Additive Decompositions in Primitive Extensions

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...

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Bibliographic Details
Published inarXiv.org
Main Authors Chen, Shaoshi, Du, Hao, Li, Ziming
Format Paper
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 07.02.2018
Subjects
Decomposition
Rational functions
Reduction
Telescoping
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Summary: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.
ISSN:2331-8422