Solvability of Equations by Quadratures and Newton's Theorem

2018/08/20 by Khovanskii, Askold
#Algebraic Geometry (math.AG) #FOS: Mathematics

paper · doi:10.48550/arxiv.1808.06328

Abstract

Picard--Vessiot theorem (1910) provides a necessary and sufficient condition for solvability of linear differential equations of order n by quadratures in terms of its Galois group. It is based on the differential Galois theory and is rather involved. J.Liouville in 1839 found an elementary criterium for such solvability for n=2. J.F.Ritt simplified Liouville's theorem (1948). In 1973 M. Rosenlicht proved a similar criterium for arbitrary n. Rosenlicht work relies on the valuation theory and is not elementary. In these notes we show that the elementary Liouville--Ritt method based on developing solutions in Puiseux series as functions of a parameter works smoothly for arbitrary n and proves the same criterium.

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