Separation of real algebraic sets and the Łojasiewicz exponent

Kurdyka, Krzysztof; Spodzieja, Stanisław

doi:10.1090/S0002-9939-2014-12061-2

Separation of real algebraic sets and the Łojasiewicz exponent
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by Krzysztof Kurdyka and Stanisław Spodzieja PDF

Proc. Amer. Math. Soc. 142 (2014), 3089-3102 Request permission

Abstract:

We discuss several aspects of Łojasiewicz inequalities, namely local and global versions, and the relations between the gradient inequality and regular separation of real algebraic sets. We give effective estimates for Łojasiewicz’s exponents in the real and complex setting.

References

Edward Bierstone and Pierre D. Milman, Canonical desingularization in characteristic zero by blowing up the maximum strata of a local invariant, Invent. Math. 128 (1997), no. 2, 207–302. MR 1440306, DOI 10.1007/s002220050141
Carles Bivià-Ausina, Injectivity of real polynomial maps and Łojasiewicz exponents at infinity, Math. Z. 257 (2007), no. 4, 745–767. MR 2342551, DOI 10.1007/s00209-007-0129-0
J. Bochnak and S. Łojasiewicz, A converse of the Kuiper-Kuo theorem, Proceedings of Liverpool Singularities—Symposium, I (1969/70), Lecture Notes in Mathematics, Vol. 192, Springer, Berlin, 1971, pp. 254–261. MR 0291971
J. Bochnak and J. J. Risler, Sur les exposants de Lojasiewicz, Comment. Math. Helv. 50 (1975), no. 4, 493–507. MR 404674, DOI 10.1007/BF02565764
Jérôme Bolte, Aris Daniilidis, Olivier Ley, and Laurent Mazet, Characterizations of Łojasiewicz inequalities: subgradient flows, talweg, convexity, Trans. Amer. Math. Soc. 362 (2010), no. 6, 3319–3363. MR 2592958, DOI 10.1090/S0002-9947-09-05048-X
W. Dale Brownawell, Bounds for the degrees in the Nullstellensatz, Ann. of Math. (2) 126 (1987), no. 3, 577–591. MR 916719, DOI 10.2307/1971361
Jacek Chądzyński, On proper polynomial mappings, Bull. Polish Acad. Sci. Math. 31 (1983), no. 3-4, 115–120 (English, with Russian summary). MR 742795
Jacek Chądzyński and Tadeusz Krasiński, Sur l’exposant de Łojasiewicz à l’infini pour les applications polynomiales de $\textbf {C}^2$ dans $\textbf {C}^2$ et les composantes des automorphismes polynomiaux de $\textbf {C}^2$, C. R. Acad. Sci. Paris Sér. I Math. 315 (1992), no. 13, 1399–1402 (French, with English and French summaries). MR 1199010
Jacek Chądzyński and Tadeusz Krasiński, The gradient of a polynomial at infinity, Kodai Math. J. 26 (2003), no. 3, 317–339. MR 2018725, DOI 10.2996/kmj/1073670612
Ewa Cygan, A note on separation of algebraic sets and the Łojasiewicz exponent for polynomial mappings, Bull. Sci. Math. 129 (2005), no. 2, 139–147 (English, with English and French summaries). MR 2123264, DOI 10.1016/j.bulsci.2004.10.001
Ewa Cygan, Tadeusz Krasiński, and Piotr Tworzewski, Separation of algebraic sets and the Łojasiewicz exponent of polynomial mappings, Invent. Math. 136 (1999), no. 1, 75–87. MR 1681109, DOI 10.1007/s002220050304
Didier D’Acunto and Krzysztof Kurdyka, Explicit bounds for the Łojasiewicz exponent in the gradient inequality for polynomials, Ann. Polon. Math. 87 (2005), 51–61. MR 2208535, DOI 10.4064/ap87-0-5
A. Gabrièlov, Multiplicities of Pfaffian intersections, and the Łojasiewicz inequality, Selecta Math. (N.S.) 1 (1995), no. 1, 113–127. MR 1327229, DOI 10.1007/BF01614074
Janusz Gwoździewicz, The Łojasiewicz exponent of an analytic function at an isolated zero, Comment. Math. Helv. 74 (1999), no. 3, 364–375. MR 1710702, DOI 10.1007/s000140050094
Janusz Gwoździewicz and Stanisław Spodzieja, The Łojasiewicz gradient inequality in a neighbourhood of the fibre, Ann. Polon. Math. 87 (2005), 151–163. MR 2208542, DOI 10.4064/ap87-0-12
Huy Vui Ha, Nombres de Lojasiewicz et singularités à l’infini des polynômes de deux variables complexes, C. R. Acad. Sci. Paris Sér. I Math. 311 (1990), no. 7, 429–432 (French, with English summary). MR 1075664
Zbigniew Jelonek, On the effective Nullstellensatz, Invent. Math. 162 (2005), no. 1, 1–17. MR 2198324, DOI 10.1007/s00222-004-0434-8
Zbigniew Jelonek, On the Łojasiewicz exponent, Hokkaido Math. J. 35 (2006), no. 2, 471–485. MR 2254661, DOI 10.14492/hokmj/1285766366
Zbigniew Jelonek and Krzysztof Kurdyka, On asymptotic critical values of a complex polynomial, J. Reine Angew. Math. 565 (2003), 1–11. MR 2024642, DOI 10.1515/crll.2003.101
Shanyu Ji, János Kollár, and Bernard Shiffman, A global Łojasiewicz inequality for algebraic varieties, Trans. Amer. Math. Soc. 329 (1992), no. 2, 813–818. MR 1046016, DOI 10.1090/S0002-9947-1992-1046016-6
János Kollár, Sharp effective Nullstellensatz, J. Amer. Math. Soc. 1 (1988), no. 4, 963–975. MR 944576, DOI 10.1090/S0894-0347-1988-0944576-7
János Kollár, An effective Łojasiewicz inequality for real polynomials, Period. Math. Hungar. 38 (1999), no. 3, 213–221. MR 1756239, DOI 10.1023/A:1004806609074
Nicolaas H. Kuiper, $C^{1}$-equivalence of functions near isolated critical points, Symposium on Infinite-Dimensional Topology (Louisiana State Univ., Baton Rouge, La., 1967) Ann. of Math. Studies, No. 69, Princeton Univ. Press, Princeton, N. J., 1972, pp. 199–218. MR 0413161
Tzee Char Kuo, On $C^{0}$-sufficiency of jets of potential functions, Topology 8 (1969), 167–171. MR 238338, DOI 10.1016/0040-9383(69)90007-X
Krzysztof Kurdyka and Adam Parusiński, $\textbf {w}_f$-stratification of subanalytic functions and the Łojasiewicz inequality, C. R. Acad. Sci. Paris Sér. I Math. 318 (1994), no. 2, 129–133 (English, with English and French summaries). MR 1260324
Krzysztof Kurdyka, On gradients of functions definable in o-minimal structures, Ann. Inst. Fourier (Grenoble) 48 (1998), no. 3, 769–783 (English, with English and French summaries). MR 1644089, DOI 10.5802/aif.1638
Krzysztof Kurdyka, Tadeusz Mostowski, and Adam Parusiński, Proof of the gradient conjecture of R. Thom, Ann. of Math. (2) 152 (2000), no. 3, 763–792. MR 1815701, DOI 10.2307/2661354
Monique Lejeune-Jalabert and Bernard Teissier, Clôture intégrale des idéaux et équisingularité, Ann. Fac. Sci. Toulouse Math. (6) 17 (2008), no. 4, 781–859 (French, with English summary). With an appendix by Jean-Jacques Risler. MR 2499856, DOI 10.5802/afst.1203
S. Łojasiewicz, Une propriété topologique des sous-ensembles analytiques réels, Les Équations aux Dérivées Partielles (Paris, 1962) Éditions du Centre National de la Recherche Scientifique (CNRS), Paris, 1963, pp. 87–89 (French). MR 0160856
S. Łojasiewicz, Ensembles semi-analytiques, preprint IHES, 1965.
S. Łojasiewicz, Sur les trajectoires du gradient d’une fonction analytique, Geometry seminars, 1982–1983 (Bologna, 1982/1983) Univ. Stud. Bologna, Bologna, 1984, pp. 115–117 (French). MR 771152
Beata Osińska, Extensions of regular mappings and the Łojasiewicz exponent at infinity, Bull. Sci. Math. 135 (2011), no. 2, 215–229. MR 2773399, DOI 10.1016/j.bulsci.2010.11.002
Adam Parusiński, On the bifurcation set of complex polynomial with isolated singularities at infinity, Compositio Math. 97 (1995), no. 3, 369–384. MR 1353280
Phạm Tiến So’n, The Łojasiewicz exponent of a continuous subanalytic function at an isolated zero, Proc. Amer. Math. Soc. 139 (2011), no. 1, 1–9. MR 2729065, DOI 10.1090/S0002-9939-2010-10683-4
A. Płoski, On the growth of proper polynomial mappings, Ann. Polon. Math. 45 (1985), no. 3, 297–309. MR 817547, DOI 10.4064/ap-45-3-297-309
Arkadiusz Płoski, Multiplicity and the Łojasiewicz exponent, Singularities (Warsaw, 1985) Banach Center Publ., vol. 20, PWN, Warsaw, 1988, pp. 353–364. MR 1101851
Tomasz Rodak and Stanisław Spodzieja, Effective formulas for the Łojasiewicz exponent at infinity, J. Pure Appl. Algebra 213 (2009), no. 9, 1816–1822. MR 2518180, DOI 10.1016/j.jpaa.2009.01.016
Tomasz Rodak and Stanisław Spodzieja, Effective formulas for the local Łojasiewicz exponent, Math. Z. 268 (2011), no. 1-2, 37–44. MR 2805423, DOI 10.1007/s00209-009-0659-8
Tomasz Rodak and Stanisław Spodzieja, Łojasiewicz exponent near the fibre of a mapping, Proc. Amer. Math. Soc. 139 (2011), no. 4, 1201–1213. MR 2748414, DOI 10.1090/S0002-9939-2010-10623-8
Grzegorz Skalski, On the Łojasiewicz exponent near the fibre of a polynomial, Bull. Pol. Acad. Sci. Math. 52 (2004), no. 3, 231–236. MR 2127060, DOI 10.4064/ba52-3-3
Stanisław Spodzieja, The Łojasiewicz exponent of subanalytic sets, Ann. Polon. Math. 87 (2005), 247–263. MR 2208551, DOI 10.4064/ap87-0-21
B. Teissier, Variétés polaires. I. Invariants polaires des singularités d’hypersurfaces, Invent. Math. 40 (1977), no. 3, 267–292 (French). MR 470246, DOI 10.1007/BF01425742