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Abstract

Valiant (1980) showed that general arithmetic circuits with negation can be exponentially more powerful than monotone ones. We give the first qualitative improvement to this classical result: we construct a family of polynomials P-n in n variables, each of its monomials has positive coefficient, such that P-n can be computed by a polynomial-size *depth-three* formula but every monotone circuit computing it has size exp(n^{1/4}/log(n)).