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

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

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

We give an expression for the {\L}ojasiewicz exponent of a wide class of n-tuples of ideals $(I_1,..., I_n)$ 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 $(\mathbb{C}^n,0)\to (\mathbb{C},0)$ for which the {\L}ojasiewicz of its gradient map $\nabla f$ attains the maximum possible value.Comment: 25 pages. Updated with minor change

Integral closure and bounds for quotients of multiplicities of monomial ideals

[EN] Given a pair of monomial ideals $I$ and $J$ of finite colength of the ring of analytic function germs (\C^n,0)\to \C, we prove that some power of $I$ admits a reduction formed by homogeneous polynomials with respect to the Newton filtration induced by $J$ if and only if the quotient of multiplicities $e(I)/e(J)$ attains a suitable upper bound expressed in terms of the Newton polyhedra of $I$ and $J$. We also explore other connections between mixed multiplicities, Newton filtrations and the integral closure of ideals.The author was partially supported by DGICYT Grant MTM2015-64013-P.Bivià-Ausina, C. (2018). Integral closure and bounds for quotients of multiplicities of monomial ideals. Journal of Algebra. 501:233-254. https://doi.org/10.1016/j.jalgebra.2017.12.030S23325450

Invariants for bi-Lipschitz equivalence of ideals

[EN] We introduce the notion of bi-Lipschitz equivalence of ideals and derive numerical invariants for such equivalence. In particular, we show that the log canonical threshold of ideals is a bi-Lipschitz invariant. We apply our method to several deformations ft:,0,0 (. n). (.) and show that they are not bi-Lipschitz trivial, specially focusing on several known examples of non-m*-constant deformations.The first author was partially supported by DGICYT Grant MTM2015-64013-P.Bivià-Ausina, C.; Fukui, T. (2017). Invariants for bi-Lipschitz equivalence of ideals. The Quarterly Journal of Mathematics. 68(3):791-815. https://doi.org/10.1093/qmath/hax002S79181568

Mixed Lojasiewicz exponents and log canonical thresholds of ideals

We study the Lojasiewicz exponent and the log canonical threshold of ideals of O-n when restricted to generic subspaces of C-n of different dimensions. We obtain effective formulas of the resulting numbers for ideals with monomial integral closure. An inequality relating these numbers is also proven.The authors thank S. Ishii and T. Krasinski for helpful comments. Part of this work was developed during the stay of the first author at the Max Planck Institute for Mathematics (Bonn, Germany) in April 2011 and the Department of Mathematics of Saitama University (Japan) in March 2012. The first author wishes to thank these institutions for hospitality and financial support.Bivià-Ausina, C.; Fukui, T. (2016). Mixed Lojasiewicz exponents and log canonical thresholds of ideals. Journal of Pure and Applied Algebra. 220(1):223-245. doi:10.1016/j.jpaa.2015.06.007S223245220