Given a polynomially bounded structure S that defines restricted
power functions,
we prove that a all intermediate structures between S and the expansion
S^IR of S by all
real power functions are of the form S^L where L is an intermediate
field between IR and
the field of exponents of S. This result can be understood as a
polynomially bounded
version of a conjecture of van den Dries and Miller.
(Joint work with Gareth Jones, Manchester).