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Weakly o‐minimal expansions of ordered fields of finite transcendence degree
Author(s) -
Wencel Roman
Publication year - 2009
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/blms/bdn112
Subject(s) - mathematics , rational number , bounded function , degree (music) , transcendental number , algebraic number , field (mathematics) , finite field , transcendence (philosophy) , pure mathematics , discrete mathematics , mathematical analysis , philosophy , physics , epistemology , acoustics
Using a work of Diaz concerning algebraic independence of certain sequences of numbers, we prove that if K ⊆ 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.