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
Depth-3 arithmetic circuits over fields of characteristic zero
Computational Complexity
Affine projections of symmetric polynomials
Journal of Computer and System Sciences - Complexity 2001
Separating the polynomial-time hierarchy by oracles
SFCS '85 Proceedings of the 26th Annual Symposium on Foundations of Computer Science
Algebraic Complexity Theory
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We derive quadratic lower bounds on the *-complexity of sum-of-products-of-sums (ΣΠΣ) formulas for classes of polynomials f that have too few partial derivatives for the techniques of Shpilka and Wigderson [10,9]. This involves a notion of "resistance" which connotes full-degree behavior of f under any projection to an affine space of sufficiently high dimension. They also show stronger lower bounds over the reals than the complex numbers or over arbitrary fields. Separately, by applying a special form of the Baur-Strassen Derivative Lemma tailored to ΣΠΣ formulas, we obtain sharper bounds on +, *-complexity than those shown for *-complexity by Shpilka and Wigderson [10], most notably for the lowest-degree cases of the polynomials they consider.