Complete Reduction for Derivatives in a Transcendental Liouvillian Extension
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arXiv
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| Autori principali: | , , , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866908809560064000 |
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| author | Chen, Shaoshi Du, Hao Gao, Yiman huang, Hui Li, Wenqiao Li, Ziming |
| author_facet | Chen, Shaoshi Du, Hao Gao, Yiman huang, Hui Li, Wenqiao Li, Ziming |
| contents | Transcendental Liouvillian extensions are differential fields, in which one can model poly-logarithmic, hyperexponential, and trigonometric functions, logarithmic integrals, and their (nested) rational expressions. For such an extension $(F, \, ^\prime)$ with the subfield $C$ of constants, we construct a complementary subspace $W$ for the $C$-subspace of derivatives in $F$, and develop an algorithm that, for every $f \in F$, computes a pair $(g,r) \in F \times W$ such that $f = g^\prime + r$. Moreover, $f$ is a derivative in $F$ if and only if $r=0$. The algorithm enables us to determine elementary integrability over $F$ by computing parametric logarithmic parts, and leads to a reduction-based approach to constructing telescopers for functions that can be represented by elements in $F$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_03592 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Complete Reduction for Derivatives in a Transcendental Liouvillian Extension Chen, Shaoshi Du, Hao Gao, Yiman huang, Hui Li, Wenqiao Li, Ziming Symbolic Computation 68U01 I.1.2 Transcendental Liouvillian extensions are differential fields, in which one can model poly-logarithmic, hyperexponential, and trigonometric functions, logarithmic integrals, and their (nested) rational expressions. For such an extension $(F, \, ^\prime)$ with the subfield $C$ of constants, we construct a complementary subspace $W$ for the $C$-subspace of derivatives in $F$, and develop an algorithm that, for every $f \in F$, computes a pair $(g,r) \in F \times W$ such that $f = g^\prime + r$. Moreover, $f$ is a derivative in $F$ if and only if $r=0$. The algorithm enables us to determine elementary integrability over $F$ by computing parametric logarithmic parts, and leads to a reduction-based approach to constructing telescopers for functions that can be represented by elements in $F$. |
| title | Complete Reduction for Derivatives in a Transcendental Liouvillian Extension |
| topic | Symbolic Computation 68U01 I.1.2 |
| url | https://arxiv.org/abs/2602.03592 |