Polynomial-rational bijections of 𝑅ⁿ

Abstract

It is shown in this note that every invertible polynomial transformation of R n {{\mathbf {R}}^n} of degree two has a rational inverse defined on the whole space R n {{\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.