A Dichotomy Theorem for Homomorphism Polynomials

In the present paper we show a dichotomy theorem for the complexity of polynomial evaluation. We associate to each graph H a polynomial that encodes all graphs of a fixed size homomorphic to H . We show that this family is computable by arithmetic circuits in constant depth if H has a loop or no edge and that it is hard otherwise (i.e., complete for VNP, the arithmetic class related to #P ). We also demonstrate the hardness over Q of cut eliminator, a polynomial defined by Bürgisser which is known to be neither VP nor VNP-complete in F2, if VP , VNP (VP is the class of polynomials computable by arithmetic circuit of polynomial size).