Liouville's theorem on integration in finite terms for \(\mathrm{D}_\infty , \mathrm{SL}_2\), and Weierstrass field extensions (Q6077852)

The article deals with the question whether elementary integrals over differential extensions can be decended to elementary integrals over the base differential field. More precisely, if \((k,\partial)\) is a differential field of characteristic zero with algebraically closed field of constants \(C\) (e.g. \((k,\partial)=(\mathbb{C}(x),\partial_x)\)), an element \(f\in k\) is said to have an \emph{elementary integral} over \(k\), if the exist \(c_1,\ldots, c_n\in C\), \(u_1,\ldots, u_n\in k\setminus \{0\}\) and \(v\in k\) such that \[ f = \sum_{i=1}^n c_i\frac{\partial(u_i)}{u_i} + \partial(v). \] Let \((E,\partial_E)\) be a differential extension of \((k,\partial)\) with the same field of constants, one can ask whether an element \(f\in k\) which has an elementary integral over \(E\) (i.e. the elements \(u_i\) and \(v\) from above are in \(E\)), also has an elementary integral over \(k\). In the article, the authors solve this question to the affirmative for certain types of extensions \(E/k\). Namely (see Theorem 1.1) for those extensions \(E/k\) that are a successive extension of \begin{enumerate} \item[a)] elementary extensions (i.e. algebraic, logarithmic or exponential), \item[b)] Picard-Vessiot extensions with Galois group isomorphic to the special linear group \(\mathop{SL}_2(C)\) or the infinite dihedral group \(\mathop{D}_\infty\), or \item[c)] an elliptic extension, i.e. generated by a transcendental element \(\theta\) and its derivative \(\theta'\) fulfilling a Weierstrass equation \((\theta')^2=\alpha^2\cdot (4\theta^3-g_1\theta-g_0)\) for constant elements \(g_0,g_1\in C\) and \(\alpha\) in the smaller differential field. \end{enumerate} Of course the general case of a successive extension follows inductively from the three cases of extensions, and case a) is a result of \textit{M. Rosenlicht} [Pac. J. Math. 24, 153--161 (1968; Zbl 0155.36702)]. The other two cases, however, are really new contributions to the subject.