Symmetrically-Closed Valuations On K(X1,..., Xn)

In the domain of valued fields K and, more specifically, the extensions of valuations on K to K(X1,…,Xn), an important category of extensions, namely the symmetric extensions with respect to the indeterminates X1,…,Xn, came recently into attention. This paper deals with the characterization of the symmetric extensions of valuations on K to K(X1,…,Xn) that are not symmetrically-open, thus not having a simple closed-form expression for definition. We will define the notions of symmetry degree and the most relevant restriction (symmetrically-wise) of an extension of a valuation from K to K(X1,…,Xn) and we will use these quantities to characterize this unfriendly class of symmetrically-closed extensions.