Abstract
The problem of minimal local realization of analytical systems (affine in the command) has been theoretically studied and solved by M.Fliess and C. Reutenauer. However, their results does not give, in general, a complete computation of the minimal realization. Our attempt is here to give such a complete computation for the restricted case of systems that have finite generating series. Moreover, we shall see that the obtained realization is "completely polynomial": the observation and components of the vector fields are commutative polynomials.First, we recall the notion of generating series and of Lie rank, and the main concepts and properties used by Reutenauer concerning Lie polynomials, Lyndon words and the (commutative) shuffle algebra of non commutative power series. Then, we explain the structure of our algorithm and the detailed execution of an example. A complete MACSYMA version is avalaible from the authors.
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