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.

Authors

• Hao Du, Beijing University of Posts and Telecommunications

  • Contact: [email protected]
  • Previously at: Johannes Kepler University, Linz, Austria

• Jing Guo

  • Previously at: Chinese Academy of Sciences, KLMM

• Ziming Li, Chinese Academy of Sciences, KLMM

  • Contact: [email protected]

• Elaine Wong, Oak Ridge National Laboratory, USA

  • Contact: [email protected]
  • Previously at: Austrian Academy of Sciences, RICAM

Paper

• This paper has been accepted to ISSAC 2020 with assigned DOI for the corresponding proceedings.

• The preprint can be found at arXiv:2002.02355.

• A more recent preprint can be found as a RICAM Report.

• The appendix to our paper is here. The references here.

• Cite our paper by using this bib.

Package Download

• The Mathematica package AdditiveDecomposition.m (Version 0.2) is available for download.

• 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).

• For those without a Mathematica installation, we offer a pdf version of the example notebook for convenience.

ISSAC 2020

• Here are the slides from our pre-recorded talk.

• Here is the PDF version of Mathematica demo from our talk.

Followup Work

• Check out this paper from ISSAC 2023.