Finite determination of regular (a,b)-modules

The concept of (a,b)-module comes from the study the Gauss-Manin lattices of an isolated singularity of a germ of an holomorphic function. It is a very simple ‘‘abstract algebraic structure’’, but very rich, whose prototype is the formal completion of the Brieskorn-module of an isolated singularity. The aim of this article is to prove a very basic theorem on regular (a,b)-modules showing that a given regular (a,b)-module is completely characterized by some ‘‘finite order jet’’ of its structure. Moreover a very simple bound for such a sufficient order is given in term of the rank and of two very simple invariants : the regularity order which count the number of times you need to apply
$b^{-1}\mathrm{.}a\simeq {\mathrm{\partial }}_z\mathrm{.}z$ in order to reach a simple pole (a,b)-module. The second invariant is the ‘‘width’’ which corresponds, in the simple pole case, to the maximal integral difference between to eigenvalues of $b^{-1}\mathrm{.}a$ (the logarithm of the monodromy). In the computation of examples this theorem is quite helpfull because it tells you at which power of $b$ in the expansions you may stop without loosing any information.