Jelonek, Zbigniew; Lason, Michael (2018). Quantitative properties of the nonproperness set of a polynomial map. Manuscripta mathematica, 156(34), pp. 383397. SpringerVerlag 10.1007/s0022901709650

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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 nonproperness 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 nonproperness 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 BiałynickiBirula theorem: Let G be a real, nontrivial, 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: 
00252611 
Publisher: 
SpringerVerlag 
Language: 
English 
Submitter: 
Olivier Bernard Mila 
Date Deposited: 
24 Apr 2019 17:05 
Last Modified: 
02 Sep 2020 03:36 
Publisher DOI: 
10.1007/s0022901709650 
BORIS DOI: 
10.7892/boris.125496 
URI: 
https://boris.unibe.ch/id/eprint/125496 