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Zbl 1159.13301
Knaf, Hagen; Kuhlmann, Franz-Viktor
Abhyankar places admit local uniformization in any characteristic. (English)
[J] Ann. Sci. Éc. Norm. Supér. (4) 38, No. 6, 833-846 (2005). ISSN 0012-9593

Summary: We prove that every place $P$ of an algebraic function field $F|K$ of arbitrary characteristic admits local uniformization, provided that the sum of the rational rank of its value group and the transcendence degree of its residue field $FP$ over $K$ is equal to the transcendence degree of $F|K$, and the extension $FP|K$ is separable. We generalize this result to the case where $P$ dominates a regular local Nagata ring $R\subseteq K$ of Krull dimension $\dim R\leq 2$, assuming that the valued field $(K,v_P)$ is defectless, the factor group $v_P F/v_P K$ is torsion-free and the extension of residue fields $FP|KP$ is separable. The results also include a form of monomialization. Further, we show that in both cases, finitely many Abhyankar places admit simultaneous local uniformization on an affine scheme if they have value groups isomorphic over $v_P K$.

MSC 2000:: *13A18 Valuations and their generalizations
12Jxx Topological fields
13C40 Linkage, complete intersections and determinantal ideals

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