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arXiv · 2510.13456

Complete Reduction for Derivatives in a Primitive Tower

Hao Du·Yiman Gao·Wenqiao Li·Ziming Li
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Abstract

A complete reduction $\phi$ 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 $\phi(f)$. A direct application of $\phi$ is that $f$ is in-field integrable if and only if $\phi(f) = 0.$ 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.

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Hao Du, Yiman Gao, Wenqiao Li, Ziming Li. 2025-10-15. Complete Reduction for Derivatives in a Primitive Tower. https://arxiv.org/abs/2510.13456

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