Saddle point braids of braided fibrations and pseudo-fibrations
Let gt be a loop in the space of monic complex polynomials in one variable of fixed degree n. If the roots of gt are distinct for all t, they form a braid B1 on n strands. Likewise, if the critical points of gt are distinct for all t, they form a braid B2 on n-1 strands. In this paper we study the r...
| Autores: | , |
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| Tipo de recurso: | artículo |
| Estado: | Versión publicada |
| Fecha de publicación: | 2024 |
| País: | España |
| Institución: | Consejo Superior de Investigaciones Científicas (CSIC) |
| Repositorio: | DIGITAL.CSIC. Repositorio Institucional del CSIC |
| OAI Identifier: | oai:digital.csic.es:10261/381553 |
| Acceso en línea: | http://hdl.handle.net/10261/381553 https://www.scopus.com/inward/record.uri?eid=2-s2.0-85190696117&doi=10.1007%2fs40687-024-00446-x&partnerID=40&md5=9c80691e3947731c99f4c5eadafef7f1 |
| Access Level: | acceso abierto |
| Palabra clave: | 14J17 14P25 30C10 32S55 57K10 Braided open book Homogeneous braid Isolated singularity P-fibered braid Real algebraic link Saddle point braid |
| Sumario: | Let gt be a loop in the space of monic complex polynomials in one variable of fixed degree n. If the roots of gt are distinct for all t, they form a braid B1 on n strands. Likewise, if the critical points of gt are distinct for all t, they form a braid B2 on n-1 strands. In this paper we study the relationship between B1 and B2. Composing the polynomials gt with the argument map defines a pseudo-fibration map on the complement of the closure of B1 in C×S1, whose critical points lie on B2. We prove that for B1 a T-homogeneous braid and B2 the trivial braid this map can be taken to be a fibration map. In the case of homogeneous braids we present a visualization of this fact. Our work implies that for every pair of links L1 and L2 there is a mixed polynomial f:C2→C in complex variables u, v and the complex conjugate v¯ such that both f and the derivative fu have a weakly isolated singularity at the origin with L1 as the link of the singularity of f and L2 as a sublink of the link of the singularity of fu. © The Author(s) 2024. |
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