Additive Decompositions in Primitive Extensions

Abstract

This paper extends the classical Ostrogradsky-Hermite 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.