Matching is as easy as matrix inversion
Combinatorica
A deterministic algorithm for sparse multivariate polynomial interpolation
STOC '88 Proceedings of the twentieth annual ACM symposium on Theory of computing
Fast parallel algorithms for sparse multivariate polynomial interpolation over finite fields
SIAM Journal on Computing
PP is as hard as the polynomial-time hierarchy
SIAM Journal on Computing
Journal of Computer and System Sciences
SIAM Journal on Computing
P = BPP if E requires exponential circuits: derandomizing the XOR lemma
STOC '97 Proceedings of the twenty-ninth annual ACM symposium on Theory of computing
Checking polynomial identities over any field: towards a derandomization?
STOC '98 Proceedings of the thirtieth annual ACM symposium on Theory of computing
An exponential lower bound for depth 3 arithmetic circuits
STOC '98 Proceedings of the thirtieth annual ACM symposium on Theory of computing
Fast Probabilistic Algorithms for Verification of Polynomial Identities
Journal of the ACM (JACM)
Extractors and pseudo-random generators with optimal seed length
STOC '00 Proceedings of the thirty-second annual ACM symposium on Theory of computing
Pseudorandom generators without the XOR lemma
Journal of Computer and System Sciences - Special issue on the fourteenth annual IEE conference on computational complexity
Randomness efficient identity testing of multivariate polynomials
STOC '01 Proceedings of the thirty-third annual ACM symposium on Theory of computing
Depth-3 arithmetic circuits over fields of characteristic zero
Computational Complexity
Lower Bounds for Matrix Product in Bounded Depth Circuits with Arbitrary Gates
SIAM Journal on Computing
Affine projections of symmetric polynomials
Journal of Computer and System Sciences - Complexity 2001
In search of an easy witness: exponential time vs. probabilistic polynomial time
Journal of Computer and System Sciences - Complexity 2001
Probabilistic algorithms for sparse polynomials
EUROSAM '79 Proceedings of the International Symposiumon on Symbolic and Algebraic Computation
Primality and identity testing via Chinese remaindering
Journal of the ACM (JACM)
Simple extractors for all min-entropies and a new pseudorandom generator
Journal of the ACM (JACM)
Deterministic polynomial identity testing in non-commutative models
Computational Complexity
Derandomizing polynomial identity tests means proving circuit lower bounds
Computational Complexity
Polynomial Identity Testing for Depth 3 Circuits
Computational Complexity
Locally Decodable Codes with Two Queries and Polynomial Identity Testing for Depth 3 Circuits
SIAM Journal on Computing
Proving lower bounds via pseudo-random generators
FSTTCS '05 Proceedings of the 25th international conference on Foundations of Software Technology and Theoretical Computer Science
Read-once polynomial identity testing
STOC '08 Proceedings of the fortieth annual ACM symposium on Theory of computing
Improved Polynomial Identity Testing for Read-Once Formulas
APPROX '09 / RANDOM '09 Proceedings of the 12th International Workshop and 13th International Workshop on Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques
On Lower Bounds for Constant Width Arithmetic Circuits
ISAAC '09 Proceedings of the 20th International Symposium on Algorithms and Computation
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In this paper we show that lower bounds for bounded depth arithmetic circuits imply derandomization of polynomial identity testing for bounded depth arithmetic circuits. More formally, if there exists an explicit polynomial f(x1,...,xm) that cannot be computed by a depth d arithmetic circuit of small size then there exists an efficient deterministic algorithm to test whether a given depth d-8 circuit is identically zero or not (assuming the individual degrees of the tested circuit are not too high). In particular, if we are guaranteed that the circuit computes a multilinear polynomial then we can perform the identity test efficiently. To the best of our knowledge this is the first hardness-randomness tradeoff for bounded depth arithmetic circuits. The above results are obtained using the arithmetic Nisan-Wigderson generator of Impagliazzo and Kabanets together with a new theorem on bounded depth circuits, which is the main technical contribution of our work. This theorem deals with polynomial equations of the form P(x1,...,xn,y) ≡ 0 and shows that if P has a circuit of depth d and size s and if the polynomial f(x1,...,xn) satisfies P(x1,...,xn,f(x1,...,xn))≡ 0 then f has a circuit of depth d+3 and size O(s • r + mr), where m is the degree of f and r is the highest degree of the variable y appearing in P. In the other direction we observe that the methods of Impagliazzo and Kabanets imply that if we can derandomize polynomial identity testing for bounded depth circuits then NEXP does not have bounded depth arithmetic circuits. That is, either NEXP ⊄ P/poly or the Permanent is not computable by polynomial size bounded depth arithmetic circuits.