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

Jelonek, Zbigniew; Lason, Michael (2018). Quantitative properties of the non-properness set of a polynomial map. Manuscripta mathematica, 156(3-4), pp. 383-397. Springer-Verlag 10.1007/s00229-017-0965-0

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Let f be a generically finite polynomial map f : ℂⁿ → ℂᵐ of algebraic degree d. Motivated by the study of the Jacobian Conjecture, we prove that the set Sf 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⊂ℝⁿ is a closed algebraic set covered by parametric curves, and f : X→ℝᵐ is a generically finite polynomial map, then the set Sf of non-properness of f is also covered by parametric curves. Moreover, if X is covered by parametric curves of degree at most d₁, and the map f has degree d₂, then the set Sf is covered by parametric curves of degree at most 2d₁d₂. 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⊂ℝⁿ. Then the set Fix(G) of fixed points has no isolated points.

Item Type:

Journal Article (Original Article)

Division/Institute:

08 Faculty of Science > Department of Mathematics and Statistics > Institute of Mathematics

UniBE Contributor:

Lason, Michael

Subjects:

500 Science > 510 Mathematics

ISSN:

0025-2611

Publisher:

Springer-Verlag

Language:

English

Submitter:

Olivier Bernard Mila

Date Deposited:

24 Apr 2019 17:05

Last Modified:

24 Oct 2019 11:35

Publisher DOI:

10.1007/s00229-017-0965-0

BORIS DOI:

10.7892/boris.125496

URI:

https://boris.unibe.ch/id/eprint/125496

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