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# Real closed exponential fields

Type of Publication: | Journal article |

Author: | D'Aquino, Paola; Knight, Julia F.; Kuhlmann, Salma; Lange, Karen |

Year of publication: | 2012 |

Published in: | Fundamenta Mathematicae ; 219 (2012), 2. - pp. 163-190. - ISSN 0016-2736. - eISSN 1730-6329 |

DOI (citable link): | https://dx.doi.org/10.4064/fm219-2-6 |

Summary: |
Ressayre considered real closed exponential fields and “exponential” integer parts, i.e., integer parts that respect the exponential function. In 1993, he outlined a proof that every real closed exponential field has an exponential integer part. In the present paper, we give a detailed account of Ressayre's construction and then analyze the complexity. Ressayre's construction is canonical once we fix the real closed exponential field R, a residue field section k, and a well ordering ≺ on R. The construction is clearly constructible over these objects. Each step looks effective, but there may be many steps. We produce an example of an exponential field R with a residue field section k and a well ordering ≺ on R such that D
^{c}(R) is low and k and ≺ are Δ 0 3, and Ressayre's construction cannot be completed in L_{ωCK1}. |

Subject (DDC): | 510 Mathematics |

Bibliography of Konstanz: | Yes |

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D'AQUINO, Paola, Julia F. KNIGHT, Salma KUHLMANN, Karen LANGE, 2012. Real closed exponential fields. In: Fundamenta Mathematicae. 219(2), pp. 163-190. ISSN 0016-2736. eISSN 1730-6329. Available under: doi: 10.4064/fm219-2-6

@article{DAquino2012close-23418, title={Real closed exponential fields}, year={2012}, doi={10.4064/fm219-2-6}, number={2}, volume={219}, issn={0016-2736}, journal={Fundamenta Mathematicae}, pages={163--190}, author={D'Aquino, Paola and Knight, Julia F. and Kuhlmann, Salma and Lange, Karen} }