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D-minimal expansions of the real field have the zero set property


Authors: Chris Miller and Athipat Thamrongthanyalak
Journal: Proc. Amer. Math. Soc. 146 (2018), 5169-5179
MSC (2010): Primary 26B05; Secondary 03C64
DOI: https://doi.org/10.1090/proc/14144
Published electronically: September 10, 2018
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Abstract: If $ E\subseteq \mathbb{R}^n$ is closed and the structure $ (\mathbb{R},+,\cdot ,E)$ is d-minimal (that is, in every structure elementarily equivalent to $ (\mathbb{R},+,\cdot ,E)$, every unary definable set is a disjoint union of open intervals and finitely many discrete sets), then for each $ p\in \mathbb{N}$, there exist $ C^p$ functions $ f\colon \mathbb{R}^n\to \mathbb{R}$ definable in $ (\mathbb{R},+,\cdot ,E)$ such that $ E$ is the zero set of $ f$.


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Additional Information

Chris Miller
Affiliation: Department of Mathematics, The Ohio State University, 231 West 18th Avenue, Columbus, Ohio 43210
Email: miller@math.osu.edu

Athipat Thamrongthanyalak
Affiliation: Department of Mathematics, The Ohio State University, 231 West 18th Avenue, Columbus, Ohio 43210
Address at time of publication: Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University, Bangkok, 10330, Thailand
Email: athipat.th@chula.ac.th

DOI: https://doi.org/10.1090/proc/14144
Keywords: Expansion of the real field, definable sets, d-minimal, $C^p$ functions, zero sets
Received by editor(s): January 17, 2017
Received by editor(s) in revised form: December 26, 2017
Published electronically: September 10, 2018
Additional Notes: The research of the second author was conducted while he was a Zassenhaus Assistant Professor at the Department of Mathematics of The Ohio State University.
Communicated by: Ken Ono
Article copyright: © Copyright 2018 American Mathematical Society

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