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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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On bifurcation points of a complex polynomial
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by Zbigniew Jelonek PDF
Proc. Amer. Math. Soc. 131 (2003), 1361-1367 Request permission

Abstract:

Let $f: \mathbb {C}^n \to \mathbb {C}$ be a polynomial of degree $d$. Assume that the set $\tilde {K}_\infty (f)=\{ y \in \mathbb {C} :$ there is a sequence $x_l\rightarrow \infty$ s.t. $f(x_l)\rightarrow y$ and $\Vert d f(x_l)\Vert \rightarrow 0\}$ is finite. We prove that the set $\tilde {K} (f)= K_0(f)\cup \tilde {K}_\infty (f)$ of generalized critical values of $f$ (hence in particular the set of bifurcation points of $f$) has at most $(d-1)^n$ points. Moreover, $\#\tilde {K}_\infty (f)\le (d-1)^{n-1}.$ We also compute the set $\tilde {K} (f)$ effectively.
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Additional Information
  • Zbigniew Jelonek
  • Affiliation: Instytut Matematyczny, Polska Akademia Nauk, Św. Tomasza 30, 31-027 Kraków, Poland
  • Address at time of publication: Max-Planck-Institut für Mathematik, Vivatsgasse 7, 53111 Bonn, Germany
  • Email: najelone@cyf-kr.edu.pl
  • Received by editor(s): April 17, 2001
  • Received by editor(s) in revised form: January 8, 2002
  • Published electronically: December 16, 2002
  • Additional Notes: The author was partially supported by KBN grant number 2P03A01722
  • Communicated by: Michael Stillman
  • © Copyright 2002 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 131 (2003), 1361-1367
  • MSC (2000): Primary 14D06, 14Q20, 14R25
  • DOI: https://doi.org/10.1090/S0002-9939-02-06822-3
  • MathSciNet review: 1949865