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

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 (Amer. J. Math. 75(4):753–824, 1953)) 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 $\textrm{SL}_2(C)$ or the infinite dihedral subgroup $\textrm{D}_\infty$ of $\textrm{SL}_2(C).$ In this article, we prove that Liouville’s theorem on integration in finite terms (Rosenlicht in Pac J Math 24(1):153–161, 1968, Theorem) 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

$$\begin{aligned} \eta '=\sum ^n_{i=1}c_i\frac{u'_i}{u_i}+v'. \end{aligned}$$