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Authors
Ehud Hrushovski
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We study finite imaginaries in certain valued fields, and prove a conjecture of Cluckers and Denef.
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In this paper we study elimination of imaginaries in some classes of 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 in the case where the ordered abelian group has finite spines, and then prove a better result for the dp-minimal case. An ordered abelian with finite spines weakly eliminates imaginaries once we add sorts for the quotient groups $Gamma/ Delta$ for each definable convex subgroup $Delta$, and sorts for the quotient groups $Gamma/ Delta+ lGamma$ where $Delta$ is a definable convex subgroup and $l in mathbb{N}_{geq 2}$. We refer to these sorts as the quotient sorts. We prove the following two theorems: Theorem: Let $K$ be a valued field of equicharacteristic zero, residue field algebraically closed and value group with finite spines. Then $K$ admits weak elimination of imaginaries once we add codes for all the definable $mathcal{O}$-submodules of $K^{n}$ for each $n in mathbb{N}$, and the quotient sorts for the value group. Theorem: Let $K$ be a henselian valued field of equicharacteristic zero, residue field algebraically closed and dp-minimal value group. Then $K$ eliminates imaginaries once we add codes for all the definable $mathcal{O}$-submodules of $K^{n}$ for each $n in mathbb{N}$, the quotient sorts for the value group and constants to distinguish representatives of the cosets of $Delta+lGamma$ in $Gamma$, where $Delta$ is a convex definable subgroup and $l in mathbb{N}_{geq 2}$.
The text is based on notes from a class entitled {em Model Theory of Berkovich Spaces}, given at the Hebrew University in the fall term of 2009, and retains the flavor of class notes. It includes an exposition of material from cite{hhmcrelle}, cite{hhm} and cite{HL}, regarding definable types in the model completion of the theory of valued fields, and the classification of imaginary sorts. The latter is given a new proof, based on definable types rather than invariant types, and on the notion of {em generic reparametrization}. I also try to bring out the relation to the geometry of cite{HL} - stably dominated definable types as the model theoretic incarnation of a Berkovich point.
We answer two open questions about the model theory of valued differential fields introduced by Scanlon. We show that they eliminate imaginaries in the geometric language introduced by Haskell, Hrushovski and Macpherson and that they have the invariant extension property. These two result follow from an abstract criterion for the density of definable types in enrichments of algebraically closed valued fields. Finally, we show that this theory is metastable.
In this paper we study elimination of imaginaries in some classes of pure ordered abelian groups. For the class of ordered abelian groups with bounded regular rank (equivalently with finite spines) we obtain weak elimination of imaginaries once we add sorts for the quotient groups $Gamma/ Delta$ for each definable convex subgroup $Delta$, and sorts for the quotient groups $Gamma/ Delta+ lGamma$ where $Delta$ is a definable convex subgroup and $l in mathbb{N}_{geq 2}$. We refer to these sorts as the emph{quotient sorts}. For the dp-minimal case we obtain a complete elimination of imaginaries, if we also add constants to distinguish the cosets of $Delta+nGamma$ in $Gamma$, where $Delta$ is a definable convex subgroup and $n in mathbb{N}_{geq 2}$.
In previous work, we have combined computable structure theory and algorithmic learning theory to study which families of algebraic structures are learnable in the limit (up to isomorphism). In this paper, we measure the computational power that is needed to learn finite families of structures. In particular, we prove that, if a family of structures is both finite and learnable, then any oracle which computes the Halting set is able to achieve such a learning. On the other hand, we construct a pair of structures which is learnable but no computable learner can learn it.
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