  The aim of this paper is to study in details the regular holonomic $D-$module
introduced in \cite{[B.19]} whose local solutions outside the polar
hyper-surface $\{\Delta(\sigma).\sigma_k = 0 \}$ are given by the local system
generated by the local branches of the multivalued function which is the root
of the universal degree $k$ equation $z^k + \sum_{h=1}^k
(-1)^h.\sigma_h.z^{k-h} = 0 $. Note that it is surprising that this regular
holonomic $D-$module is given by the quotient of $D$ by a left ideal which has
very simple explicit generators despite the fact it necessary encodes the
analogous systems for any root of the universal degree $l$ equation for each $l
\leq k$. Our main result is to relate this $D-$module with the minimal
extension of the irreducible local system associated to the difference of two
branches of the multivalued function defined above. Then we obtain again a very
simple explicit description of this minimal extension in term of the generators
of its left ideal in the Weyl algebra. As an application we show how these
results allow to compute the Taylor expansion of the root near $-1$ of the
equation $z^k + \sum_{h=-1}^k (-1)^h.\sigma_h.z^{k-h} - (-1)^k = 0 $.
