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Defining integer valued functions in rings of continuous definable functions over a topological field
Logic · Submitted 30 Oct 2018
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim 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).
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The abstract, sentence by sentence
Let K be an expansion of either an ordered field or a valued field. Given a definable set X $\subseteq$ K<sup>m</sup> 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 $\in$ C(X) which take values in Z is definable in C(X).
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | Let K be an expansion of either an ordered field or a valued field. | background 1.00 | 0.09 | 0.01 | asserted 0.87 |
| s1 | Given a definable set X $\subseteq$ K<sup>m</sup> let C(X) be the ring of continuous definable functions from X to K. | background 0.97 | 0.18 | 0.02 | asserted 0.78 |
| s2 | 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). | result 0.91 | 0.94 | 0.85 | asserted 1.00 |
| s3 | If K is o-minimal and X is definably connected of pure dimension 2, then C(X) defines the subring Z. | result 0.99 | 0.89 | 0.20 | asserted 1.00 |
| s4 | 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 $\in$ C(X) which take values in Z is definable in C(X). | result 0.99 | 0.90 | 0.17 | asserted 1.00 |
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Original abstract
Let K be an expansion of either an ordered field or a valued field. Given a definable set X $\subseteq$ K<sup>m</sup> 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 $\in$ C(X) which take values in Z is definable in C(X).
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