A dichotomy theorem for polynomial evaluation

Abstract. A dichotomy theorem for counting problems due to Creignou and Hermann states that or any finite set S of logical relations, the count-ing problem #SAT(S) is either in FP, or #P-complete. In the present paper we show a dichotomy theorem for polynomial evaluation. That is, we show that for a given set S, either there exists a VNP-complete family of polynomials associated to S, or the associated families of polynomi-als are all in VP. We give a concise characterization of the sets S that give rise to “easy ” and “hard ” polynomials. We also prove that several problems which were known to be #P-complete under Turing reductions only are in fact #P-complete under many-one reductions.