Complete Reduction for Derivatives in a Primitive Tower
Hao Du, Yiman Gao, Wenqiao Li, Ziming Li
TL;DR
The paper develops an algorithmic complete reduction for derivatives inside primitive towers, enabling a clean decomposition $f = g' + r$ with a unique remainder $r$ and a derivative part $g'$. By leveraging remainders and residues, it provides a practical criterion for in-field integrability and constructs elementary integrals within these transcendental extensions. A central contribution is the CompleteReduction framework, which extends reductions from a base field to primitive tower extensions and yields $R$-pairs for elements, facilitating both integration and the construction of telescopers for certain non-D-finite functions. These advancements advance symbolic integration in Liouvillian-type extensions and support effective computations in differential algebra and creative telescoping.
Abstract
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.