Undefinability results in o-minimal expansions of the real numbers

We show that if β ∈ R is not in the field generated by α 1 , … , α n , then no restriction of the function x β to an interval is definable in 〈 R , + , − , ⋅ , 0 , 1 , < , x α 1 , … , x α n 〉 . We also prove that if the real and imaginary parts of a complex analytic function are definable in R exp or in the expansion of R ̄ (definitions in the text) by functions x α , for irrational α , then they are already definable in R ̄ . We conclude with some conjectures and open questions.