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A sequence of polynomials with optimal condition number


Authors: Carlos Beltrán, Ujué Etayo, Jordi Marzo and Joaquim Ortega-Cerdà
Journal: J. Amer. Math. Soc. 34 (2021), 219-244
MSC (2010): Primary 65Y20
DOI: https://doi.org/10.1090/jams/956
Published electronically: December 8, 2020
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Abstract: We find an explicit sequence of univariate polynomials of arbitrary degree with optimal condition number. This solves a problem posed by Michael Shub and Stephen Smale in 1993.


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Additional Information

Carlos Beltrán
Affiliation: Departamento de Matematicas Estadistica y Computacion, University of Cantabria, 39005 Santander, Spain
Email: carlos.beltran@unican.es

Ujué Etayo
Affiliation: Institut für Analyse und Zahlentheorie, Technische Universität Graz , Kopernikusgasse 24/II 8010 Graz, Austria
Email: etayo@math.tugraz.at

Jordi Marzo
Affiliation: Departament de Matemàtiques i Informàtica, Universitat de Barcelona, Gran Via 585, 08007 Barcelona, Spain

Joaquim Ortega-Cerdà
Affiliation: Departament de Matemàtiques i Informàtica, Universitat de Barcelona, Gran Via 585, 08007 Barcelona, Spain

DOI: https://doi.org/10.1090/jams/956
Received by editor(s): March 5, 2019
Received by editor(s) in revised form: March 19, 2020, and March 30, 2020
Published electronically: December 8, 2020
Additional Notes: The first and second authors were partially supported by Ministerio de Economía y Competitividad, Gobierno de España, through grants MTM2017-83816-P and MTM2017-90682-REDT, and by the Banco de Santander and Universidad de Cantabria grant 21.SI01.64658.
The second author was also supported by the Austrian Science Fund FWF project F5503 (part of the Special Research Program (SFB) Quasi-Monte Carlo Methods: Theory and Applications).
The third and fourth authors have been partially supported by grant MTM2017-83499-P by the Ministerio de Economía y Competitividad, Gobierno de España and by the Generalitat de Catalunya (project 2017 SGR 358).
Article copyright: © Copyright 2020 American Mathematical Society