Liouville's Theorem on integration in finite terms for $\mathrm D_\infty,$ $ \mathrm{SL}_2$ and Weierstrass field extensions

TL;DR

This paper proves that Liouville's theorem on integration in finite terms applies to certain differential field extensions involving Weierstrass functions, SL2, and D_infinity groups, extending classical results to these complex cases.

Contribution

It extends Liouville's theorem to differential field extensions with specific algebraic and differential Galois groups, including Weierstrass and special linear group cases.

Findings

Liouville's theorem holds for these complex differential field extensions.

The paper characterizes integrability in finite terms for extensions with SL2 and D_infinity Galois groups.

It provides a constructive criterion for expressing derivatives as sums of logarithmic derivatives and elements in the base field.

Abstract

Let $k$ be a differential field of characteristic zero and the field of constants $C$ of $k$ be an algebraically closed field. Let $E$ be a differential field extension of $k$ having $C$ as its field of constants and that $E=E_m\supseteq E_{m-1}\supseteq\cdots\supseteq E_1\supseteq E_0=k,$ where $E_i$ is either an elementary extension of $E_{i-1}$ or $E_i=E_{i-1}(t_i, t'_i)$ and $t_i$ is weierstrassian (in the sense of Kolchin ([Page 803, Kolchin1953]) over $E_{i-1}$ or $E_i$ is a Picard-Vessiot extension of $E_{i-1}$ having a differential Galois group isomorphic to either the special linear group $\mathrm{SL}_2(C)$ or the infinite dihedral subgroup $\mathrm{D}_\infty$ of $\mathrm{SL}_2(C).$ In this article, we prove that Liouville's theorem on integration in finite terms ([Theorem, Rosenlicht1968]) holds for $E$. That is, if $\eta\in E$ and $\eta'\in k$ then there is a positive integer $n$ and for $i=1,2,\dots,n,$ there are elements $c_i\in C,$ $u_i\in k\setminus \{0\}$ and $v\in k$ such that $$\eta'=\sum^n_{i=1}c_i\frac{u'_i}{u_i}+v'.$$