Polynomial-rational bijections of $\textbf {R}^ n$

Abstract:

It is shown in this note that every invertible polynomial transformation of ${{\mathbf {R}}^n}$ of degree two has a rational inverse defined on the whole space ${{\mathbf {R}}^n}$. The same is true for polynomial transformations of higher degrees, satisfying some differential condition which is a real analogue of Jagžev’s condition considered in [3, 4, and 6]. The proofs of these statements are based on the Bialynicki-Birula and Rosenlicht surjectivity theorem [2] and on standard properties of complex dominant polynomial mappings.