On the elementary theory of the real exponential field
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2026
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| _version_ | 1866918379998150656 |
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| author | Berarducci, Alessandro Gallinaro, Francesco |
| author_facet | Berarducci, Alessandro Gallinaro, Francesco |
| contents | Assuming Schanuel's conjecture, we prove that the complete theory $T_{\exp}$ of the real exponential field is axiomatized by the axioms of definably complete exponential fields satisfying $\exp' = \exp$. This implies the result of Macintyre and Wilkie that, under the same conjecture, $T_{\exp}$ is decidable. Our approach is based on the model completeness of a similar set of axioms for the exponential function restricted to $(-1,1)$, which we prove unconditionally. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_08365 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On the elementary theory of the real exponential field Berarducci, Alessandro Gallinaro, Francesco Logic 03C10, 03C64, 12L12 Assuming Schanuel's conjecture, we prove that the complete theory $T_{\exp}$ of the real exponential field is axiomatized by the axioms of definably complete exponential fields satisfying $\exp' = \exp$. This implies the result of Macintyre and Wilkie that, under the same conjecture, $T_{\exp}$ is decidable. Our approach is based on the model completeness of a similar set of axioms for the exponential function restricted to $(-1,1)$, which we prove unconditionally. |
| title | On the elementary theory of the real exponential field |
| topic | Logic 03C10, 03C64, 12L12 |
| url | https://arxiv.org/abs/2603.08365 |