Résume | Local uniformization can be understood as a local form ofresolution of singularities. In 1940, Zariski proved the LocalUniformization Theorem for places of algebraic function fields overbase fields of characteristic 0. In our talk, we will sketch theproof of the following theorem:Let F|K be a function field of arbitrary characteristic, and P anAbhyankar place of F|K such that FP|K is separable. Further, take anyelements \zeta_1,\ldots,\zeta_m in the valuation ring of P on F. Thenthere exists a model of F|K on which P is centered at a smooth pointwhose local ring contains \zeta_1,...,\zeta_m.Abhyankar places of F|K are places of F which are trivial on K andfor which equality holds in the Abhyankar inequality. They are very"representative" as they lie dense in the Zariski space of allplaces, with respect to topologies much finer than the Zariskitopology. (This fact can be proved using the model theory of valuedfields.)We also present the arithmetic version of the above theorem, wherethe Abhyankar places are allowed to be extensions of p-adic places onQ. Moreover, we discuss the possibility of simultaneousuniformization of finitely many Abhyankar places (work in progress),this would amount to the simultaneous local resolution of finitelymany singularities.The proofs use the theory of henselian elements which wasdeveloped in joint work with Peter Roquette. An element z in analgebraic extension of a field K equipped with a place P is calledhenselian if there is a polynomial h with coefficients in thevaluation ring of K and such that h(z)=0 and the residue zP is asimple root of the reduction of h modulo P.Finally, we will discuss local uniformization for arbitrary placesin arbitrary characteristic after a finite Galois extension of thefunction field. Alternatively, one can also seek to minimize theextensions of value group and residue field induced by the extensionof the function field. We will give a corresponding result. |