On Partial Differential Operators Which Annihilate the Roots of the Universal Equation of Degree k

The aim of this paper is to study in details the regular holonomic \(D-\)module introduced in Barlet (Math Z 302 \(n^03\): 1627–1655, 2022 arXiv:1911.09347 [math]) whose local solutions outside the polar hyper-surface \(\{\Delta (\sigma ).\sigma _k = 0 \}\) are given by the local system generated by the power \(\lambda \) of the local branches of the multivalued function which is the root of the universal degree k equation \(z^k + \sum _{h=1}^k (-1)^h\sigma _hz^{k-h} = 0 \). We show that for \(\lambda \in \mathbb {C} {\setminus } \mathbb {Z}\) this D-module is the minimal extension of the holomorphic vector bundle with an integrable meromorphic connection with a simple pole which is its restriction on the open set \(\{\sigma _k\Delta (\sigma ) \not = 0\}\). We then study the structure of these D-modules in the cases where \(\lambda = 0, 1, -1\) which are a little more complicated, but which are sufficient to determine the structure of all these D-modules when \(\lambda \) is in \(\mathbb {Z}\). As an application we show how these results allow to compute, for instance, the Taylor expansion of the root near \(-1\) of the equation:

$$\begin{aligned} z^k + \sum _{h=-1}^k (-1)^h\sigma _hz^{k-h} - (-1)^k = 0. \end{aligned}$$

near \(z^k - (-1)^k = 0\).