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Effective formulas for the local Łojasiewicz exponent

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Abstract

We give an effective formula for the local Łojasiewicz exponent of a polynomial mapping. Moreover, we give an algorithm for computing the local dimension of an algebraic variety.

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References

  1. Abderrahmane O.M.: On the Łojasiewicz exponent and Newton polyhedron. Kodai Math. J. 28(1), 106–110 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  2. Achilles R., Tworzewski P., Winiarski T.: On improper isolated intersection in complex analytic geometry. Ann. Polon. Math. 51, 21–36 (1990)

    MathSciNet  MATH  Google Scholar 

  3. Acquistapace F., Broglia F., Shiota M.: The finiteness property and Łojasiewicz inequality for global semianalytic sets. Adv. Geom. 5(3), 377–390 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bochnak, J., Łojasiewicz, S.: A converse of the Kuiper-Kuo theorem. In: Proceedings of Liverpool Singularities—Symposium, I (1969/70), pp. 254–261. Lecture Notes in Math., vol. 192. Springer, Berlin (1971)

  5. Bochnak J., Risler J.J.: Sur les exposants de Łojasiewicz. Comment. Math. Helv. 50(4), 493–507 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bogusławska M.: On the Łojasiewicz exponent of the gradient of holomorphic functions. Bull. Polish Acad. Sci. Math. 47(4), 337–343 (1999)

    MathSciNet  MATH  Google Scholar 

  7. Chang S.H., Lu Y.C.: On C 0-sufficiency of complex jets. Can. J. Math. 25, 874–880 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  8. Chądzyński J., Krasiński T.: On the Łojasiewicz exponent at infinity for polynomial mappings of \({\mathbb {C}^2}\) into \({\mathbb {C}^2}\) and components of polynomial automorphisms of \({\mathbb {C}^2}\) . Ann. Polon. Math. 57, 291–302 (1992)

    MathSciNet  Google Scholar 

  9. Chądzyński J., Krasiński T.: Resultant and the Łojasiewicz exponent. Ann. Polon. Math. 61, 95–100 (1995)

    MathSciNet  Google Scholar 

  10. Chądzyński J., Krasiński T.: A set on which the local Łojasiewicz exponent is attained. Ann. Polon. Math. 67(3), 297–301 (1997)

    MathSciNet  Google Scholar 

  11. Cygan E., Krasiński T., Tworzewski P.: Separation of algebraic sets and the Łojasiewicz exponent of polynomial mappings. Invent. Math. 136(1), 75–87 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  12. D’Acunto D., Kurdyka K.: Explicit bounds for the Łojasiewicz exponent in the gradient inequality for polynomials. Ann. Polon. Math. 87, 51–61 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  13. Denkowski M.P.: The Łojasiewicz exponent of c-holomorphic mappings. Ann. Polon. Math. 87, 63–81 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  14. Fukui T.: Łojasiewicz type inequalities and Newton diagrams. Proc. Am. Math. Soc. 112(4), 1169–1183 (1991)

    MathSciNet  MATH  Google Scholar 

  15. Gabrielov A.: Multiplicities of Pfaffian intersections, and the Łojasiewicz inequality. Selecta Math. (N.S.) 1(1), 113–127 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  16. García Barroso E., Krasiński T., Płoski A.: The Łojasiewicz numbers and plane curve singularities. Ann. Polon. Math. 87, 127–150 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  17. García Barroso E., Krasiński T., Płoski A.: On the Łojasiewicz numbers. II. C. R. Math. Acad. Sci. Paris 341(6), 357–360 (2005)

    MathSciNet  MATH  Google Scholar 

  18. Greuel G.-M., Pfister G.: A Singular Introduction to Commutative Algebra. Springer, Berlin (2002)

    MATH  Google Scholar 

  19. Gunning, R.C.: Introduction to holomorphic functions of several variables, vol. II. Local theory. Wadsworth & Brooks/Cole Advanced Books & Software, Pacific Grove (1990)

  20. Hörmander L.: On the division of distributions by polynomials. Ark. Mat. 3, 555–568 (1958)

    Article  MathSciNet  MATH  Google Scholar 

  21. Kollár J.: An effective Łojasiewicz inequality for real polynomials. Period. Math. Hungar. 38(3), 213–221 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  22. Kollár J.: Effective Nullstellensatz for arbitrary ideals. J. Eur. Math. Soc. 1(3), 313–337 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  23. Kuiper, N.H.: C 1-equivalence of functions near isolated critical points. In: Proceedings of Symposium in Infinite Dimensional Topology. (Baton Rouge, 1967). Ann. of Math. Studies, vol. 69, pp. 199–218. Princeton University Press, Princeton (1972)

  24. Kuo T.C.: On C 0-sufficiency of jets of potential functions. Topology 8, 167–171 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  25. Kuo T.C.: Computation of Łojasiewicz exponent of f(x, y). Comment. Math. Helv. 49, 201–213 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  26. Kuo T.C., Parusiński A.: Newton-Puiseux roots of Jacobian determinants. J. Algebraic Geom. 13(3), 579–601 (2004)

    MathSciNet  MATH  Google Scholar 

  27. Kurdyka K., Parusiński A.: w f -stratification of subanalytic functions and the Łojasiewicz inequality. C. R. Acad. Sci. Paris Sér. I Math. 318(2), 129–133 (1994)

    MATH  Google Scholar 

  28. Lejeune-Jalabertm, M., Teissier, B.: Clôture intégrale des idéaux et équisingularité. Centre de Mathématiques Ecole Polytechnique Palaiseau (1974)

  29. Lichtin B.: Estimation of Łojasiewicz exponents and Newton polygons. Invent. Math. 64(3), 417–429 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  30. Loi T.L.: Łojasiewicz inequalities for sets definable in the structure R exp. Ann. Inst. Fourier (Grenoble) 45(4), 951–971 (1995)

    MathSciNet  MATH  Google Scholar 

  31. Łojasiewicz S.: Sur le problème de la division. Studia Math. 18, 87–136 (1959)

    MathSciNet  MATH  Google Scholar 

  32. Melle-Hernández A.: On polar invariants of hypersurface singularities. Ann. Fac. Sci. Toulouse Math. (6) 9(4), 671–688 (2000)

    MathSciNet  MATH  Google Scholar 

  33. Merle M.: Invariants polaires des courbes planes. Invent. Math. 41(2), 103–111 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  34. Płoski A.: Une évaluation pour les sous-ensembles analytiques complexes. Bull. Pol. Acad. Sci. Math. 31, 259–262 (1983)

    MATH  Google Scholar 

  35. Płoski A.: Sur l’exposant d’une application analytique, II. Bull. Pol. Acad. Sci. Math. 33, 123–127 (1985)

    MATH  Google Scholar 

  36. Płoski, A.: Multiplicity and the Łojasiewicz exponent. Banach Center Publications 20, pp. 353–364. Warsaw (1988)

  37. Płoski A.: On the maximal polar quotient of an analytic plane curve. Kodai Math. J. 24(1), 120–133 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  38. Rodak T., Spodzieja S.: Effective formulas for the Łojasiewicz exponent at infinity. J. Pure Appl. Algebra 213, 1816–1822 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  39. So’n P.T.: On the effective computation of Łojasiewicz exponents via Newton polyhedra. Period. Math. Hungar. 54(2), 201–213 (2007)

    MathSciNet  Google Scholar 

  40. Spodzieja S.: Multiplicity and the Łojasiewicz exponent. Ann. Polon. Math. 73, 257–267 (2000)

    MathSciNet  MATH  Google Scholar 

  41. Spodzieja S.: The Łojasiewicz exponent of subanalytic sets. Ann. Polon. Math. 87, 247–263 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  42. Teissier B.: Variétés polaires. I. Invariants polaires des singularités d’hypersurfaces. Invent. Math. 40(3), 267–292 (1977)

    Article  MathSciNet  MATH  Google Scholar 

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Correspondence to Stanisław Spodzieja.

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Rodak, T., Spodzieja, S. Effective formulas for the local Łojasiewicz exponent. Math. Z. 268, 37–44 (2011). https://doi.org/10.1007/s00209-009-0659-8

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