## 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

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