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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 VPnb and VNPnb. Over rings of positive characteristic the same kind of results hold in this case, and we also prove that VP=VNP iff VPnb=VNPnb. 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.