On the calculation of some differential Galois groups
type: Item
instance of: scholarly article
title: On the calculation of some differential Galois groups
zbMATH Open document ID: 0609.12025
DOI: 10.1007/BF01389152
author: Nicholas M. Katz
published in: Inventiones Mathematicae
publication date: 1987-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/143412
review text: Let \(k\) be an ordinary differential field of characteristic 0 with derivation operator \(\delta\) and let \(L\in k[\delta]\). How can one ``tell at a glance'' what the Galois group of \(L\) over \(k\) is? More than twenty years ago E. Kolchin put this question at the IMC in Moscow. In spite of the separate results received in the last years by I. Kovačić, M. Singer, F. Baldassarri, V. Salikhov, the reviewer and others, there is not any hope now to receive some satisfactory answer on this question in the near future. The author has solved the above problem in the case when \(k={\mathbb C}(x)\) and \(\delta =d/dx\) for the following operators \(L\): (A) \(L=P(\delta)+Q(x)\), where \(P(\delta)\in\mathbb C[\delta]\), Q(x)\(\in\mathbb C[x]\), \(\deg P=n\), \(\deg Q=m\), \(m\) and \(n\) are relatively prime and \(n\geq 2;\) (B) \(L=P(x\delta)+Q(x)\), if \(P\) and \(Q\) as above also satisfy that all roots of \(P\) are rational numbers with denominator prime to \(n\) and \(Q(0)=0.\) He proves that the Galois group of such operators is sufficiently large, i.e. it is caught between \(\mathrm{SL}(n)\) and \(\mathrm{GL}(n)\) or (if \(n\) is even), between \(\mathrm{Sp}(n)\) and \(\mathrm{GSp}(n)\). The author uses the theory of Tannakian categories and he considers that they are much better suited for discussing the problem than Kolchin's theory. The last is representing a disputable point of view for the reviewer.
Mathematics Subject Classification ID: 12H05
Mathematics Subject Classification ID: 34A30
Mathematics Subject Classification ID: 14A20
Mathematics Subject Classification ID: 14L17
Mathematics Subject Classification ID: 14C30
Mathematics Subject Classification ID: 14F99
zbMATH DE Number: 3985372
zbMATH Keywords: Kloosterman equations
zbMATH Keywords: affine algebraic groups
zbMATH Keywords: motifs
zbMATH Keywords: Hodge theory
zbMATH Keywords: differential Galois group
zbMATH Keywords: ordinary differential field
zbMATH Keywords: Tannakian categories
Wikidata QID: Q100164717
reviewed by: Mykola Grygorenko
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3985372