Multiplicities of Pfaffian intersections, and the Łojasiewicz inequality. (English) Zbl 0889.32005

Gabrielov, A.

Sel. Math., New Ser. 1, No. 1, 113-127 (1995).

Summary: An effective estimate for the local multiplicity of a complete intersection of complex algebraic and Pfaffian varieties is given, based on a local complex analog of the Rolle-Khovanskii theorem. The estimate is valid also for the properly defined multiplicity of a non-isolated intersection. It implies, in particular, effective estimates for the exponents of the polar curves, and the exponents in the Łojasiewicz inequalities for Pfaffian functions. For the intersections defined by sparse polynomials, the multiplicities outside the coordinate hyperplanes can be estimated in terms of the number of non-zero monomials, independent of degrees of the monomials.

MSC:

32B20	Semi-analytic sets, subanalytic sets, and generalizations
14P10	Semialgebraic sets and related spaces
58A17	Pfaffian systems

Keywords:

algebraic variety; local multiplicity; complete intersection; Pfaffian varieties; effective estimates; Łojasiewicz inequalities; Pfaffian functions

References:

[1]	A.G. Khovanskii,On a class of systems of transcendental equations, Soviet Math. Dokl.,22, 762–765, 1980. · Zbl 0569.32004
[2]	A.G. Khovanskii,Fewnomials, AMS Translation of mathematical monographs, v.88, AMS, Providence, RI, 1991. (Russian original: Malochleny, Moscow, 1987)
[3]	A. Gabrielov,Projections of semi-analytic sets, Functional Anal. Appl., v. 2, n. 4, p. 282–291, 1968. · Zbl 0179.08503
[4]	H. Hironaka,Subanalytic sets, in: Number Theory, Algebraic Geometry and Commutative Algebra (in honor of Y. Akizuki), Kinokunya, Tokyo, p. 453–493, 1973.
[5]	R.M. Hardt.Stratification of real analytic mappings and images, Invent. Math.,28, 193–208, 1975. · Zbl 0298.32003
[6]	Z. Denkowska, S. Łojasiewicz, and J. Stasica,Certaines propriétés élémentaires des ensembles sous-analytiques, Bull. Polish Acad. Sci. Math.27, 529–536, 1979. · Zbl 0435.32006
[7]	E. Bierstone and P.D. Milman.Semianalytic and subanalytic sets, Inst. Hautes Etudes Sci. Publ. Math.,67, 5–42, 1988. · Zbl 0674.32002
[8]	J. Denef and L. van den Dries,P-adic and real subanalytic sets, Ann. Math.,128, 79–138, 1988. · Zbl 0693.14012
[9]	A. Gabrielov,Existential formulas for analytic functions, preprint Cornell MSI, 1993.
[10]	A. Gabrielov and N. Vorobjov,Complexity of stratifications of semi-Pfaffian sets, preprint Cornell MSI, 1993. To appear in J. Discrete Comput. Geom., 1995. · Zbl 0832.68056
[11]	A.J. Wilkie,Smooth o-minimal theories and the model completeness of the real exponential field, preprint, 1992.
[12]	L. van den Dries, A. Macintyre, and D. Marker.The elementary theory of restricted analytic fields with exponentiation, Ann. Math.,140, n. 1, 183–205, 1994. · Zbl 0837.12006
[13]	E. Bierstone, P.D. Milman.A simple constructive proof of canonical resolution of singularities, In: Effective Methods in Algebraic Geometry, Progress in Math.,94, Birkhäuser, Boston, p. 11–30, 1991. · Zbl 0743.14012
[14]	E. Bierstone, P.D. Milman,Canonical desingularization in characteristic zero: an elementary proof, preprint, 1994.
[15]	A. Gabrielov, J.-M. Lion, R. Moussu,Ordre de contact de courbes intégrales du plan, CR Ac. Sci. Paris,319, 219–221, 1994. · Zbl 0836.14014
[16]	B. Teissier,Variétés polaires I, Inv. Math.,40, n. 3, 267–292, 1977. · Zbl 0446.32002
[17]	B. Teissier,Cycles évanescents, sections planes et conditions de Whitney, in: Singularités à Cargèse, Astérisque7–8, 285–362, 1972.
[18]	S. Łojasiewicz, Sur les trajectories du gradient d’une fonction analytique, Univ. Stud. Bologna Sem. Geom. p. 115–117, 1983.