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The aim of this paper is to study in details the regular holonomic $D-$moduleintroduced in \cite{[B.19]} whose local solutions outside the polarhyper-surface $\{\Delta(\sigma).\sigma_k = 0 \}$ are given by the local systemgenerated by the local branches of the multivalued function which is the rootof 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 regularholonomic $D-$module is given by the quotient of $D$ by a left ideal which hasvery simple explicit generators despite the fact it necessary encodes theanalogous 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 minimalextension of the irreducible local system associated to the difference of twobranches of the multivalued function defined above. Then we obtain again a verysimple explicit description of this minimal extension in term of the generatorsof its left ideal in the Weyl algebra. As an application we show how theseresults allow to compute the Taylor expansion of the root near $-1$ of theequation $z^k + \sum_{h=-1}^k (-1)^h.\sigma_h.z^{k-h} - (-1)^k = 0 $.

On partial differential operators which annihilate the roots of the universal equation of degree k