Semianalytic and subanalytic sets. (English) Zbl 0674.32002

Bierstone, Edward; Milman, Pierre D.

Publ. Math., Inst. Hautes Étud. Sci. 67, 5-42 (1988).

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

Keywords:

semialgebraic sets; semianalytic sets; uniformization; rectilinearization; subanalytic sets; resolution of singularities

References:

[1]	E. Bierstone, Differentiable functions,Bol. Soc. Brasil Mat.,11 (1980), 139–180. · Zbl 0584.58006 · doi:10.1007/BF02584636
[2]	E. Bierstone andP. D. Milman, Algebras of composite differentiable functions,Proc. Sympos. Pure Math.,40, Part 1 (1983), 127–136. · Zbl 0519.58004
[3]	P. J. Cohen, Decision procedures for real andp-adic fields,Comm. Pure Appl. Math.,22 (1969), 131–151. · Zbl 0167.01502 · doi:10.1002/cpa.3160220202
[4]	M. Coste, Ensembles semi-algébriques, Géométrie algébrique réelle et formes quadratiques (Proceedings, Rennes, 1981),Lecture Notes in Math., No. 959, Berlin-Heidelberg-New York, Springer, 1982, p. 109–138.
[5]	J. Denef andL. van den Dries,p-adic and real subanalytic sets,Ann. of Math. (2) (to appear).
[6]	Z. Denkowska, S. Łojasiewicz andJ. Stasica, Certaines propriétés élémentaires des ensembles sous-analytiques.Bull. Acad. Polon. Sci. Sér. Sci. Math.,27 (1979), 529–536. · Zbl 0435.32006
[7]	Z. Denkowska, S. Łojasiewicz andJ. Stasica, Sur le théorème du complémentaire pour les ensembles sous-analytiques,Bull. Acad. Polon. Sci. Sér. Sci. Math.,27 (1979), 537–539. · Zbl 0457.32003
[8]	Z. Denkowska, S. Łojasiewicz andJ. Stasica, Sur le nombre des composantes connexes de la section d’un sous-analytique,Bull. Acad. Polon. Sci. Sér. Sci. Math.,30 (1982), 333–335. · Zbl 0527.32007
[9]	Z. Denkowska andJ. Stasica,Ensembles sous-analytiques à la polonaise (preprint, 1985). · Zbl 0584.32013
[10]	G. Efroymson, Substitution in Nash functions,Pacific J. Math.,63 (1976), 137–145. · Zbl 0335.14002
[11]	A. M. Gabrielov, Projections of semi-analytic sets,Functional Anal. Appl.,2 (1968), 282–291 =Funkcional. Anal. i Prilozen. 2, No. 4 (1968), 18–30. · Zbl 0179.08503 · doi:10.1007/BF01075680
[12]	A. M. Gabrielov, Formal relations between analytic functions,Math. USSR Izvestija,7 (1973), 1056–1088 =Izv. Akad. Nauk SSSR Ser. Mat.,37 (1973), 1056–1090. · Zbl 0297.32007 · doi:10.1070/IM1973v007n05ABEH001992
[13]	R. M. Hardt, Stratification of real analytic mappings and images,Invent. Math.,28 (1975), 193–208. · Zbl 0298.32003 · doi:10.1007/BF01436073
[14]	R. M. Hardt, Triangulation of subanalytic sets and proper light subanalytic maps,Invent. Math.,38 (1977), 207–217. · doi:10.1007/BF01403128
[15]	R. M. Hardt, Some analytic bounds for subanalytic sets,Differential Geometric Control Theory, Boston, Birkhäuser, 1983, p. 259–267.
[16]	H. Hironaka, Resolution of singularities of an algebraic variety over a field of characteristic zero: I, II,Ann. of Math. (2),79 (1964), 109–326. · Zbl 0122.38603 · doi:10.2307/1970486
[17]	H. Hironaka, Subanalytic sets,Number Theory, Algebraic Geometry and Commutative Algebra, Tokyo, Kinokuniya, 1973, p. 453–493.
[18]	H. Hironaka,Introduction to real-analytic sets and real-analytic maps, Istituto Matematico ”L. Tonelli”, Pisa, 1973. · Zbl 0297.32008
[19]	S. Łojasiewicz, Sur le problème de la division,Studia Math.,8 (1959), 87–136. · Zbl 0115.10203
[20]	S. Łojasiewicz, Triangulation of semi-analytic sets,Ann. Scuola Norm. Sup. Pisa (3),18 (1964), 449–474. · Zbl 0128.17101
[21]	S. Łojasiewicz,Ensembles semi-analytiques, Inst. Hautes Études Sci., Bures-sur-Yvette, 1964.
[22]	S. Łojasiewicz, Sur la semi-analyticité des images inverses par l’application-tangente,Bull. Acad. Polon. Sci. Sér. Sci. Math.,27 (1979), 525–527. · Zbl 0452.32005
[23]	B. Malgrange, Frobenius avec singularités, 2. Le cas général,Invent. Math.,39 (1977), 67–89. · Zbl 0375.32012 · doi:10.1007/BF01695953
[24]	R. Narasimhan, Introduction to the theory of analytic spaces,Lecture Notes in Math., No. 25, Berlin-Heidelberg-New York, Springer, 1966. · Zbl 0168.06003
[25]	J.-B. Poly andG. Raby, Fonction distance et singularités,Bull. Sci. Math. (2),108 (1984), 187–195.
[26]	M. Tamm, Subanalytic sets in the calculus of variations,Acta Math.,146 (1981), 167–199. · Zbl 0478.58010 · doi:10.1007/BF02392462
[27]	H. Whitney, Local properties of analytic varieties,Differential and Combinatorial Topology (A Symposium in Honor of Marston Morse), Princeton, Princeton Univ. Press, 1965, p. 205–244.
[28]	O. Zariski, Local uniformization theorem on algebraic varieties,Ann. of Math. (2),41 (1940), 852–896. · JFM 66.1327.02 · doi:10.2307/1968864
[29]	O. Zariski andP. Samuel,Commutative Algebra, Vol. II, Berlin-Heidelberg-New York, Springer, 1975. · Zbl 0313.13001