Global almost analytic algebraicity of analytic sets (Q1175098)

It is known that any analytic subset of a real algebraic variety can be transformed into an algebraic subset by a smooth, but in general not analytical, diffeomorphism of the variety. The author substitutes smooth and analytic equivalences for almost analytic equivalence. Main result: Let \(X\) be a compact affine smooth real algebraic variety, and \(V\) be a coherent closed analytic hypersurface in \(X\) with a finite set of singular points and \(\mathbb{Z}/2\)-homological to some algebraic subset in \(X\) (that is a necessary condition). Then there is a multiblow-up \(p:Y\to X\) along \(\hbox{Sing}(V)\) such that for any \(r\geq 0\) there exist a \(C^ r\)- diffeomorphism \(s_ r:X\to X\) and an analytic diffeomorphism \(t_ r:Y\to Y\) satisfying: (i) \(p\circ t_ r=s_ r\circ p\), (ii) \(s_ r(V)\) is algebraic.