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On Zariski’s multiplicity problem at infinity


Author: J. Edson Sampaio
Journal: Proc. Amer. Math. Soc. 147 (2019), 1367-1376
MSC (2010): Primary 14B05, 32S50, 58K30; Secondary 58K20
DOI: https://doi.org/10.1090/proc/14351
Published electronically: January 8, 2019
MathSciNet review: 3910404
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Abstract: We address a metric version of Zariski’s multiplicity conjecture at infinity that says that two complex algebraic affine sets which are bi-Lipschitz homeomorphic at infinity must have the same degree. More specifically, we prove that the degree is a bi-Lipschitz invariant at infinity when the bi-Lipschitz homeomorphism has Lipschitz constants close to 1. In particular, we have that a family of complex algebraic sets bi-Lipschitz equisingular at infinity has constant degree. Moreover, we prove that if two polynomials are weakly rugose equivalent at infinity, then they have the same degree. In particular, we obtain that if two polynomials are rugose equivalent at infinity or bi-Lipschitz contact equivalent at infinity or bi-Lipschitz right-left equivalent at infinity, then they have the same degree.


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Additional Information

J. Edson Sampaio
Affiliation: Departamento de Matemática, Universidade Federal do Ceará, Rua Campus do Pici, s/n, Bloco 914, Pici, 60440-900, Fortaleza-CE, Brazil – and – BCAM - Basque Center for Applied Mathematics, Mazarredo, 14 E48009 Bilbao, Basque Country, Spain
MR Author ID: 1144437
Email: edsonsampaio@mat.ufc.br,esampaio@bcamath.org

Keywords: Bi-Lipschitz contact at infinity, degree, Zariski’s conjecture
Received by editor(s): September 7, 2017
Received by editor(s) in revised form: January 12, 2018, and March 27, 2018
Published electronically: January 8, 2019
Additional Notes: The author was partially supported by the ERCEA 615655 NMST Consolidator Grant and also by the Basque government through the BERC 2014–2017 program and by Spanish Ministry of Economy and Competitiveness MINECO: BCAM Severo Ochoa excellence accreditation SEV-2013-0323.
Communicated by: Michael Wolf
Article copyright: © Copyright 2019 American Mathematical Society