Abstract
We give an expression for the Łojasiewicz exponent of a wide class of n-tuples of ideals (I 1,…,I n ) in \(\mathcal {O}_{n}\) using the information given by a fixed Newton filtration. In order to obtain this expression we consider a reformulation of Łojasiewicz exponents in terms of Rees mixed multiplicities. As a consequence, we obtain a wide class of semi-weighted homogeneous functions f:(ℂn,0)→(ℂ,0) for which the Łojasiewicz of its gradient map ∇f attains the maximum possible value.
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Arnold, V.I., Gusein-Zade, S.M., Varchenko, A.N.: Singularities of Differentiable Maps. Vol. I: The Classification of Critical Points, Caustics and Wave Fronts. Monogr. Math., vol. 82. Birkhäuser, Boston (1985)
Bivià-Ausina, C.: Łojasiewicz exponents, the integral closure of ideals and Newton polyhedra. J. Math. Soc. Jpn. 55(3), 655–668 (2003)
Bivià-Ausina, C.: Jacobian ideals and the Newton non-degeneracy condition. Proc. Edinb. Math. Soc. 48, 21–36 (2005)
Bivià-Ausina, C.: Joint reductions of monomial ideals and multiplicity of complex analytic maps. Math. Res. Lett. 15(2), 389–407 (2008)
Bivià-Ausina, C.: Local Łojasiewicz exponents, Milnor numbers and mixed multiplicities of ideals. Math. Z. 262(2), 389–409 (2009)
Bivià-Ausina, C., Encinas, S.: Łojasiewicz exponents and resolution of singularities. Arch. Math. 93(3), 225–234 (2009)
Bivià-Ausina, C., Encinas, S.: The Łojasiewicz exponent of a set of weighted homogeneous ideals. J. Pure Appl. Algebra 215(4), 578–588 (2011)
Bivià-Ausina, C., Fukui, T., Saia, M.J.: Newton graded algebras and the codimension of non-degenerate ideals. Math. Proc. Camb. Philos. Soc. 133, 55–75 (2002)
Damon, J., Gaffney, T.: Topological triviality of deformations of functions and Newton filtrations. Invent. Math. 72, 335–358 (1983)
Fukui, T.: Łojasiewicz type inequalities and Newton diagrams. Proc. Am. Math. Soc. 114(4), 1169–1183 (1991)
Greuel, G.-M., Pfister, G., Schönemann, H.: Singular 3.0. A computer algebra system for polynomial computations. Centre for Computer Algebra, University of Kaiserslautern (2005). http://www.singular.uni-kl.de
Herrmann, M., Ikeda, S., Orbanz, U.: Equimultiplicity anb Blowing Up. An Algebraic Study with an Appendix by B. Moonen. Springer, Berlin (1988)
Hochster, M.: Rings of invariants of tori, Cohen-Macaulay rings generated by monomials, and polytopes. Ann. Math. 96, 318–337 (1972)
Huneke, C., Swanson, I.: Integral Closure of Ideals, Rings, and Modules. London Math. Soc. Lecture Note Series, vol. 336. Cambridge University Press, Cambridge (2006)
Kouchnirenko, A.G.: Polyèdres de Newton et nombres de Milnor. Invent. Math. 32, 1–31 (1976)
Krasiński, T., Oleksik, G., Płoski, A.: The Łojasiewicz exponent of an isolated weighted homogeneous surface singularity. Proc. Am. Math. Soc. 137(10), 3387–3397 (2009)
Lejeune, M., Teissier, B.: Clôture intégrale des idéaux et equisingularité, with an appendix by J.J. Risler. Centre de Mathématiques, École Polytechnique (1974) and Ann. Fac. Sci. Toulouse Math. (6) 17(4), 781–859 (2008)
Lenarcik, A.: On the Łojasiewicz exponent of the gradient of a holomorphic function. In: Jakubczyk, B. (ed.) Singularities Symposium–Łojasiewicz 70. Banah Center Publications, vol. 44, pp. 149–166. Institute of Mathematics of the Polish Academy of Science, Warsaw (1998)
Łojasiewicz, S.: Sur le problème de la division. Studia Math. 18, 87–136 (1959)
Milnor, J., Orlik, P.: Isolated singularities defined by weighted homogeneous map polynomials. Topology 9, 385–393 (1970)
Nagata, M.: Note on a paper of Samuel concerning asymptotic properties of ideals. Mem. Coll. Sci., Univ. Kyoto, Ser. A: Math. 30, 165–175 (1957)
Oleksik, G.: The Łojasiewicz exponent of nondegenerate singularities. Univ. Iagel. Acta Math. 47, 301 (2009)
Parusiński, A.: Topological triviality of μ-constant deformations of type f(x)+tg(x). Bull. Lond. Math. Soc. 31(6), 686–692 (1999)
Płoski, A.: Multiplicity and the Łojasiewicz exponent. In: Singularities (Warsaw, 1985). Banach Center Publ., vol. 20, pp. 353–364. PWN, Warsaw (1988)
Płoski, A.: Sur l’exposant d’une application analytique II. Bull. Pol. Acad. Sci., Math. 33(3–4), 123–127 (1985)
Płoski, A.: Semicontinuity of the Łojasiewicz exponent. Univ. Iagel. Acta Math. 48, 103–110 (2010)
Saeki, O.: Topological invariance of weights for weighted homogeneous isolated singularities in ℂ3. Proc. Am. Math. Soc. 103(3), 905–909 (1988)
Saia, M.J.: The integral closure of ideals and the Newton filtration. J. Algebr. Geom. 5, 1–11 (1996)
Teissier, B.: Cycles évanescents, sections planes et conditions of Whitney. Singularités à Cargèse, Astérisque 7–8, 285–362 (1973)
Teissier, B.: Variétés polaires I. Invariants polaires des singularités d’hypersurfaces. Invent. Math. 40(3), 267–292 (1977)
Teissier, B.: Monomial ideals, binomial ideals, polynomial ideals. Math. Sci. Res. Inst. Publ. 51, 211–246 (2004)
Yau, S.S.-T.: Topological types and multiplicities of isolated quasihomogeneous surface singularities. Bull. Am. Math. Soc. 19, 447–454 (1988)
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The first author was partially supported by DGICYT Grant MTM2009-08933. The second author was partially supported by DGICYT Grant MTM2009-07291 and CCG08-UAM/ESP-3928.
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Bivià-Ausina, C., Encinas, S. Łojasiewicz exponent of families of ideals, Rees mixed multiplicities and Newton filtrations. Rev Mat Complut 26, 773–798 (2013). https://doi.org/10.1007/s13163-012-0104-0
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DOI: https://doi.org/10.1007/s13163-012-0104-0