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Semianalytic and subanalytic sets. (English) Zbl 0674.32002
The main achievements of the article are short and elementary proofs of the following uniformization resp. rectilinearization theorem for subanalytic sets (originally due to Hironaka, using resolution of singularities): Let M be a real analytic manifold and $$X\subset M$$ a subanalytic subset.
Uniformization theorem: If X is closed then there exists a real analytic manifold N of the same dimension as X and a proper real analytic mapping $$\phi$$ : $$N\to M$$ such that $$\phi (N)=X.$$
Rectilinearization theorem: If M is of pure dimension m and $$K\subset M$$ a compact subset, then there are finitely many real analytic mappings $$\phi_ i: {\mathbb{R}}^ m\to M$$ such that: (1) For each i there is a compact subset $$L_ i\subset {\mathbb{R}}^ m$$ such that $$\cup \phi_ i(L_ i)$$ is a neighbourhood of K in M. (2) For each i, $$\phi_ i^{- 1}(X)$$ is a union of quadrants in $${\mathbb{R}}^ m.$$
As the authors say, from the point of view of analysis, these two theorems express the most important aspects of resolution of singularities. On the other hand, the theorems are still far from resolution of singularities since the morphisms are not required to be bimeromorphic. Moreover, they are actually local. Using the fact that each subanalytic set is locally the image of a closed analytic set under a projection, the proof of the uniformization theorem is reduced to the case where $$X\subset M$$ is analytic. By considering the product of the defining equations for X one can further reduce to the case of a hypersurface.
The authors give an easy proof of the fact, that any analytic function on an analytic manifold (real or complex) can be transformed to normal crossings by blowings-up. The proof uses only the Weierstrass preparation theorem and elementary but clever induction on dim M. (The induction hypothesis guarantees that the coefficients of the Weierstrass polynomial in question can be transformed by blowings-up to monomials (up to a factor by units) such that the exponents are totally ordered with respect to the induced partial ordering from $${\mathbb{N}}^{m-1}$$, $$m=\dim M.)$$
The rectilinearization theorem follows by applying the normal crossing theorem to appropriate distance functions.
In the last two sections the authors apply the uniformization theorem to give a proof of Łojasiewicz’s inequality and of the fact (due to Tamm) that the set of smooth points of a subanalytic set is subanalytic. The first three sections of the article consists of a useful selfcontained treatment of semialgebraic resp. semianalytic resp. subanalytic sets.
In addition to the bibliography the referee likes to mention that the first complete treatment of semialgebraic sets (even for real ordered fields) including the triangulation theorem appeared in 1954 in the thesis of H. Brakhage. This thesis was never officially published but a copy is available from Prof. H. Brakhage (Univ. Kaiserslautern).
Reviewer: G.M.Greuel

##### MSC:
 32B20 Semi-analytic sets, subanalytic sets, and generalizations 32B05 Analytic algebras and generalizations, preparation theorems 32C05 Real-analytic manifolds, real-analytic spaces 32S45 Modifications; resolution of singularities (complex-analytic aspects) 14Pxx Real algebraic and real-analytic geometry
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