Feasible arithmetic computations: Valiant's hypothesis
Journal of Symbolic Computation
Lower bounds for non-commutative computation
STOC '91 Proceedings of the twenty-third annual ACM symposium on Theory of computing
PP is as hard as the polynomial-time hierarchy
SIAM Journal on Computing
Lower bounds on arithmetic circuits via partial derivatives
Computational Complexity
Cook's versus Valiant's hypothesis
Theoretical Computer Science - Selected papers in honor of Manuel Blum
Depth-3 arithmetic circuits over fields of characteristic zero
Computational Complexity
Multi-linear formulas for permanent and determinant are of super-polynomial size
STOC '04 Proceedings of the thirty-sixth annual ACM symposium on Theory of computing
Multilinear- NC" " Multilinear- NC"
FOCS '04 Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science
A quadratic lower bound for the permanent and determinant problem over any characteristic ≠ 2
STOC '08 Proceedings of the fortieth annual ACM symposium on Theory of computing
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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 , and the symmetric power sum polynomial $\sum_{i=1}^n x_i^d$, for any constant d 1. A 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 ***(n d /(2d *** r )), for constants d and r with 2 ≤ d + 1 ≤ r d . In the most restrictive setting an $n^{\Omega(\epsilon n^{1/5-\epsilon})}$ lower bound is observed, for any *** *** (0, 1/5) and $d = \lfloor n^{1/5-\epsilon}\rfloor$.