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Local Łojasiewicz exponents, Milnor numbers and mixed multiplicities of ideals

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Abstract

Let \({g: (\mathbb{C}^n, 0)\to (\mathbb{C}^n, 0)}\) be a finite analytic map. We give an expression for the local Łojasiewicz exponent and for the multiplicity of g when the component functions of g satisfy certain condition with respect to a set of n monomial ideals I 1,..., I n . We give an effective method to compute Łojasiewicz exponents based on the computation of mixed multiplicities. As a consequence of our study, we give a numerical characterization of a class of functions that includes semi-weighted homogenous functions and Newton non-degenerate functions.

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Authors and Affiliations

  1. Institut Universitari de Matemàtica Pura i Aplicada, Universitat Politècnica de València, Camí de Vera s/n, 46022, València, Spain

    Carles Bivià-Ausina

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  1. Carles Bivià-Ausina
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Correspondence to Carles Bivià-Ausina.

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Work supported by DGICYT Grant MTM2006-06027.

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Bivià-Ausina, C. Local Łojasiewicz exponents, Milnor numbers and mixed multiplicities of ideals. Math. Z. 262, 389–409 (2009). https://doi.org/10.1007/s00209-008-0380-z

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