A note on the local Lipschitz triviality of values of complex polynomial functions
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Mathematische Zeitschrift
A note on the local Lipschitz triviality of values of complex polynomial functions Alexandre Fernandes1 · Vincent Grandjean1 · Humberto Soares2 Received: 18 July 2019 / Accepted: 14 November 2019 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract We address here the question of the bi-Lipschitz local triviality of a complex polynomial function over a complex value. Our main result states that a non constant complex polynomial admits a locally bi-Lipschitz trivial value if and only if it is a polynomial in one complex variable. Keywords Complex polynomials · Regular value · Bi-Lipschitz trivialization Mathematics Subject Classification 14B05 · 32S15
1 Introduction One of the results of Thom’s famous Ensembles et Morphismes Stratifiés [5] is that, given any complex polynomial function f : Cn → C, there exists a smallest finite subset B( f ) of values, called the bifurcation set of f , such that the space f −1 (C \ B( f )) is a smooth fiber bundle over C \ B( f ) with model fiber f −1 (a) for any value a taken outside B( f ). The bifurcation locus B( f ) always contains the critical values of f , but may contain also regular values. In particular for any value a not in B( f ), we always find a neighbourhood U of a in
The authors are very grateful to the anonymous referee for helpful remarks and comments. A. Fernandes was partially supported by FUNCAP/CAPES/CNPq grant 304221/2017-1; V. Grandjean was partially supported by FUNCAP/CAPES/CNPq grant 305614/2015-0; C.H. Soares was partially supported by CNPq grant 113058/2016-0.
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Alexandre Fernandes [email protected] Vincent Grandjean [email protected] Humberto Soares [email protected]
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Departamento de Matemática, Universidade Federal do Ceará (UFC), Campus do Pici, Bloco 914, Cep. 60455-760 Fortaleza-Ce, Brasil
2
Departamento de Matemática, Universidade Federal do Piaui (UFPi), Campus Universitário Ministro Petrônio Portella, Cep. 64049-550 Teresina-Pi, Brasil
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C such that f −1 (U ) is diffeomorphic, as a fiber bundle, to the trivial bundle f −1 (a) × U . A value a at which this local triviality condition of the function f is satisfied will be called typical value of the function. On the other hand, given an algebraic family {St }t∈P of complex algebraic sets of Cn or PCn , many results have been produced in the last fifty years to guarantee the local topological constancy of St in a neighbourhood of a given parameter. Most often it is controlled by the regularity of a tailored stratification of the parameter space. Any subset S of a metric space (E, d) is a metric space in its own when equipped with the ambient metric: the distance between any pair of points of S is taken in the ambient space (E, d). Two subsets S, S of a metric space (E, d) are bi-Lipschitz equivalent or have the same bi-Lipschitz type, if S, S are bi-Lipschitz homeomorphic metric spaces. As a consequence of Mostowski’s result about the existence of Lipschitz stratification of affine complex algebraic sets [4],
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