14 research outputs found

    The {\L}ojasiewicz exponent of a set of weighted homogeneous ideals

    Get PDF
    We give an expression for the {\L}ojasiewicz exponent of a set of ideals which are pieces of a weighted homogeneous filtration. We also study the application of this formula to the computation of the {\L}ojasiewicz exponent of the gradient of a semi-weighted homogeneous function (\C^n,0)\to (\C,0) with an isolated singularity at the origin.Comment: 15 page

    Łojasiewicz exponents and resolution of singularities

    Get PDF

    Lojasiewicz exponent of families of ideals, Rees mixed multiplicities and Newton filtrations

    Full text link
    We give an expression for the {\L}ojasiewicz exponent of a wide class of n-tuples of ideals (I1,...,In)(I_1,..., I_n) in \O_n using the information given by a fixed Newton filtration. In order to obtain this expression we consider a reformulation of {\L}ojasiewicz exponents in terms of Rees mixed multiplicities. As a consequence, we obtain a wide class of semi-weighted homogeneous functions (Cn,0)(C,0)(\mathbb{C}^n,0)\to (\mathbb{C},0) for which the {\L}ojasiewicz of its gradient map f\nabla f attains the maximum possible value.Comment: 25 pages. Updated with minor change

    Global multiplicity, special closure and non-degeneracy of gradient maps

    Get PDF
    [EN] Given a polynomial map F : Cn --> Cp with finite zero set, p (sic)n, we introduce the notion of global multiplicity m(F) associated to F, which is analogous to the multiplicity of ideals in Noetherian local rings. This notion allows to characterize numerically the Newton non-degeneracy at infinity of F. This fact motivates us to study a combinatorial inequality concerning the normalized volume of global Newton polyhedra and to characterize the corresponding equality using special closures. We also study the Newton non-degeneracy at infinity of gradient maps.CRUE-CSIC agreement with Springer NatureBivià-Ausina, C.; Huarcaya, JAC. (2023). Global multiplicity, special closure and non-degeneracy of gradient maps. Revista de la Real Academia de Ciencias Exactas Físicas y Naturales Serie A Matemáticas. 117(3). https://doi.org/10.1007/s13398-023-01452-4117

    Polynomial maps with maximal multiplicity and the special closure

    Full text link
    [EN] In this article we characterize the polynomialmaps F : Cn. Cn for which F -1(0) is finite and their multiplicity mu(F) is equal to n! Vn( +(F)), where +(F) is the global Newton polyhedron of F. As an application, we derive a characterization of those polynomial maps whose multiplicity is maximal with respect to a fixed Newton filtration.Carles Bivia-Ausina was partially supported by DGICYT Grant MTM2015-64013-P. Jorge A. C. Huarcaya was partially supported by FAPESP-BEPE 2012/22365-8.Bivià-Ausina, C.; Huarcaya, JAC. (2019). Polynomial maps with maximal multiplicity and the special closure. Monatshefte für Mathematik. 188(3):413-429. https://doi.org/10.1007/s00605-018-1204-9S4134291883Artal Bartolo, E., Luengo, I.: On the topology of a generic fibre of a polynomial function. Commun. Algebra 28(4), 1767–1787 (2000)Artal Bartolo, E., Luengo, I., Melle-Hernández, A.: Milnor number at infinity, topology and Newton boundary of a polynomial function. Math. Z. 233(4), 679–696 (2000)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)Bivià-Ausina, C., Huarcaya, J.A.C.: The special closure of polynomial maps and global non-degeneracy, Mediterr. J. Math. 14(2), Art. 71 (2017)Bivià-Ausina, C., Huarcaya, J.A.C.: Growth at infinity and index of polynomial maps. J. Math. Anal. Appl. 422, 1414–1433 (2015)Broughton, S.A.: Milnor numbers and the topology of polynomial hypersurfaces. Invent. Math. 92(2), 217–241 (1988)Cima, A., Gasull, A., Mañosas, F.: Injectivity of polynomial local homeomorphisms of Rn\mathbb{R}^n. Nonlinear Anal. 26(4), 877–885 (1996)Cox, D., Little, J., O’Shea, D.: Using Algebraic Geometry, Graduate Texts in Mathematics, vol. 185, 2nd edn. Springer, Berlin (2005)Cygan, E., Krasiński, T., Tworzewski, P.: Separation of algebraic sets and the Łojasiewicz exponent of polynomial mappings. Invent. Math. 136(1), 75–87 (1999)Furuya, M., Tomari, M.: A characterization of semi-quasihomogeneous functions in terms of the Milnor number. Proc. Am. Math. Soc. 132(7), 1885–1889 (2004)Grothendieck, A.: Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. III. Inst. Hautes Études Sci. Publ. Math 28, 255 (1966)Hà Huy, V., Zaharia, A.: Families of polynomials with total Milnor number constant. Math. Ann. 304(3), 481–488 (1996)Hochster, M.: Rings of invariants of tori, Cohen-Macaulay rings generated by monomials, and polytopes. Ann. Math. (2) 96, 318–337 (1972)Huneke, C., Swanson, I.: Integral Closure of Ideals, Rings, and Modules. London Math. Soc. Lecture Note Series 336. Cambridge University Press, Cambridge (2006)Kouchnirenko, A.G.: Polyèdres de Newton et nombres de Milnor. Invent. Math. 32, 1–31 (1976)Li, T.Y., Wang, X.: The BKK root count in Cn\mathbb{C}^n. Math. Comput. 65(216), 1477–1484 (1996)Matsumura, H.: Commutative Ring Theory. Cambridge Studies in Advanced Mathematics 8, Cambridge University Press, Cambridge (1986)Rojas, J.M.: A convex geometric approach to counting the roots of a polynomial system. Theor. Comput. Sci. 133(1), 105–140 (1994)Saia, M.J.: Pre-weighted homogeneous map germs-finite determinacy and topological triviality. Nagoya Math. J. 151, 209–220 (1998)Vasconcelos, W.: Integral Closure. Rees Algebras, Multiplicities, Algorithms. Springer Monographs in Mathematics. Springer, Berlin (2005

    The special closure of polynomial maps and global non-degeneracy

    Full text link
    [EN] Let F : C-n -> C-n be a polynomial map such that F-1 (0) is finite. We analyze the connections between the multiplicity of F, the Newton polyhedron of F and the set of special monomials with respect to F, which is a notion motivated by the integral closure of ideals in the ring of analytic function germs (C-n, 0) -> C. In particular, we characterize the polynomial maps whose set of special monomials is maximal.The first author was partially supported by DGICYT Grant MTM2015-64013-P. The second author was partially supported by FAPESP-BEPE 2012/22365-8.Bivià-Ausina, C.; Huarcaya, JAC. (2017). The special closure of polynomial maps and global non-degeneracy. Mediterranean Journal of Mathematics. 14(2):1-21. https://doi.org/10.1007/s00009-017-0879-9S121142Arnol’d, V.I., Gusein-Zade, S.M., Varchenko, A.N.: Singularities of Differentiable Maps. Vol. I. The Classification of Critical Points, Caustics and Wave Fronts. Monographs in Mathematics. Birkhäuser, Boston (1985)Bivià-Ausina, C.: Łojasiewicz exponents, the integral closure of ideals and Newton polyhedrons. J. Math. Soc. Jpn. 55(3), 655–668 (2003)Bivià-Ausina, C.: Non-degenerate ideals in formal power series rings. Rocky Mt. J. Math. 34(2), 495–511 (2004)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)Bivià-Ausina, C., Huarcaya, J.A.C.: Growth at infinity and index of polynomial maps. J. Math. Anal. Appl. 422, 1414–1433 (2015)Broughton, S.A.: Milnor numbers and the topology of polynomial hypersurfaces. Invent. Math. 92(2), 217–241 (1988)Cox, D., Little, J., O’Shea, D.: Using Algebraic Geometry. Graduate Texts in Mathematics, vol. 185, 2nd edn. Springer, Berlin (2005)D’Angelo, J.P.: Several Complex Variables and the Geometry of Real Hypersurfaces. Studies in Advanced Mathematics. CRC Press, Boca Raton, FL (1993)Ewald, G.: Combinatorial Convexity and Algebraic Geometry. Graduate Texts in Mathematics. Springer, Berlin (1996)Gaffney, T.: Integral closure of modules and Whitney equisingularity. Invent. Math. 107, 301–322 (1992)Greuel, G.-M., Pfister, G.: A Singular Introduction to Commutative Algebra, 2nd edn. Springer, Berlin (2008)Herrmann, M., Ikeda, S., Orbanz, U.: Equimultiplicity and Blowing Up. An Algebraic Study. With An Appendix by B. Moonen. Springer, Berlin (1988)Kouchnirenko, A.G.: Polyèdres de Newton et nombres de Milnor. Invent. Math. 32, 1–31 (1976)Krasiński, T.: On the Łojasiewicz exponent at infinity of polynomial mappings. Acta Math. Vietnam 32(2–3), 189–203 (2007)Huneke, C., Swanson, I.: Integral Closure of Ideals, Rings, and Modules. London Mathematical Society Lecture Note Series, vol. 336. Cambridge University Press, Cambridge (2006)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)Li, T.Y., Wang, X.: The BKK root count in Cn{\mathbb{C}}^{n} C n . Math. Comput. 65(216), 1477–1484 (1996)Lloyd, N.G.: Degree Theory, Cambridge Tracts in Mathematics, vol. 73. Cambridge University Press, Cambridge (1978)Outerelo, E., Ruiz, J.M.: Mapping Degree Theory, Graduate Studies in Mathematics, vol. 108. American Mathematical Society, Real Sociedad Matemática Española, Madrid, Providence, RI (2009)Palamodov, V.P.: The multiplicity of a holomorphic transformation. Funkcional. Anal. i Priložen 1(3), 54–65 (1967)Pham, T.S.: On the effective computation of Łojasiewicz exponents via Newton polyhedra. Period. Math. Hung. 54(2), 201–213 (2007)Saia, M.J.: The integral closure of ideals and the Newton filtration. J. Algebraic Geom. 5, 1–11 (1996)Vasconcelos, W.: Integral Closure. Rees Algebras, Multiplicities, Algorithms. Monographs in Mathematics. Springer, Berlin (2005)Yoshinaga, E.: Topologically principal part of analytic functions. Trans. Am. Math. Soc. 314(2), 803–814 (1989

    The Łojasiewicz exponent over a field of arbitrary characteristic

    Get PDF
    Let K be an algebraically closed field and let K((XQ)) denote the field of generalized series with coefficients in K. We propose definitions of the local Łojasiewicz exponent of F = ( f1, . . . , fm) ∈ K[[X, Y ]]m as well as of the Łojasiewicz exponent at infinity of F = ( f1, . . . , fm) ∈ K[X, Y ]m, which generalize the familiar case of K = C and F ∈ C{X, Y }m (resp. F ∈ C[X, Y ]m), see Cha˛dzy´nski and Krasi´nski (In: Singularities, 1988; In: Singularities, 1988; Ann Polon Math 67(3):297–301, 1997; Ann Polon Math 67(2):191–197, 1997), and prove some basic properties of such numbers. Namely, we show that in both cases the exponent is attained on a parametrization of a component of F (Theorems 6 and 7), thus being a rational number. To this end, we define the notion of the Łojasiewicz pseudoexponent of F ∈ (K((XQ))[Y ])m for which we give a description of all the generalized series that extract the pseudoexponent, in terms of their jets. In particular, we show that there exist only finitely many jets of generalized series giving the pseudoexponent of F (Theorem 5). The main tool in the proofs is the algebraic version of Newton’s Polygon Method. The results are illustrated with some explicit examples
    corecore