Abhyankar places admit local uniformization in any characteristic

We prove that every place P of an algebraic function field F vertical bar 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 vertical bar K, and the extension FP vertical bar K is separable. We generalize this result to the case where P dominates a regular local Nagata ring R subset of K of Krull dimension dim R <= 2, assuming that the valued field (K, upsilon(P)) is defectless, the factor group v(P)F/upsilon K-P is torsion-free and the extension of residue fields FP vertical bar KP is separable. The results also include a form of monomialization. (c) 2005 Published by Elsevier SAS.