# Formal Solutions and Factorization of Differential Operators with Power Series Coefficients

The purpose of this paper is to list the technical ingredients about local (i.e. over the power series) differential operators that are needed in [jsc1997c]. Furthermore an efficient algorithm is given for factorization in the ring of differential operators k((x))[\delta] where \delta = x d/dx. This can be used to compute formal solutions more efficiently. Note: the algorithm in [jsc1997c] uses factorization in k((x))[\delta] but does not use formal solutions; the formal solutions are only used to explain the algorithm.

In this paper the following concepts are introduced:

* The coprime index (w.r.t. a valuation) of two elements in a ring. Coprime index 1 means that the traditional Hensel lifting can be applied. Higher coprime index means that lifting becomes more complicated.

* Exponential parts. Note that these resemble normalized eigenvalues introduced by Ron Sommeling. However, our treatment of this subject is quite different, it is based on tools needed for factorization in k((x))[\delta].

* Multiplicities of exponential parts, semi-regular part, and the relation with factorization and with formal solutions.

For the implementation see the file factor_op in the diffop package.