2020/02/06 by Hao Du, Jing Guo, Ziming Li +1 · 1 citation
Computer Science · #cs.SC
paper · pdf · doi:10.1145/3373207.3404025
published as ISSAC 2020: Proceedings of the 45th International Symposium on Symbolic and Algebraic Computation, July 2020, Pages 146-153 · This article has been submitted to ISSAC2020 for review. Supplementary material at https://wongey.github.io/add-decomp-sprimitive/
arxiv created 2020/02/06 · arxiv updated 2020/10/20
We consider the additive decomposition problem in primitive towers and present an algorithm to decompose a function in an S-primitive tower 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 solving any differential equations. We also show that a kind of S-primitive towers, known as logarithmic towers, can be embedded into a particular extension where we can obtain a finer remainder.