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

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.