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4 Bulletin of the Polish Academy of Sciences. Mathematics

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Birational finite extensions of mappings from a smooth variety

Karaś M.

Bulletin of the Polish Academy of Sciences. Mathematics

2009

Vol. 57, no 2

117-120

We present an example of finite mappings of algebraic varieties ƒ : V → W, where V ⊂ kn, W ⊂ kn+1, and F : kn → kn+1 such that F¦v = ƒ and gdeg F = 1 < gdeg/ (gdeg h means the number of points in the generic fiber of h). Thus, in some sense, the result of this note improves our result in J. Pure Appl. Algebra 148 (2000) where it was shown that this phenomenon can occur when V ⊂ kn, W ⊂ km with m ≥ n + 2. In the case V, W ⊂ kn a similar example does not exist.

A note on geometric degree of finite extensions of mappings from a smooth variety

Karaś M.

Bulletin of the Polish Academy of Sciences. Mathematics

2008

Vol. 56, no 2

105-108

Let ƒ : V → W be a finite polynomial mapping of algebraic subsets V, W of k[sup]n and k[sup]m, respectively, with n ≤ m. Kwieciński [J. Pure Appl. Algebra 76 (1991)] proved that there exists a finite polynomial mapping F : k[sup]n → k[sup]m such that F| v = ƒ. In this note we prove that, if V, W ⊂ k[sup] are smooth of dimension k with 3k + 2 ≤ n, and ƒ : V → W is finite, dominated and dominated on every component, then there exists a finite polynomial mapping F : k[sup]n → [sup]n such that F|v = ƒ and gdeg F ≤ (gdegƒ)[sup]k+1. This improves earlier results of the author.

Finite extensions of mappings of finite sets

Karaś M.

Bulletin of the Polish Academy of Sciences. Mathematics

2002

Vol. 50, no 2

237-239

In this note we prove that for every finite sets V, W [is a subset of] [C^k] with k, #V, #W > 1 and for every surjective mapping f : V --> W there exists a finite mapping F : [C^k] --> [C^k] such that F\v = f, gdegF = gdegf and degF [is less than or equal to (#V - 1)^2].

On some characterization of proper polynomial mappings

Rodak T., Spodzieja S.

Bulletin of the Polish Academy of Sciences. Mathematics

2000

Vol. 48, no 2

157-164

It is well known that a proper, in the classical topology, polynomial mapping is closed in the Zariski topology. In the paper we prove that the inverse is true. Namely, any non-constant polynomial mapping from [C^n] into [C^m] which is closed in the Zariski topology is proper in the classical topology.