Extensions of regular mappings and the Łojasiewicz exponent at infinity

Let F:V→Cm be a regular mapping, where V⊂Cn is an algebraic set of positive dimension and m⩾n⩾2, and let L∞(F) be the Łojasiewicz exponent at infinity of F. We prove that F has a polynomial extension G:Cn→Cm such L∞(G)=L∞(F). Moreover, we give an estimate of the degree of the extension G. Additionally, we prove that if then for any β∈Q, β⩽L∞(F), the mapping F has a polynomial extension G with L∞(G)=β. We also give an estimate of the degree of this extension.