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
Locally decodable codes with 2 queries and polynomial identity testing for depth 3 circuits
Proceedings of the thirty-seventh annual ACM symposium on Theory of computing
On obtaining pseudorandomness from error-correcting codes
FSTTCS'06 Proceedings of the 26th international conference on Foundations of Software Technology and Theoretical Computer Science
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We give a deterministic polynomial time algorithm forpolynomial identity testing in the following two cases:1. Non Commutative Arithmetic Formulas: The algorithmgets as an input an arithmetic formula in thenon-commuting variables x1, ..., xn and determineswhether or not the output of the formula is identically0 (as a formal expression).2. Pure Arithmetic Circuits: The algorithm gets as aninput a pure arithmetic circuit (as defined by Nisanand Wigderson [3]) in the variables x1, ..., xn and determineswhether or not the output of the circuitidentically 0 (as a formal expression).We also give a deterministic polynomial time identity testingalgorithm for non commutative algebraic branching programsas defined by Nisan [2]. One application is a deterministicpolynomial time identity testing for multilineararithmetic circuits of depth 3.Finally, we observe an exponential lower bound for thesize of pure arithmetic circuits for the permanent and for thedeterminant. (Only lower bounds for the depth of pure circuitswere previously known [3]).