Elimination of imaginaries in C((Γ))$\mathbb {C}((\Gamma ))$

Abstract

AbstractIn this paper, we study elimination of imaginaries in henselian valued fields of equicharacteristic zero and residue field algebraically closed. The results are sensitive to the complexity of the value group. We focus first on the case where the ordered abelian group has finite spines, and then prove a better result for the dp‐minimal case. In previous work the author proved that an ordered abelian with finite spines weakly eliminates imaginaries once one adds sorts for the quotient groups for each definable convex subgroup , and sorts for the quotient groups where is a definable convex subgroup and . We refer to these sorts as the quotient sorts. Jahnke, Simon, and Walsberg (J. Symb. Log. 82 (2017) 151–165) characterized ‐minimal ordered abelian groups as those without singular primes, that is, for every prime number one has .We prove the following two theorems: