Exponential complexity lower bounds for depth 3 arithmetic circuits in algebras of functions over finite fields

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: Dⁿ→F for subsets D⊂F. In particular, we prove an exponential lower bound on the complexity of a depth 3 arithmetic circuit which computes the determinant or the permanent of a matrix considered as functions f:(F*)n²→F