Complete Reduction for Derivatives in a Primitive Tower
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912648895922176 |
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| author | Du, Hao Gao, Yiman Li, Wenqiao Li, Ziming |
| author_facet | Du, Hao Gao, Yiman Li, Wenqiao Li, Ziming |
| contents | A complete reduction $ϕ$ for derivatives in a differential field is a linear operator on the field over its constant subfield. The reduction enables us to decompose an element $f$ as the sum of a derivative and the remainder $ϕ(f)$. A direct application of $ϕ$ is that $f$ is in-field integrable if and only if $ϕ(f) = 0.$
In this paper, we present a complete reduction for derivatives in a primitive tower algorithmically. Typical examples for primitive towers are differential fields generated by (poly-)logarithmic functions and logarithmic integrals. Using remainders and residues, we provide a necessary and sufficient condition for an element from a primitive tower to have an elementary integral, and discuss how to construct telescopers for non-D-finite functions in some special primitive towers. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_13456 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Complete Reduction for Derivatives in a Primitive Tower Du, Hao Gao, Yiman Li, Wenqiao Li, Ziming Symbolic Computation 68U01 I.1.2 A complete reduction $ϕ$ for derivatives in a differential field is a linear operator on the field over its constant subfield. The reduction enables us to decompose an element $f$ as the sum of a derivative and the remainder $ϕ(f)$. A direct application of $ϕ$ is that $f$ is in-field integrable if and only if $ϕ(f) = 0.$ In this paper, we present a complete reduction for derivatives in a primitive tower algorithmically. Typical examples for primitive towers are differential fields generated by (poly-)logarithmic functions and logarithmic integrals. Using remainders and residues, we provide a necessary and sufficient condition for an element from a primitive tower to have an elementary integral, and discuss how to construct telescopers for non-D-finite functions in some special primitive towers. |
| title | Complete Reduction for Derivatives in a Primitive Tower |
| topic | Symbolic Computation 68U01 I.1.2 |
| url | https://arxiv.org/abs/2510.13456 |