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In this paper we introduce the word {\em fresco} to denote a monogenicgeometric (a,b)-module. This "basic object" (generalized Brieskorn module withone generator) corresponds to the formal germ of the minimal filtered (regular)differential equation. Such an equation is satisfied by a relative de Rhamcohomology class at a critical value of a holomorphic function on a smoothcomplex manifold. In [B.09] the first structure theorems are proved. Then in[B.10] we introduced the notion of {\em theme} which corresponds in the \$[\lambda]-$primitive case to frescos having a unique Jordan-H{\"o}ldersequence (a unique Jordan block for the monodromy). Themes correspond toasymptotic expansion of a given vanishing period, so to an image of a fresco inthe module of asymptotic expansions. For a fixed relative de Rham cohomologyclass (for instance given by a smooth differential form $d-$closed and$df-$closed) each choice of a vanishing cycle in the spectral eigenspace of themonodromy for the eigenvalue \ $exp(2i\pi.\lambda)$ \ produces a \$[\lambda]-$primitive theme, which is a quotient of the fresco associated tothe given relative de Rham class itself. \\ We show that for any fresco thereexists an {\em unique} Jordan-H{\"o}lder sequence, called the {\em principalJ-H. sequence}, with corresponding quotients giving the opposite of the rootsof the Bernstein polynomial in increasing order. We study the semi-simple partof a given fresco and we characterize the semi-simplicity of a fresco by thefact for any given order on the roots of its Bernstein polynomial we may find aJ-H. sequence making them appear with this order. Then we construct a numericalinvariant, called the \ $\beta-$invariant, and we show that it producesnumerical criteria in order to give a necessary and sufficient condition on afresco to be semi-simple. We show that these numerical invariants define anatural algebraic stratification on the set of isomorphism classes of frescowith given fundamental invariants (or equivalently with given roots of theBernstein polynomial).

Asymptotics of a vanishing period : characterization of semi-simplicity