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manuscripta mathematica

, Volume 156, Issue 3–4, pp 383–397 | Cite as

Quantitative properties of the non-properness set of a polynomial map

  • Zbigniew JelonekEmail author
  • Michał Lasoń
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Abstract

Let f be a generically finite polynomial map \(f: \mathbb {C}^n\rightarrow \mathbb {C}^m\) of algebraic degree d. Motivated by the study of the Jacobian Conjecture, we prove that the set \(S_f\) of non-properness of f is covered by parametric curves of degree at most \(d-1\). This bound is best possible. Moreover, we prove that if \(X\subset \mathbb {R}^n\) is a closed algebraic set covered by parametric curves, and \(f: X\rightarrow \mathbb {R}^m\) is a generically finite polynomial map, then the set \(S_f\) of non-properness of f is also covered by parametric curves. Moreover, if X is covered by parametric curves of degree at most \(d_1\), and the map f has degree \(d_2\), then the set \(S_f\) is covered by parametric curves of degree at most \(2d_1d_2\). As an application of this result we show a real version of the Białynicki-Birula theorem: Let G be a real, non-trivial, connected, unipotent group which acts effectively and polynomially on a connected smooth algebraic variety \(X\subset \mathbb {R}^n\). Then the set Fix(G) of fixed points has no isolated points.

Mathematics Subject Classification

14R25 14P10 14R99 

References

  1. 1.
    Bass, H., Connell, E., Wright, D.: The Jacobian conjecture: reduction of degree and formal expansion of the inverse. Bull. Am. Math. Soc. (N.S.) 7(2), 287–330 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Białynicki-Birula, A.: On fixed point schemes of actions of multiplicative and additive groups. Topology 12, 99–103 (1973)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Drużkowski, L.: An effective approach to Keller’s Jacobian conjecture. Math. Ann. 264(3), 303–313 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Hà, H.V., Pham, T.S.: Representations of positive polynomials and optimization on noncompact semialgebraic sets. SIAM J. Optim. 20, 3082–3103 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Jelonek, Z.: The set of points at which the polynomial mapping is not proper. Ann. Polon. Math. 58, 259–266 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Jelonek, Z.: Testing sets for properness of polynomial mappings. Math. Ann. 315, 1–35 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Jelonek, Z.: Geometry of real polynomial mappings. Math. Z. 239, 321–333 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Jelonek, Z.: On the Russell problem. J. Algebra 324(12), 3666–3676 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Jelonek, Z., Kurdyka, K.: On asymptotic critical values of a complex polynomial. J. Reine Angew. Math. 565, 1–11 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Jelonek, Z., Lasoń, M.: The set of fixed points of a unipotent group. J. Algebra 322, 2180–2185 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Kollar, J.: Rational Curves on Algebraic Varieties. Springer, Berlin (1999)zbMATHGoogle Scholar
  12. 12.
    Shafarevich, I.: Basic Algebraic Geometry 1. Springer, Berlin (1994)CrossRefzbMATHGoogle Scholar
  13. 13.
    Stasica, A.: Geometry of the Jelonek set. J. Pure Appl. Algebra 198, 317–327 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    van den Dries, L., McKenna, K.: Surjective polynomial maps, and a remark on the Jacobian problem. Manuscr. Math. 67(1), 1–15 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    van den Essen, A.: Polynomial Automorphisms and the Jacobian Conjecture. Birkhauser Verlag, Basel (2000)CrossRefzbMATHGoogle Scholar

Copyright information

© The Author(s) 2017

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Institute of Mathematics of the Polish Academy of SciencesWarszawaPoland

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