Kolloquien und Oberseminare Übersicht Detail

Vortragende Person/Vortragende Personen:
Mickaël Matusinski

Let $K$ be a field of characteristic zero and  $x=(x_1,...,x_r)$. We consider the algebraic closure of $K[[x]]$ as a subfield of the so-called field of rational polyhedral Puiseux series, and call its elements algebroid Puiseux series. We deal with the two following problems:

- given a polynomial equation P(x, y) = 0 for P ∈ K[[x]][y], provide a closed form formula for the coefficients of an algebroid Puiseux series solution y(x) in terms of the coefficients of P;

- given an algebroid Puiseux series y(x), reconstruct algorithmically the coefficients of a vanishing polynomial P ∈ K[[x]][y] using the coefficients of the series.

Our strategy involves the answers that we recently obtained to the same type of questions about algebraic Puiseux series, i.e. for the algebraic closure of K(x). Note that this field of algebroid series contains the field of fractions of $K[[x]]$ which was recently studied by Sebastian, Salma and Michele in the more general context of Hahn power series. I will discuss some of the questions they raise. This is about a joint work in progress with Michel Hickel (U. of Bordeaux).