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

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.