lower bounds for arithmetic circuits. From this, we obtain the lower bound by giving an explicit construction, computable in the polynomial hierarchy, of a hitting set for arithmetic circuits.

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Suppose f is a univariate polynomial of degree r = r(n) that is computed by a size n arithmetic circuit. It is a basic fact of algebra that a nonzero univariate polynomial of degree r can vanish on at most r points. This implies that for checking whether f is identically zero, it suffices to query f on an arbitrary test set of r + 1 points. Could this brute-force method be improved upon by a single point? We develop a framework where such a marginal improvement implies that Permanent does not have polynomial size arithmetic circuits.More formally, we formulate the following hypothesis for any field of characteristic zero: There is a fixed depth d and some function s(n) = O(n), such that for arbitrarily small > 0, there exists a hitting set Hn ⊂ Z of size at most 2 s(n ) against univariate polynomials of degree at most 2 s(n ) computable by size n constant-free 1 arithmetic circuits, where Hn can be encoded by uniform TC 0 circuits of size 2and depth d. We prove that the hypothesis implies that Permanent does not have polynomial size constant-free arithmetic circuits.Our hypothesis provides a unifying perspective on several important complexity theoretic conjectures, as it follows from these conjectures for different degree ranges as determined by the function s(n). We will show that it follows for s(n) = n from the widely-believed assumption that poly size Boolean circuits cannot compute the Permanent of a 0, 1-matrix over Z. The hypothesis can also be easily derived from the Shub-Smale τ -conjecture [21], for any s(n) * Supported by EPSRC Grant H05068X/1. † Supported by EPSRC Grant H05068X/1.

Abstract. Asymptotically tight lower bounds are proven for the determinantal complexity of the elementary symmetric polynomial S d n of degree d in n variables, 2d-fold iterated matrix multiplication of the form u| X 1 X 2 . . . X 2d |v , and the symmetric power sum polynomialA restriction of determinantal computation is considered in which the underlying affine linear map must satisfy a rank lowerability property. In this model strongly nonlinear and exponential lower bounds are proven for several polynomial families. For example, for S 2d n it is proved that the determinantal complexity using so-called r-lowerable maps is Ω(n d/(2d−r) ), for constants d and r with 1 < d < r < 2d. In the most restrictive setting an n Ω(ǫn 1/5−ǫ ) lower bound is observed, for any ǫ ∈ (0, 1/5) and d = ⌊n 1/5−ǫ ⌋.