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.

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