An additive decomposition in logarithmic towers and beyond

Hao Du, Jing Guo, Ziming Li, Elaine Wong

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

3 Scopus citations

Abstract

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.

Original languageEnglish
Title of host publicationISSAC 2020 - Proceedings of the 45th International Symposium on Symbolic and Algebraic Computation
EditorsAngelos Mantzaflaris
PublisherAssociation for Computing Machinery
Pages146-153
Number of pages8
ISBN (Electronic)9781450371001
DOIs
StatePublished - Jul 20 2020
Externally publishedYes
Event45th International Symposium on Symbolic and Algebraic Computation, ISSAC 2020 - Kalamata, Virtual, Greece
Duration: Jul 20 2020Jul 23 2020

Publication series

NameProceedings of the International Symposium on Symbolic and Algebraic Computation, ISSAC

Conference

Conference45th International Symposium on Symbolic and Algebraic Computation, ISSAC 2020
Country/TerritoryGreece
CityKalamata, Virtual
Period07/20/2007/23/20

Keywords

  • additive decomposition
  • elementary integral
  • logarithmic tower
  • primitive tower
  • symbolic integration

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