Liouville's Theorem on integration in finite terms for \(\mathrm D_\infty,\) \( \mathrm{SL}_2\) 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 ([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 \[\eta'=\sum^n_{i=1}c_i\frac{u'_i}{u_i}+v'.\]