## On the calculation of some differential Galois groups.(English)Zbl 0609.12025

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.

### MSC:

 12H05 Differential algebra 34A30 Linear ordinary differential equations and systems 14A20 Generalizations (algebraic spaces, stacks) 14L17 Affine algebraic groups, hyperalgebra constructions 14C30 Transcendental methods, Hodge theory (algebro-geometric aspects) 14F99 (Co)homology theory in algebraic geometry
Full Text:

### References:

 [1] [Bour-1] Bourbaki, N.: Groupes et Algèbres de Lie. Chapitres 4, 5, et 6,, Paris: Masson 1981 · Zbl 0483.22001 [2] [Bour-2] Bourbaki, N.: Groupes et algèbres de Lie. Chapitres 7 et 8, Paris: Diffusion CCLS 1975 · Zbl 0329.17002 [3] [Chev] Chevalley, C.: Théorie des groupes de Lie. Groupes Algébriques, Théorèmes généraux sur les Algèbres de Lie. Paris: Hermann 1968 [4] [De-1] Deligne, P.: Equations différentielles à points singuliers reguliers. Lect. Notes Math.163, 1970 · Zbl 0244.14004 [5] [De-Mi] Deligne, P., Milne, J.: Tannakian categories. In: Deligne, P., Milne, J., Ogus, A., Kuang-yer, Sh.: Hodge cycles, motives, and Shimura varieties. Lect. Notes Math.900, 101-228 (1982) · Zbl 0477.14004 [6] [Dw-1] Dwork, B.: Bessel functions asp-adic functions of the argument. Duke Math. J.41, 711-738 (1974) · Zbl 0302.14008 [7] [Ince] Ince, E.L.: Ordinary differential equations. Dover: 1956 · Zbl 0063.02971 [8] [Kapl] Kaplansky, I.: An introduction to differential algebra. Deuxième édition, Paris: Hermann 1976 [9] [Ka-1] Katz, N.: Algebraic solutions of differential equations;p-curvature and the Hodge filtration. Invent. Math.18, 1-118 (1972) · Zbl 0278.14004 [10] [Ka-2] Katz, N.: A conjecture in the arithmetic theory of differential equations. Bull. Soc. Math. Fr110, 203-239 (1982) · Zbl 0504.12022 [11] [Ka-3] Katz, N.: Gauss sums, Kloosterman sums, and monodromy groups. Ann. Math. Study113, (to appear) · Zbl 0675.14004 [12] [Ka-4] Katz, N.: A simple algorithm for cyclic vectors. Am. J. Math. (to appear) · Zbl 0621.13003 [13] [Ka-5] Katz, N.: Local-to-global extensions of representations of fundamental groups. Ann. Inst. Fourier (to appear) · Zbl 0564.14013 [14] [Ka-6] Katz, N.: Nilpotent connections and the monodromy theorem?applications of a result of Turitin. Publ. Math., Inst. Hautes Etud. Sci.39, 355-412 (1970) · Zbl 0221.14007 [15] [Le-1] Levelt, A.H.M.: Jordan decomposition for a class of singular differential operators. Ark. Math.13, 1-27 (1975) · Zbl 0305.34008 [16] [Ra] Ramis, J.-P.: Théorèmes d’indices Gevrey pour les équations différentielles ordinaires. Memoirs of the A. M. S.,48, No 296, 1984 [17] [Rob] Robba, P.: Lemme de Hensel pour les opérateurs différentielles. Application à la réduction formelle des équations différentielles, L’Enseignement Math., 2ième série,26, 279-311 (1980) [18] [Saa] Saavedra Rivano, N.: Categories Tanakiennes. Lecture Notes Math.265, 1972 · Zbl 0241.14008 [19] [Se-1] Serre, J.P.: Corps Locaux. Deuxième édition, Paris: Hermann, 1968 [20] [Sp-1] Sperber, S.: Congruence properties of the hyperkloosterman sum. Compos. Math.,40, 3-33 (1980) · Zbl 0444.12014 [21] [Wat] Watson, G.N.: Theory of Bessel functions. Cambridge University Press, 1966
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.