Additive Decompositions in Primitive Extensions

Abstract

This paper extends the classical Hermite-Ostrogradsky reduction for rational functions to more general functions in primitive extensions of certain types. For an element f in such an extension K , the extended reduction decomposes f as the sum of a derivative in K and another element r such that f has an antiderivative in K if and only if r=0; and f has an elementary antiderivative over K if and only if r is a linear combination of logarithmic derivatives over the constants when K is a logarithmic extension. Moreover, r is minimal in some sense. Additive decompositions may lead to reduction-based creative-telescoping methods for nested logarithmic functions, which are not necessarily D -finite.