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Bulletin of the London Mathematical Society Advance Access originally published online on February 5, 2009
Bulletin of the London Mathematical Society 2009 41(1):109-116; doi:10.1112/blms/bdn112
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© 2009 London Mathematical Society

Weakly o-minimal expansions of ordered fields of finite transcendence degree

Roman Wencel

Mathematical Institute
University of Wroclaw
pl. Grunwaldzki 2/4
Wroclaw 50-384
Poland

Received 20 March 2008.

Using a work of Diaz concerning algebraic independence of certain sequences of numbers, we prove that if K {subseteq} R is a field of finite transcendence degree over the rationals, then every weakly o-minimal expansion of (K,≤,+,·) is polynomially bounded. In the special case where K is the field of all real algebraic numbers, we give a proof which makes use of a much weaker result from transcendental number theory, namely, the Gelfond–Schneider theorem. Apart from this we make a couple of observations concerning weakly o-minimal expansions of arbitrary ordered fields of finite transcendence degree over the rationals. The strongest result we prove says that if K is a field of finite transcendence degree over the rationals, then all weakly o-minimal non-valuational expansions of (K,≤,+,·) are power bounded.

2000 Mathematics Subject Classification 03C64.

This research was supported by a Marie Curie Intra-European Fellowships within the 6th European Community Framework Programme (contract number: MEIF-CT-2003-501326), by a Marie Curie European Reintegration Grant (contract number MERG-CT-2005-031095) and by a Polish government grant N201 018 32/0800.


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