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
The monotone circuit complexity of Boolean functions
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Lower bounds for non-commutative computation
STOC '91 Proceedings of the twenty-third annual ACM symposium on Theory of computing
Apolarity and canonical forms for homogeneous polynomials
European Journal of Combinatorics - Special issue dedicated to Bernt Lindstro¨m
Threshold circuits of bounded depth
Journal of Computer and System Sciences
Size--Depth Tradeoffs for Threshold Circuits
SIAM Journal on Computing
An exponential lower bound for depth 3 arithmetic circuits
STOC '98 Proceedings of the thirtieth annual ACM symposium on Theory of computing
Lower bounds on arithmetic circuits via partial derivatives
Computational Complexity
Lower bounds for matrix product, in bounded depth circuits with arbitrary gates
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
Completeness classes in algebra
STOC '79 Proceedings of the eleventh annual ACM symposium on Theory of computing
Hardness-randomness tradeoffs for bounded depth arithmetic circuits
STOC '08 Proceedings of the fortieth annual ACM symposium on Theory of computing
Arithmetic Circuits: A survey of recent results and open questions
Foundations and Trends® in Theoretical Computer Science
Affine projections of polynomials: extended abstract
STOC '12 Proceedings of the forty-fourth annual ACM symposium on Theory of computing
"Resistant" polynomials and stronger lower bounds for depth-three arithmetical formulas
COCOON'07 Proceedings of the 13th annual international conference on Computing and Combinatorics
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In this paper, we introduce a new model for computing polynomials--a depth-2 circuit with a symmetric gate at the top and plus gates at the bottom, i.e. the circuit computes a symmetric function in linear functions--Smd(l1,l2 ...,lm) (Smd is the dth elementary symmetric polynomial in m variables, and the li's are linear functions). We refer to this model as the symmetric model. This new model is related to standard models of arithmetic circuits, especially to depth-3 circuits. In particular, we show that in order to improve the results of Shpilka and Wigderson (in: CCC, Vol. 14, 1999, pp. 87-96), i.e. to prove super-quadratic lower bounds for depth-3 circuits, one must first prove a super-linear lower bound for the symmetric model.We prove two non-trivial linear lower bounds for our model. The first lower bound is for computing the determinant, and the second is for computing the sum of two monomials. The main technical contribution relates the maximal dimension of linear subspaces on which Smd vanishes to lower bounds in the symmetric model. In particular, we show that an answer of the following problem (which is very natural, and of independent interest) will imply lower bound on symmetric circuits for many polynomials: What is the maximal dimension of a linear subspace of Cm, on which Smd vanishes? We give two partial solutions to the problem above, each enables us to prove a different lower bound. Using our techniques we also prove quadratic lower bounds for depth-3 circuits computing the elementary symmetric polynomials of degree ∞n (where 0 α 1 is a constant), thus extending the result of Shpilka and Wigderson (in: CCC, Vol. 14, 1999, pp. 87-96). These are the best lower bounds known for depth-3 circuits over fields of characteristic zero.