The Complexity of Polynomials and Their Coefficient Functions

We study the link between the complexity of a polynomial and that of its coefficient functions. Valiant´s theory is a good setting for this, and we start by generalizing one of Valiant´s observations, showing that the class VNP is stable for coefficient functions, and that this is true of the class VP iff VP=VNP, an eventuality which would be as surprising as the equality of the classes P and NP in Boolean complexity. We extend the definition of Valiant´s classes to polynomials of unbounded degree, thus defining the classes VP_nb and VNP_nb. Over rings of positive characteristic the same kind of results hold in this case, and we also prove that VP=VNP iff VP_nb=VNP_nb. Finally, we use our extension of Valiant´s results to show that iterated partial derivatives can be efficiently computed iff VP=VNP. This is also true for the case of polynomials of unbounded degree, if the characteristic of the ring is positive.