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Homogeneous minimal polynomials with prescribed interpolation conditions


Authors: Leokadia Białas-Cież and Jean-Paul Calvi
Journal: Trans. Amer. Math. Soc. 368 (2016), 8383-8402
MSC (2010): Primary 41A29, 41A05, 41A50, 41A63
DOI: https://doi.org/10.1090/tran/6604
Published electronically: January 6, 2016
MathSciNet review: 3551575
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Abstract: Given a compact set $ E$ in $ \mathbb{C}^{N+1}$, we consider the problem of finding a homogeneous polynomial of degree $ d$ on $ \mathbb{C}^{N+1}$ which deviates the least from zero on $ E$ with respect to the uniform norm among all those satisfying interpolation conditions of the form $ p(a)=f(a)$ where $ a$ belongs to a given finite subset of $ \mathbb{C}^{N+1}$ and $ f$ is any function on such set. We show that this formalism enables one to recover several types of minimal polynomials previously studied and to prove a general invariance property of such minimal polynomials under polynomial mappings.


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Additional Information

Leokadia Białas-Cież
Affiliation: Faculty of Mathematics and Computer Science, Jagiellonian University, Institute of Mathematics, 30-059 Kraków, Poland
Email: Leokadia.Bialas-Ciez@im.uj.edu.pl

Jean-Paul Calvi
Affiliation: Institut de Mathématiques, Université de Toulouse III and CNRS (UMR 5219), 31062, Toulouse Cedex 9, France
Email: jean-paul.calvi@math.univ-toulouse.fr

DOI: https://doi.org/10.1090/tran/6604
Keywords: Best polynomial approximation, multivariate polynomial interpolation, polynomial mappings
Received by editor(s): December 26, 2013
Received by editor(s) in revised form: October 4, 2014
Published electronically: January 6, 2016
Additional Notes: The work of the first author was partially supported by the grant NCN Harmonia UMO-2013/08/M/ST1/00986.
Article copyright: © Copyright 2016 American Mathematical Society