Additive Decompositions in Primitive Extensions

7 Feb 2018  ·  Shaoshi Chen, Hao Du, Ziming Li ·

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.

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Symbolic Computation 33F10, 68W30, 12H05


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