DDC - Valuation Theory: Essential dimension of diophantine sets
Arno Fehm (TU Dresden)
Valuation theory plays a major part in the interaction between number theory and logic. In this seminar, a variety of topics from valuation theory, and in particular connections to model theory, will be discussed. There will be a strong emphasis on results involving the definability of valuations.
I will report on ongoing joint work with N. Daans and P. Dittmann in which we study the minimal number of existential quantifiers needed to define a diophantine set and relate this number to essential dimension in the sense of Merkurjev. In the case of the field Q this is connected to questions like whether Z is diophantine in Q. After surveying our general results I will focus on one whose proof requires understanding the relation between p-th powers (p the characteristic) and elements whose value at every valuation is divisible by p.