Analytic continuations of log-exp-analytic germs

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2019
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Transactions of the American Mathematical Society (TRAN). 2019, 371(7), pp. 5203-5246. ISSN 0002-9947. eISSN 1088-6850. Available under: doi: 10.1090/tran/7748
Zusammenfassung

We describe maximal, in a sense made precise, L-analytic continuations of germs at +∞ of unary functions definable in the o-minimal structure Ran,exp on the Riemann surface L of the logarithm. As one application, we give an upper bound on the logarithmic-exponential complexity of the compositional inverse of an infinitely increasing such germ, in terms of its own logarithmic-exponential complexity and its level. As a second application, we strengthen Wilkie’s theorem on definable complex analytic continuations of germs belonging to the residue field Rpoly of the valuation ring of all polynomially bounded definable germs.

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510 Mathematik
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ISO 690KAISER, Tobias, Patrick SPEISSEGGER, 2019. Analytic continuations of log-exp-analytic germs. In: Transactions of the American Mathematical Society (TRAN). 2019, 371(7), pp. 5203-5246. ISSN 0002-9947. eISSN 1088-6850. Available under: doi: 10.1090/tran/7748
BibTex
@article{Kaiser2019Analy-40967.2,
  year={2019},
  doi={10.1090/tran/7748},
  title={Analytic continuations of log-exp-analytic germs},
  number={7},
  volume={371},
  issn={0002-9947},
  journal={Transactions of the American Mathematical Society (TRAN)},
  pages={5203--5246},
  author={Kaiser, Tobias and Speissegger, Patrick}
}
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    <dcterms:abstract xml:lang="eng">We describe maximal, in a sense made precise, L-analytic continuations of germs at +∞ of unary functions definable in the o-minimal structure R&lt;sub&gt;an,exp&lt;/sub&gt; on the Riemann surface L of the logarithm. As one application, we give an upper bound on the logarithmic-exponential complexity of the compositional inverse of an infinitely increasing such germ, in terms of its own logarithmic-exponential complexity and its level. As a second application, we strengthen Wilkie’s theorem on definable complex analytic continuations of germs belonging to the residue field R&lt;sub&gt;poly&lt;/sub&gt; of the valuation ring of all polynomially bounded definable germs.</dcterms:abstract>
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