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This article presents two constructions motivated by a conjecture of L. vanden Dries and C. Miller concerning the restricted analytic field withexponentiation. The first construction provides an example of two o-minimalexpansions of a real closed field that possess the same field of germs atinfinity of one-variable functions and yet define different global one-variablefunctions. The second construction gives an example of a family of infinitelymany distinct polynomially bounded reducts (all this in the sense ofdefinability) of the restricted analytic field with exponentiation.

Two remarks on polynomially bounded reducts of the restricted analytic field with exponentiation