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Advances in Geometry

Managing Editor: Grundhöfer, Theo / Joswig, Michael

Editorial Board: Bamberg, John / Bannai, Eiichi / Cavalieri, Renzo / Coskun, Izzet / Duzaar, Frank / Eberlein, Patrick / Gentili, Graziano / Henk, Martin / Kantor, William M. / Korchmaros, Gabor / Kreuzer, Alexander / Lagarias, Jeffrey C. / Leistner, Thomas / Löwen, Rainer / Ono, Kaoru / Ratcliffe, John G. / Scharlau, Rudolf / Scheiderer, Claus / Van Maldeghem, Hendrik / Weintraub, Steven H. / Weiss, Richard

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Volume 14, Issue 4


Bifurcation values and stability of algebras of bounded polynomials

Krzysztof Kurdyka
  • Laboratoire de Mathematiques (LAMA) Université de Savoie, UMR-5127 de CNRS, 73-376 Le Bourget-du-Lac cedex, France
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/ Maria Michalska / Stanisław Spodzieja
  • This research was partially supported by NCN (Poland), grant 2012/07/B/ST1/03293 and by ANR (France), grant STAAVF.
  • Faculty of Mathematics and Computer Science, University of Łód´z, S. Banacha 22, 90-238 Łód´z, Poland
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Published Online: 2014-10-08 | DOI: https://doi.org/10.1515/advgeom-2014-0006


We show a tight relation between bifurcation values of a fixed polynomial f ∈ R[X; Y] and the family of polynomials bounded on a set { x ∈ R2 | f(x) ≤ c}. If g ∈ R[X; Y] is bounded on {x ∈ R2 | f(x) ≤ c}, then it is bounded on the bigger set {x ∈ R2 | f(x) ≤ c̃} with c < c̃ provided that the interval [c; c̃] does not contain any complex bifurcation value at infinity of f.

Keywords: Bounded polynomials; bifurcation values; generalised critical values.

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Published Online: 2014-10-08

Published in Print: 2014-10-01

Citation Information: Advances in Geometry, Volume 14, Issue 4, Pages 631–646, ISSN (Online) 1615-7168, ISSN (Print) 1615-715X, DOI: https://doi.org/10.1515/advgeom-2014-0006.

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