The minimal cone of an algebraic Laurent series

We study the algebraic closure of $\mathbb K(\!(x)\!)$, the field of power series in several indeterminates over a field $\mathbb K$. In characteristic zero we show that the elements algebraic over $\mathbb K(\!(x)\!)$ can be expressed as Puiseux series such that the convex hull of its support is essentially a polyhedral rational cone, strengthening the known results. In positive characteristic we construct algebraic closed fields containing the field of power series and we give examples showing that the results proved in characteristic zero are longer valid in positive characteristic.