Franz-Viktor Kuhlmann, University of Szczecin
Two longstanding open problems in positive characteristic and their relation to valuation theory
Since Hironaka proved resolution of singularities over base fields of characteristic zero in 1964, the corresponding problem in positive characteristic has remained open, and so has its local form, called local uniformization. The latter is in fact a valuation theoretical problem, due to ideas of Zariski. I will present these ideas and show the connection of local uniformization with the structure theory of valued function fields. The positive characteristic case is so much harder than the characteristic zero case because of the phenomenon of the defect. I will define it and sketch strategies to either avoid it or work around it; these led to some partial solutions to the local uniformization problem.
In 1965, one year after Hironaka, Ax and Kochen used the model theory of valued fields to prove a corrected version of Artin's Conjecture. Thereafter they, and independently Ershov, proved the decidability of the elementary theory of the fields of p-adic numbers. The problem for their counterpart in positive characteristic, the Laurent series fields over finite fields, is still open. I will explain which tools can be used to prove decidability. Via general principles of model theory, the task can be reduced to proving embedding lemmas for valued function fields, which I will describe. This in turn requires a good structure theory for such valued function fields, and this is what our decidability problem has in common with the local uniformization problem. In analogy to the local uniformization problem, our theory of the defect has led to partial solutions, in the sense of new model theoretic results about certain classes of valued fields in positive characteristic.