Transasymptotic expansions of o-minimal germs
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866929319234764800 |
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| author | Rolin, Jean-Philippe Servi, Tamara Speissegger, Patrick |
| author_facet | Rolin, Jean-Philippe Servi, Tamara Speissegger, Patrick |
| contents | Given an o-minimal expansion $\mathbb{R}_{\mathcal{A}}$ of the real ordered field, generated by a generalized quasianalytic class $\mathcal{A}$, we construct an explicit truncation closed ordered differential field embedding of the Hardy field of the expansion $\mathbb{R}_{\mathcal{A},\exp}$ of $\mathbb{R}_{\mathcal{A}}$ by the unrestricted exponential function, into the field $\mathbb{T}$ of transseries. We use this to prove some non-definability results. In particular, we show that the restriction to the positive half-line of Euler's Gamma function is not definable in the structure $\mathbb{R}_{\text{an}^{*},\exp}$, generated by all convergent generalized power series and the exponential function, thus establishing the non-interdefinability of the restrictions to a neighbourhood of $+\infty$ of Euler's Gamma and of the Riemann Zeta function. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_12073 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Transasymptotic expansions of o-minimal germs Rolin, Jean-Philippe Servi, Tamara Speissegger, Patrick Logic 03C64, 26E10, 03C10, 12J15 Given an o-minimal expansion $\mathbb{R}_{\mathcal{A}}$ of the real ordered field, generated by a generalized quasianalytic class $\mathcal{A}$, we construct an explicit truncation closed ordered differential field embedding of the Hardy field of the expansion $\mathbb{R}_{\mathcal{A},\exp}$ of $\mathbb{R}_{\mathcal{A}}$ by the unrestricted exponential function, into the field $\mathbb{T}$ of transseries. We use this to prove some non-definability results. In particular, we show that the restriction to the positive half-line of Euler's Gamma function is not definable in the structure $\mathbb{R}_{\text{an}^{*},\exp}$, generated by all convergent generalized power series and the exponential function, thus establishing the non-interdefinability of the restrictions to a neighbourhood of $+\infty$ of Euler's Gamma and of the Riemann Zeta function. |
| title | Transasymptotic expansions of o-minimal germs |
| topic | Logic 03C64, 26E10, 03C10, 12J15 |
| url | https://arxiv.org/abs/2402.12073 |