Communications of the ACM
How to construct random functions
Journal of the ACM (JACM)
Almost optimal lower bounds for small depth circuits
STOC '86 Proceedings of the eighteenth annual ACM symposium on Theory of computing
Constructing a perfect matching is in random NC
Combinatorica
Algebraic methods in the theory of lower bounds for Boolean circuit complexity
STOC '87 Proceedings of the nineteenth annual ACM symposium on Theory of computing
Matching is as easy as matrix inversion
Combinatorica
Feasible arithmetic computations: Valiant's hypothesis
Journal of Symbolic Computation
A deterministic algorithm for sparse multivariate polynomial interpolation
STOC '88 Proceedings of the twentieth annual ACM symposium on Theory of computing
Matrix multiplication via arithmetic progressions
Journal of Symbolic Computation - Special issue on computational algebraic complexity
Journal of Symbolic Computation - Special issue on computational algebraic complexity
Fast parallel algorithms for sparse multivariate polynomial interpolation over finite fields
SIAM Journal on Computing
Journal of Algorithms
Lower bounds for non-commutative computation
STOC '91 Proceedings of the twenty-third annual ACM symposium on Theory of computing
Learning read-once formulas over fields and extended bases
COLT '91 Proceedings of the fourth annual workshop on Computational learning theory
Handbook of theoretical computer science (vol. A)
Lower bounds for polynomial evaluation and interpolation problems
SFCS '91 Proceedings of the 32nd annual symposium on Foundations of computer science
PP is as hard as the polynomial-time hierarchy
SIAM Journal on Computing
Algebraic methods for interactive proof systems
Journal of the ACM (JACM)
Journal of the ACM (JACM)
Cryptographic limitations on learning Boolean formulae and finite automata
Journal of the ACM (JACM)
A direct version of Shamir and Snir's lower bounds on monotone circuit depth
Information Processing Letters
Journal of Computer and System Sciences
An introduction to computational learning theory
An introduction to computational learning theory
Learning Arithmetic Read-Once Formulas
SIAM Journal on Computing
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
Journal of Computer and System Sciences - Special issue: 26th annual ACM symposium on the theory of computing & STOC'94, May 23–25, 1994, and second annual Europe an conference on computational learning theory (EuroCOLT'95), March 13–15, 1995
Complexity and real computation
Complexity and real computation
Probabilistic checking of proofs: a new characterization of NP
Journal of the ACM (JACM)
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
Proof verification and the hardness of approximation problems
Journal of the ACM (JACM)
Interpolating Arithmetic Read-Once Formulas in Parallel
SIAM Journal on Computing
A Spectral Approach to Lower Bounds with Applications to Geometric Searching
SIAM Journal on Computing
Lower bounds on arithmetic circuits via partial derivatives
Computational Complexity
On interpolating arithmetic read-once formulas with exponentiation
Journal of Computer and System Sciences
Modern computer algebra
Note on a Lower Bound on the Linear Complexity of the Fast Fourier Transform
Journal of the ACM (JACM)
Fast Probabilistic Algorithms for Verification of Polynomial Identities
Journal of the ACM (JACM)
Learning functions represented as multiplicity automata
Journal of the ACM (JACM)
Reducing Randomness via Irrational Numbers
SIAM Journal on Computing
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
Introduction to Automata Theory, Languages and Computability
Introduction to Automata Theory, Languages and Computability
Geometric Complexity Theory I: An Approach to the P vs. NP and Related Problems
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)
Nearly tight bounds on the learnability of evolution
FOCS '97 Proceedings of the 38th Annual Symposium on Foundations of Computer Science
Completeness classes in algebra
STOC '79 Proceedings of the eleventh annual ACM symposium on Theory of computing
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
Lower Bounds for Matrix Product
SIAM Journal on Computing
On the Complexity of Matrix Product
SIAM Journal on Computing
A Group-Theoretic Approach to Fast Matrix Multiplication
FOCS '03 Proceedings of the 44th Annual IEEE Symposium on Foundations of Computer Science
Combinatorics, Probability and Computing
Deterministic polynomial identity testing in non-commutative models
Computational Complexity
Derandomizing polynomial identity tests means proving circuit lower bounds
Computational Complexity
A Lower Bound on the Complexity of Polynomial Multiplication over Finite Fields
SIAM Journal on Computing
More on Noncommutative Polynomial Identity Testing
CCC '05 Proceedings of the 20th Annual IEEE Conference on Computational Complexity
Group-theoretic Algorithms for Matrix Multiplication
FOCS '05 Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science
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
An Improved Lower Bound on Polynomial Multiplication
IEEE Transactions on Computers
Algebras with Polynomial Identities and Computing the Determinant
SIAM Journal on Computing
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
Read-once polynomial identity testing
STOC '08 Proceedings of the fortieth annual ACM symposium on Theory of computing
Elusive functions and lower bounds for arithmetic circuits
STOC '08 Proceedings of the fortieth annual ACM symposium on Theory of computing
New Results on Noncommutative and Commutative Polynomial Identity Testing
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
The matching problem for bipartite graphs with polynomially bounded permanents is in NC
SFCS '87 Proceedings of the 28th Annual Symposium on Foundations of Computer Science
Geometric Complexity Theory II: Towards Explicit Obstructions for Embeddings among Class Varieties
SIAM Journal on Computing
Diagonal Circuit Identity Testing and Lower Bounds
ICALP '08 Proceedings of the 35th international colloquium on Automata, Languages and Programming, Part I
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
Efficient learning algorithms yield circuit lower bounds
Journal of Computer and System Sciences
Cryptographic hardness for learning intersections of halfspaces
Journal of Computer and System Sciences
Arithmetic Circuits: A Chasm at Depth Four
FOCS '08 Proceedings of the 2008 49th Annual IEEE Symposium on Foundations of Computer Science
Fast Modular Composition in any Characteristic
FOCS '08 Proceedings of the 2008 49th Annual IEEE Symposium on Foundations of Computer Science
Multilinear Formulas, Maximal-Partition Discrepancy and Mixed-Sources Extractors
FOCS '08 Proceedings of the 2008 49th Annual IEEE Symposium on Foundations of Computer Science
Deterministically testing sparse polynomial identities of unbounded degree
Information Processing Letters
Balancing Syntactically Multilinear Arithmetic Circuits
Computational Complexity
Multi-linear formulas for permanent and determinant are of super-polynomial size
Journal of the ACM (JACM)
Computational Complexity: A Modern Approach
Computational Complexity: A Modern Approach
On lattices, learning with errors, random linear codes, and cryptography
Journal of the ACM (JACM)
An Almost Optimal Rank Bound for Depth-3 Identities
CCC '09 Proceedings of the 2009 24th Annual IEEE Conference on Computational Complexity
Reconstruction of Generalized Depth-3 Arithmetic Circuits with Bounded Top Fan-in
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
Interpolation of Depth-3 Arithmetic Circuits with Two Multiplication Gates
SIAM Journal on Computing
Monotone separations for constant degree polynomials
Information Processing Letters
Blackbox Polynomial Identity Testing for Depth 3 Circuits
FOCS '09 Proceedings of the 2009 50th Annual IEEE Symposium on Foundations of Computer Science
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
Tensor-rank and lower bounds for arithmetic formulas
Proceedings of the forty-second ACM symposium on Theory of computing
Non-commutative circuits and the sum-of-squares problem
Proceedings of the forty-second ACM symposium on Theory of computing
Relationless Completeness and Separations
CCC '10 Proceedings of the 2010 IEEE 25th Annual Conference on Computational Complexity
On Matrix Rigidity and Locally Self-Correctable Codes
CCC '10 Proceedings of the 2010 IEEE 25th Annual Conference on Computational Complexity
On the relation between polynomial identity testing and finding variable disjoint factors
ICALP'10 Proceedings of the 37th international colloquium conference on Automata, languages and programming
Hardness-Randomness Tradeoffs for Bounded Depth Arithmetic Circuits
SIAM Journal on Computing
FOCS '10 Proceedings of the 2010 IEEE 51st Annual Symposium on Foundations of Computer Science
Algebraic Complexity Theory
Proving lower bounds via pseudo-random generators
FSTTCS '05 Proceedings of the 25th international conference on Foundations of Software Technology and Theoretical Computer Science
Black-box identity testing of depth-4 multilinear circuits
Proceedings of the forty-third annual ACM symposium on Theory of computing
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
Algebraic independence and blackbox identity testing
ICALP'11 Proceedings of the 38th international conference on Automata, languages and programming - Volume Part II
On identity testing of tensors, low-rank recovery and compressed sensing
STOC '12 Proceedings of the forty-fourth annual ACM symposium on Theory of computing
Short proofs for the determinant identities
STOC '12 Proceedings of the forty-fourth annual ACM symposium on Theory of computing
STOC '12 Proceedings of the forty-fourth annual ACM symposium on Theory of computing
Separating multilinear branching programs and formulas
STOC '12 Proceedings of the forty-fourth annual ACM symposium on Theory of computing
Reconstruction of depth-4 multilinear circuits with top fan-in 2
STOC '12 Proceedings of the forty-fourth annual ACM symposium on Theory of computing
Affine projections of polynomials: extended abstract
STOC '12 Proceedings of the forty-fourth annual ACM symposium on Theory of computing
Software Engineering and complexity in effective Algebraic Geometry
Journal of Complexity
Algebraic independence and blackbox identity testing
Information and Computation
Quasi-polynomial hitting-set for set-depth-Δ formulas
Proceedings of the forty-fifth annual ACM symposium on Theory of computing
Verifiable delegation of computation on outsourced data
Proceedings of the 2013 ACM SIGSAC conference on Computer & communications security
Arithmetic circuit lower bounds via maxrank
ICALP'13 Proceedings of the 40th international conference on Automata, Languages, and Programming - Volume Part I
Tensor-Rank and Lower Bounds for Arithmetic Formulas
Journal of the ACM (JACM)
Non-commutative arithmetic circuits with division
Proceedings of the 5th conference on Innovations in theoretical computer science
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A large class of problems in symbolic computation can be expressed as the task of computing some polynomials; and arithmetic circuits form the most standard model for studying the complexity of such computations. This algebraic model of computation attracted a large amount of research in the last five decades, partially due to its simplicity and elegance. Being a more structured model than Boolean circuits, one could hope that the fundamental problems of theoretical computer science, such as separating P from NP, will be easier to solve for arithmetic circuits. However, in spite of the appearing simplicity and the vast amount of mathematical tools available, no major breakthrough has been seen. In fact, all the fundamental questions are still open for this model as well. Nevertheless, there has been a lot of progress in the area and beautiful results have been found, some in the last few years. As examples we mention the connection between polynomial identity testing and lower bounds of Kabanets and Impagliazzo, the lower bounds of Raz for multilinear formulas, and two new approaches for proving lower bounds: Geometric Complexity Theory and Elusive Functions. The goal of this monograph is to survey the field of arithmetic circuit complexity, focusing mainly on what we find to be the most interesting and accessible research directions. We aim to cover the main results and techniques, with an emphasis on works from the last two decades. In particular, we discuss the recent lower bounds for multilinear circuits and formulas, the advances in the question of deterministically checking polynomial identities, and the results regarding reconstruction of arithmetic circuits. We do, however, also cover part of the classical works on arithmetic circuits. In order to keep this monograph at a reasonable length, we do not give full proofs of most theorems, but rather try to convey the main ideas behind each proof and demonstrate it, where possible, by proving some special cases.