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

- 137 Downloads

## 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.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.Białynicki-Birula, A.: On fixed point schemes of actions of multiplicative and additive groups. Topology
**12**, 99–103 (1973)MathSciNetCrossRefzbMATHGoogle Scholar - 3.Drużkowski, L.: An effective approach to Keller’s Jacobian conjecture. Math. Ann.
**264**(3), 303–313 (1983)MathSciNetCrossRefzbMATHGoogle Scholar - 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.Jelonek, Z.: The set of points at which the polynomial mapping is not proper. Ann. Polon. Math.
**58**, 259–266 (1993)MathSciNetCrossRefzbMATHGoogle Scholar - 6.Jelonek, Z.: Testing sets for properness of polynomial mappings. Math. Ann.
**315**, 1–35 (1999)MathSciNetCrossRefzbMATHGoogle Scholar - 7.Jelonek, Z.: Geometry of real polynomial mappings. Math. Z.
**239**, 321–333 (2002)MathSciNetCrossRefzbMATHGoogle Scholar - 8.Jelonek, Z.: On the Russell problem. J. Algebra
**324**(12), 3666–3676 (2010)MathSciNetCrossRefzbMATHGoogle Scholar - 9.Jelonek, Z., Kurdyka, K.: On asymptotic critical values of a complex polynomial. J. Reine Angew. Math.
**565**, 1–11 (2003)MathSciNetCrossRefzbMATHGoogle Scholar - 10.Jelonek, Z., Lasoń, M.: The set of fixed points of a unipotent group. J. Algebra
**322**, 2180–2185 (2009)MathSciNetCrossRefzbMATHGoogle Scholar - 11.Kollar, J.: Rational Curves on Algebraic Varieties. Springer, Berlin (1999)zbMATHGoogle Scholar
- 12.Shafarevich, I.: Basic Algebraic Geometry 1. Springer, Berlin (1994)CrossRefzbMATHGoogle Scholar
- 13.Stasica, A.: Geometry of the Jelonek set. J. Pure Appl. Algebra
**198**, 317–327 (2005)MathSciNetCrossRefzbMATHGoogle Scholar - 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.van den Essen, A.: Polynomial Automorphisms and the Jacobian Conjecture. Birkhauser Verlag, Basel (2000)CrossRefzbMATHGoogle Scholar

## Copyright information

**Open Access**This 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.