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Testing sets for properness of polynomial mappings. (English) Zbl 0946.14039
Fix $$m,n\in \mathbb{N}$$. A set $$S\subset \mathbb{C}^{m}$$ is a testing set for polynomial mappings $$f:\mathbb{C}^{n}\rightarrow \mathbb{C}^{m}$$ if and only if for every polynomial mapping $$f:\mathbb{C}^{n}\rightarrow \mathbb{C} ^{m}$$ if $$f|f^{-1}(S):f^{-1}(S)\rightarrow S$$ is proper then $$f$$ is proper. The first aim of the author is to study testing sets. He gives many characterizations and concrete classes of testing sets, even in the more general case of arbitrary affine varieties (instead of $$\mathbb{C}^{n}$$ and $$\mathbb{C}^{m}$$). The second aim of the author is to study the sets $$S_{f}\subset Y$$ where a regular mapping $$f:X\rightarrow Y$$ is not proper ($$y\in S_{f}$$ if and only if there is no neighbourhood $$U$$ of $$y$$ such that $$f^{-1}(\bar{U})$$ is compact). The author proves that if $$X,Y$$ are affine sets then $$S_{f}$$ is empty or a hypersurface in $$Y$$ and (when $$X$$ is dominated by $$\mathbb{C}^{n}$$) is always $$\mathbb{C}$$-uniruled.
As applications the author: 1. solves the complementary conjecture of McKay and Wang on characterization of $$\mathbb{C}$$-automorphisms of $$\mathbb{C}[z_{1},\dots,z_{n}],$$
2. gives a characterization of $$\mathbb{C}^{n}$$ (it is a particular case of a Russel-Kraft conjecture),
3. shows that any irreducible affine variety $$X$$ can be always extended to another irreducible affine variety $$Y$$ of the same dimension such that $$X\varsubsetneq Y$$.

##### MSC:
 14R15 Jacobian problem 14R10 Affine spaces (automorphisms, embeddings, exotic structures, cancellation problem)
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