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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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  • [1] Thomas Bloom, On families of polynomials which approximate the pluricomplex Green function, Indiana Univ. Math. J. 50 (2001), no. 4, 1545-1566. MR 1889070 (2003a:32055), https://doi.org/10.1512/iumj.2001.50.1951
  • [2] Thomas Bloom and Jean-Paul Calvi, On multivariate minimal polynomials, Math. Proc. Cambridge Philos. Soc. 129 (2000), no. 3, 417-431. MR 1780496 (2001j:32036), https://doi.org/10.1017/S0305004100004606
  • [3] Philip J. Davis, Interpolation and approximation, Dover Publications, Inc., New York, 1975. Republication, with minor corrections, of the 1963 original, with a new preface and bibliography. MR 0380189 (52 #1089)
  • [4] Robert C. Gunning, Introduction to holomorphic functions of several variables. Vol. II, Local theory, The Wadsworth & Brooks/Cole Mathematics Series, Wadsworth & Brooks/Cole Advanced Books & Software, Monterey, CA, 1990. MR 1057177 (92b:32001b)
  • [5] Maciej Klimek, Pluripotential theory, London Mathematical Society Monographs. New Series, vol. 6, The Clarendon Press, Oxford University Press, New York, 1991. Oxford Science Publications. MR 1150978 (93h:32021)
  • [6] Ernest Michael, Continuous selections. I, Ann. of Math. (2) 63 (1956), 361-382. MR 0077107 (17,990e)
  • [7] Stéphanie Nivoche, The pluricomplex Green function, capacitative notions, and approximation problems in $ \mathbf {C}^n$, Indiana Univ. Math. J. 44 (1995), no. 2, 489-510. MR 1355409 (96g:32030), https://doi.org/10.1512/iumj.1995.44.1998
  • [8] Thomas Ransford, Potential theory in the complex plane, London Mathematical Society Student Texts, vol. 28, Cambridge University Press, Cambridge, 1995. MR 1334766 (96e:31001)
  • [9] V. I. Smirnov and N. A. Lebedev, Functions of a complex variable: Constructive theory, Translated from the Russian by Scripta Technica Ltd, The M.I.T. Press, Cambridge, Mass., 1968. MR 0229803 (37 #5369)
  • [10] M. Tsuji, Potential theory in modern function theory, Chelsea Publishing Co., New York, 1975. Reprinting of the 1959 original. MR 0414898 (54 #2990)

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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

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