Complete Reduction for Derivatives in a Primitive Tower

2025-10-15

AI Summary

This paper addresses the complete reduction problem for derivations in elementary differential fields. We introduce a linear reduction operator that uniquely decomposes any element into a derivative part and a remainder. This decomposition yields a necessary and sufficient condition for elementary integrability over the base field: an element is elementary integrable if and only if the residues of its remainder vanish. We present the first algorithmic construction of complete reduction in elementary towers and extend the telescoping operator framework to a class of non-D-finite functions. Our approach integrates differential algebra theory, structural analysis of elementary towers, and residue/remainder computation techniques. The results resolve a long-standing problem in elementary integrability testing and broaden the applicability of operator-based methods in symbolic integration and hypergeometric summation.

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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.

Problem

• Develops complete reduction algorithm for derivatives in primitive towers
• Provides integrability condition using remainders and residues
• Constructs telescopers for non-D-finite functions in special towers

Innovation

• Complete reduction algorithm for primitive tower derivatives
• Elementary integration criterion using remainders and residues
• Telescoper construction method for non-D-finite functions