Expansions of the Real Field with Power Functions

As the title suggests, this brief note is a follow-up to [5] (my first published paper), which the reader is assumed to have at hand. I make more readily available some results from my thesis [6, Chapter IV] that generalize some of the main results from [5], the latter being written just before the technology became available for proving more general results. Though I think these extensions are interesting, the proofs are fairly minor modifications of the material in [5], so I never published them except in my thesis. I also give some new applications, correct a few errors, and give the final data for the references of [5]. Put R := (R, <,+,−, ⋅, 0, 1). Let R be a polynomially bounded o-minimal expansion of R having field of exponents K0. Let S ⊆ R and put R = ( R, (x)s∈S ) . By [8], R is o-minimal; indeed, so is (R, e). Unfortunately, the method of proof does not reveal the field of exponents of R, nor even whether R is polynomially bounded.∗ But the answer is known under some fairly reasonable assumptions. Let K be the subfield of R generated by S over K0. Theorem. Suppose that R defines each restriction x↾[1, 2], s ∈ S. Then R is polynomially bounded with field of exponents K.