an:03985372
Zbl 0609.12025
Katz, Nicholas M.
On the calculation of some differential Galois groups
EN
Invent. Math. 87, 13-61 (1987).
00151483
1987
j
12H05 34A30 14A20 14L17 14C30 14F99
Kloosterman equations; affine algebraic groups; motifs; Hodge theory; differential Galois group; ordinary differential field; Tannakian categories
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. Kova??i??, 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.
N. V. Grigorenko (Mykola Grygorenko) (Ky??v)