# The minimal cone of an algebraic Laurent series

## TL;DR

This paper investigates the structure of algebraic elements over multivariate power series fields, showing in characteristic zero they can be represented with supports forming rational polyhedral cones, while in positive characteristic the structure differs.

## Contribution

It proves that algebraic elements in characteristic zero have supports forming rational polyhedral cones, and constructs algebraically closed fields in positive characteristic with different properties.

## Key findings

- In characteristic zero, algebraic elements have supports forming rational polyhedral cones.
- In positive characteristic, the structure of algebraic elements differs, and the previous results do not hold.
- Constructed algebraically closed fields in positive characteristic with distinct properties.

## Abstract

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.