Valuations on rational function fields that are invariant under permutation of the variables

We study and characterize the class of valuations on rational functions fields that are invariant under permutation of the variables and can be extended to valuations with the same property whenever a finite number of new variables is adjoined. The Gauß valuation is in this class, which constitutes a natural generalization of the concept of Gauß valuation. Further, we apply our characterization to show that the most common ad hoc generalization of the Gauß valuation is also in this class.