We consider the additive decomposition problem in primitive towers and present an algorithm to decompose a function in a certain kind of primitive tower which we call S-primitive, as a sum of a derivative in the tower and a remainder which is minimal in some sense. Special instances of S-primitive towers include differential fields generated by finitely many logarithmic functions and logarithmic integrals. A function in an S-primitive tower is integrable in the tower if and only if the remainder is equal to zero. The additive decomposition is achieved by viewing our towers not as a traditional chain of extension fields, but rather as a direct sum of certain subrings. Furthermore, we can determine whether or not a function in an S-primitive tower has an elementary integral without the need to deal with differential equations explicitly. We also show that any logarithmic tower can be embedded into a particular extension where we can further decompose the given function. The extension is constructed using only differential field operations without introducing any new constants.
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Hao Du, Beijing University of Posts and Telecommunications
- Contact: duhao@amss.ac.cn
- Previously at: Johannes Kepler University, Linz, Austria
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Jing Guo
- Previously at: Chinese Academy of Sciences, KLMM
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Ziming Li, Chinese Academy of Sciences, KLMM
- Contact: zmli@mmrc.iss.ac.cn
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Elaine Wong, Oak Ridge National Laboratory, USA
- Contact: wongey@ornl.gov
- Previously at: Austrian Academy of Sciences, RICAM
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This paper has been accepted to ISSAC 2020 with assigned DOI for the corresponding proceedings.
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The preprint can be found at arXiv:2002.02355.
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A more recent preprint can be found as a RICAM Report.
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Cite our paper by using this bib.
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The Mathematica package AdditiveDecomposition.m (Version 0.2) is available for download.
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The Mathematica notebook AdditiveDecomposition_Examples.nb contains some examples, including this collection, that illustrate the use of package (requires that the package and example file be stored in the same directory as the notebook).
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For those without a Mathematica installation, we offer a pdf version of the example notebook for convenience.
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Here are the slides from our pre-recorded talk.
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Here is the PDF version of Mathematica demo from our talk.
- Check out this paper from ISSAC 2023.