Appears in collection : 2018 - T1 - WS2 - Model theory and valued fields

Consider the expansion of an algebraically closed field K by ? arbitrary valuation rings (encoded as unary predicates). We show that the resulting structure does not have the second tree property, and is in fact strong. Along the way, we observe that the theory of algebraically closed fields with n valuations is decidable. This talk will outline the model-theoretic analysis of the case of independent non-trivial valuation rings, and sketch how the proof generalizes to the situation of arbitrary valuation rings.