54 research outputs found

    Images of analytic map germs, and singular fibrations

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    For a map germ GG with target (Cp,0)(\mathbb C^{p}, 0) or (Rp,0)(\mathbb R^{p}, 0) with p2p\ge 2, we address two phenomena which do not occur when p=1p=1: the image of GG may be not well-defined as a set germ, and a local fibration near the origin may not exist. We show how these two phenomena are related, and how they can be characterised.Comment: revised, to appear in Europ. J. Mat

    Local linear Morsifications

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    The number of Morse points in a Morsification determines the topology of the Milnor fibre of a holomorphic function germ ff with isolated singularity. If ff has an arbitrary singular locus, then this nice relation to the Milnor fibre disappears. We show that despite this loss, the numbers of stratified Morse singularities of a general linear Morsification are effectively computable in terms of topological invariants of ff

    Limits of tangents and minimality of complex links

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    AbstractWe show that the complex link of a large class of space germs (X,x0) is characterized by its “simplicity”, among the Milnor fibres of functions with isolated singularity on X. This amounts to the minimality of the Milnor number, whenever this number is defined. Such a phenomenon has been first pointed out in case (X,x0) is an isolated hypersurface singularity, by Teissier (Cycles évanescents, sections planes et conditions de Whitney, in: Singularités à Cargèse 1972, Asterisque, Nos. 7 et 8, Soc. Math. France, Paris, 1973, pp. 285–362)

    Bouquet decomposition of the Milnor fibre

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    AbstractWe consider the Milnor fibre of an isolated singularity ƒ:(X, 0) → (C, 0) on a reduced, Whitney stratified space germ (X, 0) and prove that it is homotopy equivalent to a bouquet of repeated suspensions of complex links of strata.This is a consequence of a “cell attaching” type result which we prove first by investigating the geometric monodromy given by Lê's “carrousel method”.Our results generalise the bouquet theorems of Milnor, Hamm, Lê, Siersma and the handlebody theorem of Lê and Perron

    Detecting bifurcation values at infinity of real polynomials

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    We present a new approach for estimating the set of bifurcation values at infinity. This yields a significant shrinking of the number of coefficients in the recent algorithm introduced by Jelonek and Kurdyka for reaching critical values at infinity by rational arcs

    On singular maps with local fibration

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    We discuss the most general condition under which a singular local tube fibration exists. We give an application to composition of map germs.Comment: to appear in: Revue Roumaine de Math\'ematiques Pures et Appliqu\'ees, 2023. A volume dedicated to the memory of Mihnea Coltoi

    Polar degree of singular projective hypersurfaces, and vanishing cycles of polynomials

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    We prove that the polar degree of an arbitrarily singular projective hypersurface VV can be decomposed as a sum of non-negative numbers which represent local vanishing cycles of two different types, namely related to "special points" of VV, or related to "non-generic polar curves". We use on the one hand a non-generic slicing strategy which extends Huh's method, and on the other hand the study of the vanishing cycles of affine functions. We derive lower bounds for the polar degree of any singular VV.Comment: 26

    Morse numbers of complex polynomials

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    To a polynomial function ff with arbitrary singularities we associate the number of Morse points in a general linear Morsification ft:=ftf_{t} := f - t\ell. We produce computable algebraic formulas in terms of invariants of ff for the numbers of stratwise Morse trajectories which abut, as t0t\to 0, to some point of XX or to some point at infinity.Comment: 16
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