Local semianalytic geometry

Abstract

This paper treats the theory of semianalytic function germs over real closed fields more general than ℝ. An ordered field is microbial if it has a non-zero element whose powers converge to zero. The fields we treat are direct limits of countable microbial subfields. We define local rings of analytic function germs algebraically and use the Weierstrass preparation theory to prove an Artin-Lang property. We end by relating seminash functions to abstract semialgebraic functions on the real spectrum of the local rings.