Testing sets for properness of polynomial mappings

Abstract.

In this paper we characterize testing sets for properness of polynomial mappings $f:\mathbb{C}^n\rightarrow\mathbb{C}^m$. We also study the set of points at which such mappings are not proper. As the first application we give a proof of the Complementary Conjecture of McKay and Wang (Conjecture 12 in [16]). The second application is an answer to the problem of Kraft- Russell about a characterization of $\mathbb{C}^n.$ The third application is (given in [13]) a solution of the Problem of Van den Essen and Shpilrain about endomorphism of the ring $\mathbb{C}[x_1,...,x_n].$ The fourth application is a theorem about extensions of affine varieties.