Differential algebra and Liouvillian first integrals of foliations
Let \((k,\Delta)\) be a differential field, where \(k\) is a field and \(\Delta\) is a set of derivations of \(k\). A Liouvillian extension of \((k,\Delta)\) is a differential extension \((K,\tilde{\Delta})\) of \( (k,\Delta)\) for which there is a chain of differential extensions \(k=k_{0}\subset k_{1}\subset\dots\subset k_{m}=K\) such that \( k_{i+1}/k_{i}=k(t_{i})/k_{i}\) is either an algebraic extension or it is of the type adjunction of an integral or the exponential of an integral. A Liouville function is an element of a Liouvillian extension. Let \(\mathcal{F}\) be a codimension-one foliation on a projective non-singular manifold \(M\), defined by an integrable rational 1-form \(\Omega\) on \(M\). If one considers the differential field of rational functions on \(M\) and a Liouville function \(F\) of its corresponding extension, \(F\) is a Liouvillian first integral if \(dF\wedge\Omega=0\). If \(M=\mathbb{CP}^{2}\) and \(\mathcal{F}\) is defined in some affine \(\mathbb{C}^{2}\subset\mathbb{CP}^{2}\) by a polynomial 1-form \(\omega=P\operatorname{d}\!y-Q\operatorname{d}\!x\), the important work of \textit{M. F. Singer} [Trans. Am. Math. Soc. 333, No. 2, 673--688 (1992; Zbl 0756.12006)] implies (among other things) that: {\parindent=5mm \begin{itemize}\item[1)]\(\mathcal{F}\) has a Liouvillian first integral if and only if \(\omega\) has an integrating factor \(h=\exp (\int \eta)\), where \(\eta\) is a closed rational 1-form, and \item[2)]it is enough that \(P \partial/\partial x + Q \partial/\partial y\) has a local solution which is not contained in an algebraic curve but satisfies a Liouvillian relation to obtain a Liouvillian first integral for \(\mathcal{F}\). \end{itemize}} In this interesting paper, the author mainly studies generalizations of these two results of Singer to compact manifolds and Stein manifolds as well as to foliations of codimension one on \(\mathbb{CP}^{n}\), \(n\geq{3}\). Relations with previous results of \textit{C. Camacho} and the author in [Ergodic Theory Dyn. Syst. 21, No. 3, 717--756 (2001; Zbl 1051.37022)] are also discussed and nice applications to the classification of nondegenerate germs of holomorphic differential 1-forms in dimension two are provided, showing further interest in the subject.
scientific article; zbMATH DE number 5853373
scientific article