Matching is as easy as matrix inversion
Combinatorica
Learning read-once formulas over fields and extended bases
COLT '91 Proceedings of the fourth annual workshop on Computational learning theory
Learning Arithmetic Read-Once Formulas
SIAM Journal on Computing
Modern computer algebra
Fast Probabilistic Algorithms for Verification of Polynomial Identities
Journal of the ACM (JACM)
Randomness efficient identity testing of multivariate polynomials
STOC '01 Proceedings of the thirty-third annual ACM symposium on Theory of computing
Deterministic identity testing for multivariate polynomials
SODA '03 Proceedings of the fourteenth annual ACM-SIAM symposium on Discrete algorithms
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)
Testing polynomials which are easy to compute (Extended Abstract)
STOC '80 Proceedings of the twelfth annual ACM symposium on Theory of computing
Polynomial factorization: a success story
ISSAC '03 Proceedings of the 2003 international symposium on Symbolic and algebraic computation
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
Deterministic polynomial identity testing in non-commutative models
Computational Complexity
Derandomizing polynomial identity tests means proving circuit lower bounds
Computational Complexity
On the complexity of factoring bivariate supersparse (Lacunary) polynomials
Proceedings of the 2005 international symposium on Symbolic and algebraic computation
Who was who in polynomial factorization: 1
Proceedings of the 2006 international symposium on Symbolic and algebraic computation
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
Read-once polynomial identity testing
STOC '08 Proceedings of the fortieth annual ACM symposium on Theory of computing
Lower Bounds and Separations for Constant Depth Multilinear Circuits
CCC '08 Proceedings of the 2008 IEEE 23rd Annual Conference on Computational Complexity
CCC '08 Proceedings of the 2008 IEEE 23rd Annual Conference on Computational Complexity
Derandomizing the Isolation Lemma and Lower Bounds for Circuit Size
APPROX '08 / RANDOM '08 Proceedings of the 11th international workshop, APPROX 2008, and 12th international workshop, RANDOM 2008 on Approximation, Randomization and Combinatorial Optimization: Algorithms and Techniques
Arithmetic Circuits: A Chasm at Depth Four
FOCS '08 Proceedings of the 2008 49th Annual IEEE Symposium on Foundations of Computer Science
An Almost Optimal Rank Bound for Depth-3 Identities
CCC '09 Proceedings of the 2009 24th Annual IEEE Conference on Computational Complexity
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
A Lower Bound for the Size of Syntactically Multilinear Arithmetic Circuits
SIAM Journal on Computing
The monomial ideal membership problem and polynomial identity testing
ISAAC'07 Proceedings of the 18th international conference on Algorithms and computation
Deterministic identity testing of depth-4 multilinear circuits with bounded top fan-in
Proceedings of the forty-second ACM symposium on Theory of computing
Hardness-Randomness Tradeoffs for Bounded Depth Arithmetic 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
Arithmetic Circuits: A survey of recent results and open questions
Foundations and Trends® in Theoretical Computer Science
Black-box identity testing of depth-4 multilinear circuits
Proceedings of the forty-third annual ACM symposium on Theory of computing
STOC '12 Proceedings of the forty-fourth annual ACM symposium on Theory of computing
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We say that a polynomial f(x1, ..., xn) is indecomposable if it cannot be written as a product of two polynomials that are defined over disjoint sets of variables. The polynomial decomposition problem is defined to be the task of finding the indecomposable factors of a given polynomial. Note that for multilinear polynomials, factorization is the same as decomposition, as any two different factors are variable disjoint. In this paper we show that the problem of derandomizing polynomial identity testing is essentially equivalent to the problem of derandomizing algorithms for polynomial decomposition. More accurately, we show that for any reasonable circuit class there is a deterministic polynomial time (black-box) algorithm for polynomial identity testing of that class if and only if there is a deterministic polynomial time (black-box) algorithm for factoring a polynomial, computed in the class, to its indecomposable components. An immediate corollary is that polynomial identity testing and polynomial factorization are equivalent (up to a polynomial overhead) for multilinear polynomials. In addition, we observe that derandomizing the polynomial decomposition problem is equivalent, in the sense of Kabanets and Impagliazzo [1], to proving arithmetic circuit lower bounds for NEXP. Our approach uses ideas from [2], that showed that the polynomial identity testing problem for a circuit class C is essentially equivalent to the problem of deciding whether a circuit from C computes a polynomial that has a read-once arithmetic formula.