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Fast Probabilistic Algorithms for Verification of Polynomial Identities
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Randomness efficient identity testing of multivariate polynomials
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Lower Bounds for Linear Locally Decodable Codes and Private Information Retrieval
CCC '02 Proceedings of the 17th IEEE Annual Conference on Computational Complexity
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STOC '08 Proceedings of the fortieth annual ACM symposium on Theory of computing
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Arithmetic Circuits: A Chasm at Depth Four
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FOCS '09 Proceedings of the 2009 50th Annual IEEE Symposium on Foundations of Computer Science
Deterministic identity testing of depth-4 multilinear circuits with bounded top fan-in
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Tensor-rank and lower bounds for arithmetic formulas
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Arithmetic Circuits: A survey of recent results and open questions
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Blackbox identity testing for bounded top fanin depth-3 circuits: the field doesn't matter
Proceedings of the forty-third annual ACM symposium on Theory of computing
Derandomizing Polynomial Identity Testing for Multilinear Constant-Read Formulae
CCC '11 Proceedings of the 2011 IEEE 26th Annual Conference on Computational Complexity
Algebraic independence and blackbox identity testing
ICALP'11 Proceedings of the 38th international conference on Automata, languages and programming - Volume Part II
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SIAM Journal on Computing
Tight Lower Bounds for 2-query LCCs over Finite Fields
FOCS '11 Proceedings of the 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science
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FSTTCS '05 Proceedings of the 25th international conference on Foundations of Software Technology and Theoretical Computer Science
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We study the problem of identity testing for depth-3 circuits of top fanin k and degree d. We give a new structure theorem for such identities that improves the known deterministic dkO(k)-time blackbox identity test over rationals [Kayal and Saraf, 2009] to one that takes dO(k2)-time. Our structure theorem essentially says that the number of independent variables in a real depth-3 identity is very small. This theorem affirmatively settles the strong rank conjecture posed by Dvir and Shpilka [2006]. We devise various algebraic tools to study depth-3 identities, and use these tools to show that any depth-3 identity contains a much smaller nucleus identity that contains most of the “complexity” of the main identity. The special properties of this nucleus allow us to get near optimal rank bounds for depth-3 identities. The most important aspect of this work is relating a field-dependent quantity, the Sylvester-Gallai rank bound, to the rank of depth-3 identities. We also prove a high-dimensional Sylvester-Gallai theorem for all fields, and get a general depth-3 identity rank bound (slightly improving previous bounds).