Separating the polynomial-time hierarchy by oracles
Proc. 26th annual symposium on Foundations of computer science
Almost optimal lower bounds for small depth circuits
STOC '86 Proceedings of the eighteenth annual ACM symposium on Theory of computing
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
Feasible arithmetic computations: Valiant's hypothesis
Journal of Symbolic Computation
Annual review of computer science: vol. 3, 1988
Lower bounds for non-commutative computation
STOC '91 Proceedings of the twenty-third annual ACM symposium on Theory of computing
Lower bounds for polynomial evaluation and interpolation problems
SFCS '91 Proceedings of the 32nd annual symposium on Foundations of computer science
On the complexity of bilinear forms: dedicated to the memory of Jacques Morgenstern
STOC '95 Proceedings of the twenty-seventh 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
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
FOCS '95 Proceedings of the 36th Annual Symposium on Foundations of Computer Science
Completeness classes in algebra
STOC '79 Proceedings of the eleventh annual ACM symposium on Theory of computing
Superconcentrators, generalizers and generalized connectors with limited depth
STOC '83 Proceedings of the fifteenth annual ACM symposium on Theory of computing
Lower Bounds for Polynomial Calculus: Non-Binomial Case
FOCS '01 Proceedings of the 42nd IEEE symposium on Foundations of Computer Science
On the Complexity of Matrix Product
SIAM Journal on Computing
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
Pseudorandom Generators in Propositional Proof Complexity
SIAM Journal on Computing
Multilinear- NC" " Multilinear- NC"
FOCS '04 Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science
Derandomizing polynomial identity tests means proving circuit lower bounds
Computational Complexity
Algebraic Complexity Theory
Monotone separations for constant degree polynomials
Information Processing Letters
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
Uniform derandomization from pathetic lower bounds
APPROX/RANDOM'10 Proceedings of the 13th international conference on Approximation, and 14 the International conference on Randomization, and combinatorial optimization: algorithms and techniques
Arithmetic Circuits: A survey of recent results and open questions
Foundations and Trends® in Theoretical Computer Science
Hardness-Randomness Tradeoffs for Bounded Depth Arithmetic Circuits
SIAM Journal on Computing
Permanent does not have succinct polynomial size arithmetic circuits of constant depth
ICALP'11 Proceedings of the 38th international colloquim conference on Automata, languages and programming - Volume Part I
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
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A basic fact in linear algebra is that the image of the curve f(x)=(x1,x2,x3,...,xm), say over C, is not contained in any m-1 dimensional affine subspace of Cm. In other words, the image of f is not contained in the image of any polynomial-mapping Γ:Cm-1 → Cm of degree~1 (that is, an affine mapping). Can one give an explicit example for a polynomial curve f:C → Cm, such that, the image of f is not contained in the image of any polynomial-mapping Γ:Cm-1 → Cm of degree 2? In this paper, we show that problems of this type are closely related to proving lower bounds for the size of general arithmetic circuits. For example, any explicit f as above (with the right notion of explicitness implies super-polynomial lower bounds for computing the permanent over~C. More generally, we say that a polynomial-mapping f:Fn → Fm is (s,r)-elusive, if for every polynomial-mapping Γ:Fs → Fm of degree r, Im(f) ⊄ Im(Γ). We show that for many settings of the parameters n,m,s,r, explicit constructions of elusive polynomial-mappings imply strong (up to exponential) lower bounds for general arithmetic circuits. Finally, for every r n → Fn2, of degree O(r), that is (s,r)-elusive for s = n1+Ω(1/r). We use this to construct for any r, an explicit example for an n-variate polynomial of total-degree O(r), with coefficients in {0,1,}such that, any depth r arithmetic circuit for this polynomial (over any field) is of size ≥ n1+Ω(1/r). In particular, for any constant r, this gives a constant degree polynomial, such that, any depth r arithmetic circuit for this polynomial is of size ≥ n1+Ω(1). Previously, only lower bounds of the type Ω(n • λr (n)), where λr (n) are extremely slowly growing functions (e.g., λ5(n) = log n, and λ7(n) = log* log*n), were known for constant-depth arithmetic circuits for polynomials of constant degree.