Dynamic computations inside the algebraic closure of a valued field

We explain how to compute in the algebraic closure of a valued field. These computations heavily rely on the Newton Polygon Algorithm. They are made in the same spirit as the dynamic algebraic closure of a field. They give a concrete content to the theorem saying that a valued field does have an algebraically closed valued extension. The algorithms created for that purpose can be used to perform an effective quantifier elimination for algebraically closed valued fields, which relies on a very natural geometric idea.