Liouville's Theorem on integration in finite terms for $\mathrm D_\infty,$ $ \mathrm{SL}_2$ and Weierstrass field extensions (2308.00659v1)

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 $$\eta'=\sum^n{i=1}c_i\frac{u'_i}{u_i}+v'.$$