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Exponential lower bounds for depth 3 arithmetic circuits in algebras of functions over finite fields. (English) Zbl 1040.68045

Summary: A depth 3 arithmetic circuit can be viewed as a sum of products of linear functions. We prove an exponential complexity lower bound on depth 3 arithmetic circuits computing some natural symmetric functions over a finite field \(F\). Also, we study the complexity of the functions \(f\:D^n\to F\) for subsets \(D\subset F\). In particular, we prove an exponential lower bound on the complexity of depth 3 arithmetic circuits computing some explicit functions \(f\:(F^*)^n\to F\) (in particular, the determinant of a matrix).

MSC:

68Q17 Computational difficulty of problems (lower bounds, completeness, difficulty of approximation, etc.)
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