Expansions of o-minimal structures on the real field by trajectories of linear vector fields
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Abstract:
Let $\mathfrak R$ be an o-minimal expansion of the field of real numbers that defines nontrivial arcs of both the sine and exponential functions. Let $\mathcal G$ be a collection of images of solutions on intervals to differential equations $y’=F(y)$, where $F$ ranges over all $\mathbb R$-linear transformations $\mathbb R^n\to \mathbb R^n$ and $n$ ranges over $\mathbb N$. Then either the expansion of $\mathfrak R$ by the elements of $\mathcal G$ is as well behaved relative to $\mathfrak R$ as one could reasonably hope for or it defines the set of all integers $\mathbb Z$ and thus is as complicated as possible. In particular, if $\mathfrak R$ defines any irrational power functions, then the expansion of $\mathfrak R$ by the elements of $\mathcal G$ either is o-minimal or defines $\mathbb Z$.References
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Additional Information
- Chris Miller
- Affiliation: Department of Mathematics, The Ohio State University, 231 West 18th Avenue, Columbus, Ohio 43210
- MR Author ID: 330760
- Email: miller@math.ohio-state.edu
- Received by editor(s): July 14, 2009
- Received by editor(s) in revised form: March 16, 2010
- Published electronically: July 23, 2010
- Additional Notes: This research was partially supported by the hospitality of the Fields Institute during the Thematic Program on O-minimal Structures and Real Analytic Geometry, January–June 2009.
- Communicated by: Julia Knight
- © Copyright 2010
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 139 (2011), 319-330
- MSC (2010): Primary 03C64; Secondary 34A30
- DOI: https://doi.org/10.1090/S0002-9939-2010-10506-3
- MathSciNet review: 2729094