Counterexamples to local monomialization in positive characteristic

In this paper we consider birational properties of ramification in excellent local rings. We give an example showing that local monomialization (and weak local monomialization) can fail for extensions of algebraic local rings in algebraic function fields of dimension greater than or equal to two along a valuation over a field of positive characteristic. It was earlier proven by the author that local monomialization holds within characteristic zero algebraic function fields.