Calcul de facteurs déterminants. (The calculation of determining factors) (Q1122042)

Let \({\mathcal L}=D^ n+\sum^{n}_{i=0}a_ iD^ i\), \((a_ i\in K)\), be a differential operator with coefficients in the ring K of the series with complex coefficients. A Theorem by Hukuhara and Turrittin states the existence of a generating family of n independent solutions of \({\mathcal L}\), \(u_ i(t)=\exp Q_ i(t)t^{\lambda_ i}v_ i(t)\), where \(Q_ i\) is a polynomial in \(t^{-1/e}(e\in {\mathbb{N}})\), \(i=1,...,n\). The \(Q_ i\) are the determining factors of \({\mathcal L}\). The purpose of this paper is to provide a direct formula for the \(Q_ i\), as opposed to algorithmic computations going together with proofs of Hukuhara and Turrittin theorem. It is done here in the case of operators \({\mathcal L}\) of order \(\leq 3\). A same result was previously obtained by B. Helffer and Y. Kannai for the case of double characteristics.