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In this paper we affirmatively answer a question posed by F.-V. Kuhlmann. We show that the first Artin-Schreier defect extension in Cutkosky and Piltant's counterexample to strong monomialization is a dependent extension. Our main tool is the use of generating sequences of valuations.The authors would like to thank Franz-Viktor Kuhlmann for motivating this work, and S. Dale Cutkosky for many helpful conversations regarding the material in this paper.

We generalize Zariski's Perron transforms to obtain a proof of local uniformization of an arbitrary singular variety of characteristic zero along a valuation of arbitrary rank. Furthermore, we resolve the centers of all the composite valuations as well as the prime ideals of infinite value.

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