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Defining integer valued functions in rings of continuous definable functions over a topological field

Abstract : Let K be an expansion of either an ordered field or a valued field. Given a definable set X ⊆ Km let C(X) be the ring of continuous definable functions from X to K. Under very mild assumptions on the geometry of X and on the structure K, in particular when K is o-minimal or P-minimal, or an expansion of a local field, we prove that the ring of integers Z is interpretable in C(X). If K is o-minimal and X is definably connected of pure dimension 2, then C(X) defines the subring Z. If K is P-minimal and X has no isolated points, then there is a discrete ring Z contained in K and naturally isomorphic to Z, such that the ring of functions f ∈ C(X) which take values in Z is definable in C(X).
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https://hal.archives-ouvertes.fr/hal-01907668
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Submitted on : Monday, October 29, 2018 - 1:12:21 PM
Last modification on : Wednesday, November 3, 2021 - 9:18:32 AM
Long-term archiving on: : Wednesday, January 30, 2019 - 3:29:03 PM

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  • HAL Id : hal-01907668, version 1
  • ARXIV : 1810.12562

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Luck Darnière, Marcus Tressl. Defining integer valued functions in rings of continuous definable functions over a topological field. 2018. ⟨hal-01907668⟩

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