A density theorem in parametrized differential Galois theory
Let \(F\) be a differential field of characteristic zero with an algebraically closed field of constants \(C\) and let \(t_1,\ldots ,t_n\) be constants transcendental over \(C\). In the framework of the differential Galois theory developed by Kolchin, it is difficult to investigate differential equations of the form NEWLINE\[NEWLINE\delta y=Ay,\quad A\in M(m,F(t_1,\ldots ,t_n)).NEWLINE\]NEWLINE Because most of the constructive results of the theory are obtained in the case of a closed field of constants. For example, it is difficult to prove the existence of the Picard-Vessiot extension of the field \(\mathbb{C}(\rho(z),\rho'(z),\lambda)\) corresponding to the Lamé equation \(y''-n(n+1)\rho(z)y=\lambda y\), when \(\lambda\) is a transcendental over \(\mathbb C\) constant (parameter). At the same time, in applied problems, it is often necessary to investigate equations with parameters. In 2007, \textit{P. J. Cassidy} and \textit{M. F. Singer} [in: Differential equations and quantum groups. Andrey A. Bolibrukh memorial volume. Zürich: European Mathematical Society Publishing House, 113--155 (2007; Zbl 1230.12003)] presented a special Galois theory for linear differential equations with parameters, similar to Kolchin's theory for equations without parameters. In particular, they showed that the role of weakly singular points in the new theory is the same as in the Kolchin theory. In this article, the author proves a stronger resultNEWLINENEWLINETheorem 2.20 (parametrized analog of the density theorem of Ramis). The group generated by the parametrized monodromy, the parametrized exponential torus and the parametrized Stokes operators is dense in the parametrized differential Galois group for the Kolchin topology.NEWLINENEWLINEA number of other interesting results on the inverse problem, isomonodromy and complete integrability for equations with parameters are also obtained in the article.
scientific article; zbMATH DE number 6369630