On Preparation Theorems for $\mathbb{R}_{an,exp}$-definable functions
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2021
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866912443204108288 |
|---|---|
| author | Opris, Andre |
| author_facet | Opris, Andre |
| contents | In this article we give strong versions for preparation theorems for $\mathbb{R}_{an,exp}$-definable functions outgoing from methods of Lion and Rolin ($\mathbb{R}_{an,exp}$ is the o-minimal structure generated by all restricted analytic functions and the global exponential function). By a deep model theoretic fact of Van den Dries, Macintyre and Marker every $\mathbb{R}_{an,exp}$-definable function is piecewise given by $\mathcal{L}_{an}(\exp,\log)$-terms where $\mathcal{L}_{an}(\exp,\log)$ denotes the language of ordered rings augmented by all restricted analytic functions, the global exponential and the global logarithm. The idea is to consider log-analytic functions at first, i.e. functions which are iterated compositions from either side of globally subanalytic functions and the global logarithm, and then $\mathbb{R}_{an,exp}$-definable functions as compositions of log-analytic functions and the global exponential. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2112_08161 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | On Preparation Theorems for $\mathbb{R}_{an,exp}$-definable functions Opris, Andre Logic 03C64, 32B20, 33B10, 26A09 In this article we give strong versions for preparation theorems for $\mathbb{R}_{an,exp}$-definable functions outgoing from methods of Lion and Rolin ($\mathbb{R}_{an,exp}$ is the o-minimal structure generated by all restricted analytic functions and the global exponential function). By a deep model theoretic fact of Van den Dries, Macintyre and Marker every $\mathbb{R}_{an,exp}$-definable function is piecewise given by $\mathcal{L}_{an}(\exp,\log)$-terms where $\mathcal{L}_{an}(\exp,\log)$ denotes the language of ordered rings augmented by all restricted analytic functions, the global exponential and the global logarithm. The idea is to consider log-analytic functions at first, i.e. functions which are iterated compositions from either side of globally subanalytic functions and the global logarithm, and then $\mathbb{R}_{an,exp}$-definable functions as compositions of log-analytic functions and the global exponential. |
| title | On Preparation Theorems for $\mathbb{R}_{an,exp}$-definable functions |
| topic | Logic 03C64, 32B20, 33B10, 26A09 |
| url | https://arxiv.org/abs/2112.08161 |