On rank Theorems and the Nash points of subanalytic sets

Abstract

We prove a generalization of Gabrielov's rank theorem for families of rings of power series which we call W-temperate. Examples include the family of complex analytic functions and of Eisenstein series. Then the rank theorem for Eisenstein series allows us to give new proofs of the following two results of W. Pawlucki: I) The non regular locus of a complex or real analytic map is an analytic set. II) The set of semianalytic or Nash points of a subanalytic set $X$ is a subanalytic set, whose complement has codimension two in $X$.Comment: 50 page