Quantifier Elimination and Rectilinearisation Theorem for quasi-analytic algebras

An algebra of real functions is quasi-analytic if there is an injective morphism which associates a (divergent) generalised power series to each germ at zero of a function in the algebra. In the context of quasi-analytic algebras, which generalises that of real analytic geometry, we prove an analogue to Denef and van de Dries' Quantifier Elimination Theorem and analogue to Hironaka's Rectilinearisation Theorem, which states that every bounded subanalytic set can be written, after a finite sequence of blow-ups, as a finite union of quadrants.

Every known o-minimal polynomially bounded expansion of the real field is generated by a family of quasi-analytic algebras, and hence satisfies the hypotheses of our results (joint work with J.-P. Rolin).