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Analytic continuations of log-exp-analytic germs
Analytic continuations of log-exp-analytic germs
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Date
2017
Authors
Kaiser, Tobias
Speissegger, Patrick
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Transactions of the American Mathematical Society ; 2017. - ISSN 0002-9947. - eISSN 1088-6850
Abstract
We describe maximal, in a sense made precise, analytic continuations of germs at infinity of unary functions definable in the o-minimal structure R_an,exp on the Riemann surface 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 of the valuation ring of all polynomially bounded definable germs.
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510 Mathematics
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O-minimal structures, log-exp-analytic germs, analytic continuation
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54 pages. Corollary 1.6 and 7.6 were added, to describe complex analytic continuations resulting from our Continuation theorem. Application 1.3 was changed to correspond to Corollary 8.10(1) instead of 8.5. Both changes were made to reflect applications in an upcoming paper; no changes were made to the main theorems or their proofs.