On the Properness of Polynomial Selfmaps

This paper studies the properness of polynomial selfmaps F:Rᵐ → Rᵐ F : R m → R m or F:Cᵐ→ Cᵐ F : C m → C m . By results of Druzkowski, this question is reduced to that of properness of maps of the following special form, which we call identity plus linear powers. For two vectors x,y∈ Rᵐ x , y ∈ R m , we use the notation x *y =(x₁y₁,… ,xₘyₘ) x ∗ y = ( x 1 y 1 , … , x m y m ) , and if $$x=y$$ x = y we also use the notation x²=x*x x 2 = x ∗ x and by induction xᵏ=x*(xᵏ⁻¹) x k = x ∗ ( x k - 1 ) . We use ,\, ⟨ , ⟩ for the usual inner product on Rᵐ R m . For A an m× m m × m <