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

Using a work of Diaz concerning algebraic independence of certainsequences of numbers, we prove that if K is a field of finitetranscendence degree over the rationals, then every weakly o-minimalexpansion of (K,,+,·) is polynomially bounded. In thespecial case where K is the field of all real algebraic numbers,we give a proof which makes use of a much weaker result fromtranscendental number theory, namely, the Gelfond–Schneidertheorem. Apart from this we make a couple of observations concerningweakly o-minimal expansions of arbitrary ordered fields of finitetranscendence degree over the rationals. The strongest resultwe prove says that if K is a field of finite transcendence degreeover the rationals, then all weakly o-minimal non-valuationalexpansions of (K,,+,·) are power bounded.