Phénomène de Stokes et filtration Gevrey sur le groupe de Picard- Vessiot. (Stokes phenomena and Gevrey filtration on the Picard-Vessiot group). (French) Zbl 0593.12015

Ramis, Jean-Pierre

C. R. Acad. Sci., Paris, Sér. I 301, 165-167 (1985).

Let \({\mathbb{C}}[[ x]]\) be the ring of formal power series, \({\mathbb{C}}\{x\}\) the ring of germs of convergent power series, \(\hat K={\mathbb{C}}[[ x]][x^{-1}]\) and \(K={\mathbb{C}}\{x\}[x^{-1}]\). The author considers differential equations of the form \(Y'=AY\), where \(A\in End(n,K)\). Such a differential equation is known to have a formal solution of the form \(\hat F=\hat Bt^ L\exp (Q(1/t))\), where \(x=t^ q\), q a positive integer, \(\hat B\in GL(n, {\mathbb{C}}[[ t]])\), \(L\in End(n, {\mathbb{C}})\) and Q is an \(n\times n\) diagonal matrix with diagonal entries \(q_ i\in (1/t) {\mathbb{C}}[1/t]\). We can form the Picard-Vessiot extensions \(\hat L=\hat K<\hat F>\) and \(L=K<\hat F>\) of \(\hat K\) and K with Galois groups \(\hat G\) and G. Note that we may identify \(\hat G\) with a subgroup of G. The author shows that G is determined by \(\hat G\) and the Stokes multipliers of \(Y'=AY.\)
For \(s>1\), let \(\hat K_ s\) be the field of Laurent series (with finite poles) of Gevrey order s and \(K_ s\) be the field of k-summable meromorphic germs with finite poles \((k=1/(s-1))\). Let \(\hat L_ s\) and \(L_ s\) be the corresponding Picard-Vessiot extensions of these fields and \(\hat G_ s\) and \(G_ s\) be their Galois groups. We have injections \(\hat G\hookrightarrow \hat G_ s\hookrightarrow G_ s\hookrightarrow G\). The author shows that for all \(s>1\), \(\hat G_ s=G_ s\), for large s, \(\hat G_ s=\hat G\), and for s close to 1, \(\hat G_ s=G\). He also shows that there exist numbers \(s_ 1<s_ 2<...<s_ k\) such that \(\hat G_ s=\hat G_{s'}\) for s and s’ in \([s_ i,s_{i+1})\). The \(s_ i\) are determined by the slopes of the Newton polygon associated to the differential equation. Furthermore, each \(G_ s\) is the Zariski closure of a group that is generated by a group of diagonal matrices and a finite set of matrices that can be determined from the formal solution F.
The author also shows that using a natural representation of F as an asymptotic expansion of a germ of functions analytic in a sector at the origin, one gets an isomorphism of \(K<F>\) onto \(K<F>\). Details appear in ”Filtration Gevrey sur le groupe de Picard-Vessiot d’une équation differentielle irregulière” (Inf. Mat., Ser. A).

Reviewer: M.F.Singer

MSC:

12H05	Differential algebra
34G10	Linear differential equations in abstract spaces

Keywords:

Stokes phenomenon; Gevrey filtration; formal power series; differential equations; Picard-Vessiot extensions; Galois groups; Stokes multipliers; Laurent series; Gevrey order; meromorphic germs; Newton polygon; asymptotic expansion