Factorisation d'opérateurs différentiels à coefficients dans une extension liouvillienne d'un corps valué. (Factorization of differential operators with coefficients in a Liouvillian extension of a valued field) (Q1613959)

Let \(\mathbb C\) be the field of complex numbers, \({\mathbb C}((z))\) be the field of formal Laurent series in \(z\) with coefficients in \(\mathbb C\) endowed with the derivation operator \(\delta = -z^2d/dz\). By \(L\) we denote the field \({\mathbb C}((z))((e^{1/z}))\). The goal of the author is the study of linear differential equations with coefficients in \(L\) in order to construct a differential Galois theory over \(L\). In particular, the solutions of linear differential equations with coefficients in \(L\) should be determined or, in other words, differential operators with coefficients in \(L\) should be factorized. This class of fields is of interest, because the coefficients of the differential equations admit essential singularities and not only poles. The following theorem is proven. Theorem. Consider the field \(K={\mathbb C}((z))\) endowed with the \(z\)-adic valuation and the derivation operator \(\delta=ad/dz\), where \(a\in{\mathbb C}((z))\). Let \(\mu_1,\dots,\mu_p\) be elements of \({\mathbb C}((z))\) such that \[ X_1=e^{\mu_1},\dots,X_n=e^{\mu_n},X_{n+1}=\log(\mu_{n+1}),\dots,X_p=\log(\mu_p) \] are algebraically independent over \({\mathbb C}((z))\). Let \(K\subset L=K(X_1,\dots,X_p)\). By \(F_0\) we denote the algebraic closure of \(K\) and, for any \(i\), \(1\leq i\leq p\), we define \(F_i=\bigcup_{n\geq 1}F_{i-1}((X_i^{1/n}))\). If \(P\in L[\delta]\) is a nonzero differential operator, then \(P\) can be factorized into a product of first-order operators in the ring \(F_p[\delta]\).