Valiant defines algebraic analogues of the classes P and NP. We characterize the classes VP and VQP, yielding a simplified proof of VNP = VNPe and of the VQP-completeness of the determinant, and a proof of a conjecture by Bürgisser. The classes VPo and VNPo, defined without arbitrary constants, yield a link between the complexity of a polynomial and that of its coefficient function: VNPo is stable for the operation of taking coefficient functions; claiming that this holds for VPo is equivalent to VPo = VNPo. For polynomials of unbounded degree, one needs efficient computations of binomial coefficients, which can be done in positive characteristic but are unlikely in characteristic 0. At last we study the related problem of the effect of derivation on complexity. After a new proof of a result by Kaltofen (the number of variables matters more than the derivation order) we show how to simultaneously compute partial derivatives.
