`guillaume.rond@univ-amu.fr`

It is well known that the solutions of a univariate polynomial equation with power series coefficients (in several variables over a characteristic zero field) can be expressed as Laurent Puiseux series whose support is contained in the translation of a rational strongly convex cone. We will begin this talk by explaining this sentence through examples. Then we will present partial answers to the following question: how can we characterize among Laurent series whose support is contained in a rational convex cone those which are algebraic over the field of power series? We will see that we can give some characterizations in terms of the gaps in the expansion of the Laurent power series and in terms of the "shape" of the support of the given Laurent series.

(Joint work with Fuensanta Aroca)