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arXiv · 2002.02355

An Additive Decomposition in S-Primitive Towers

Hao Du·Jing Guo·Ziming Li·Elaine Wong
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Abstract

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.

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Hao Du, Jing Guo, Ziming Li, Elaine Wong. 2020-02-06. An Additive Decomposition in S-Primitive Towers. https://doi.org/10.1145/3373207.3404025

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