Abstract:
In 1839 Liouville published his ingenious pioneering work containing an elementary criterium for solvability of second order linear differential equations by quadratures. Surprisingly beatific Liouville's theory did not get an appropriate
credit. About 70 years later Picard and Vessiot found a criterium for solvability of a linear differential equation of arbitrary order $n$ in terms of its Galois group. This result is based on their differential Galois theory which is rather involved.
In 1948 J. F. Ritt clarified the original Liouville's proof [1], [3]. In 2018 I understood that the elementary Liouville–Ritt method based on developing solutions in Puiseux series as functions of a parameter works smoothly for an arbitrary $n$ and
proved a similar criterium [2], [3]. In the talk I will discuss this criterium and ideas of its proof.
[1] Ritt J. F., Integration in Finite Terms. Liouville's Theory of Elementary
Methods. Columbia University Press, New York, 1948.
[2] Khovanskii A. G., Solvability of equations by quadratures and Newton's theorem, Arnold Math. J.4:2 (2018),
193–211.
[3] Khovanskii A. G., Comments on J. F. Ritt's book "Integration in Finite Terms", arXiv:1908.02048 [math.AG]
(Aug 2019).