Additive Decompositions in Primitive Extensions

TL;DR

This paper generalizes classical rational function reduction techniques to primitive extensions, enabling better integration and telescoping methods for complex functions beyond D-finite cases.

Contribution

It introduces an extended additive decomposition method for primitive extensions, facilitating integration and creative telescoping for broader classes of functions.

Findings

Decomposition separates functions into derivatives and residuals in primitive extensions.

Residuals determine integrability and elementary integrability conditions.

Method is applicable to nested logarithmic functions beyond D-finite functions.

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.