Iterated and Generalized Iterated Integrals

TL;DR

This work characterizes Picard-Vessiot extensions with unipotent Galois groups over differential fields with algebraically closed constants, revealing that such extensions are generated by generalized iterated integrals and that the Picard-Vessiot ring is precisely the algebra of these integrals. It provides constructive factorizations of differential operators and matrix realizations that realize unipotent groups as differential Galois groups, linking to Kovacic’s inverse problem. The authors apply these results to stability questions in integration, establishing Liouville-type criteria for elementary and higher-order integrability (including $n$- and $\infty$-integrability) and giving explicit classifications for integrable elements in canonical towers like $C(x,e^x)$, $C(x,\log x)$, and $C(x^{1/n})$. Overall, the paper advances understanding of when unipotent Galois groups arise from iterated integrals and provides practical tools for stability analysis in symbolic integration and differential Galois theory.

Abstract

For a differential field $F$ having an algebraically closed field of constants, we analyze the structure of Picard-Vessiot extensions of $F$ whose differential Galois groups are unipotent algebraic groups and apply these results to study stability problems in integration in finite terms and the inverse problem in differential Galois theory for unipotent algebraic groups.

Theorems & Definitions (30)

• Theorem 1.1
• Theorem 1.2: Generalization of Liouville's Theorem
• Proposition 2.1
• Proposition 2.2
• proof
• Theorem 2.3
• proof
• Proposition 2.4
• proof
• Remark 2.5