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1
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The set of points at which a polynomial map is not proper

100%
Jelonek Z.
Annales Polonici Mathematici
|
1993
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tom 58
|
nr 3
259-266
EN
We describe the set of points over which a dominant polynomial map $f=(f_1,...,f_n) : ℂ^n → ℂ^n$ is not a local analytic covering. We show that this set is either empty or it is a uniruled hypersurface of degree bounded by $(∏_{i=1}^n deg f_i - μ (f)) / (min_{i=1,...,n} deg f_i)$.
2
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The automorphism groups of Zariski open affine subsets of the affine plane

100%
Jelonek Z.
Annales Polonici Mathematici
|
1994-1995
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tom 60
|
nr 2
163-171
EN
We study some properties of the affine plane. First we describe the set of fixed points of a polynomial automorphism of ℂ². Next we classify completely so-called identity sets for polynomial automorphisms of ℂ². Finally, we show that a sufficiently general Zariski open affine subset of the affine plane has a finite group of automorphisms.
3
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A Few Remarks on the Mohan Kumar Theorem

100%
Jelonek Z.
Bulletin of the Polish Academy of Sciences. Mathematics
|
2010
|
tom 58
|
nr 3
197-200
EN
We give a simple geometric proof of Mohan Kumar's result about complete intersections.
4
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Local characterization of algebraic manifolds and characterization of components of the set $S_f$

100%
Jelonek Z.
Annales Polonici Mathematici
|
2000
|
tom 75
|
nr 1
7-13
EN
We show that every n-dimensional smooth algebraic variety X can be covered by Zariski open subsets $U_i$ which are isomorphic to closed smooth hypersurfaces in $ℂ^{n+1}$. As an application we show that forevery (pure) n-1-dimensional ℂ-uniruled variety $X ⊂ ℂ^m$ there is a generically-finite (even quasi-finite) polynomial mapping $f:ℂ^n → ℂ^m$ such that $X ⊂ S_f$. This gives (together with [3]) a full characterization of irreducible components of the set $S_f$ for generically-finite polynomial mappings $f:ℂ^n→ℂ^m$.
5
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On asymptotic critical values and the Rabier Theorem

100%
Jelonek Z.
Banach Center Publications
|
2004
|
tom 65
|
nr 1
125-133
EN
Let X ⊂ kⁿ be a smooth affine variety of dimension n-r and let $f = (f₁,..., f_m): X → k^m$ be a polynomial dominant mapping. It is well-known that the mapping f is a locally trivial fibration outside a small closed set B(f). It can be proved (using a general Fibration Theorem of Rabier) that the set B(f) is contained in the set K(f) of generalized critical values of f. In this note we study the Rabier function. We give a few equivalent expressions for this function, in particular we compare this function with the Kuo function and with the (generalized) Gaffney function. As a consequence we give a direct short proof of the fact that f is a locally trivial fibration outside the set K(f) (i.e., that B(f) ⊂ K(f)). This generalizes the previous results of the author for $X = k^r$ (see [2]).
6
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Manifolds with a unique embedding

100%
Jelonek Z.
Colloquium Mathematicum
|
2009
|
tom 117
|
nr 2
299-317
EN
We show that if X, Y are smooth, compact k-dimensional submanifolds of ℝⁿ and 2k+2 ≤ n, then each diffeomorphism ϕ: X → Y can be extended to a diffeomorphism Φ: ℝⁿ → ℝⁿ which is tame (to be defined in this paper). Moreover, if X, Y are real analytic manifolds and the mapping ϕ is analytic, then we can choose Φ to be also analytic. We extend this result to some interesting categories of closed (not necessarily compact) subsets of ℝⁿ, namely, to the category of Nash submanifolds (with Nash, real-analytic and smooth morphisms) and to the category of closed semi-algebraic subsets of ℝⁿ (with morphisms being semi-algebraic continuous mappings). In each case we assume that X, Y are k-dimensional and ϕ is an isomorphism, and under the same dimension restriction 2k+2 ≤ n we assert that there exists an extension Φ :ℝⁿ → ℝⁿ which is an isomorphism and it is tame. The same is true in the category of smooth algebraic subvarieties of ℂⁿ, with morphisms being holomorphic mappings and with morphisms being polynomial mappings.
7
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On ramification locus of a polynomial mapping

100%
Jelonek Z.
Banach Center Publications
|
2003
|
tom 62
|
nr 1
133-134
EN
Let X be a smooth algebraic hypersurface in ℂⁿ. There is a proper polynomial mapping F: ℂⁿ → ℂⁿ, such that the set of ramification values of F contains the hypersurface X.
8
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Characterization of diffeomorphisms that are symplectomorphisms

64%
Janeczko S., Jelonek Z.
Fundamenta Mathematicae
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2009
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tom 205
|
nr 2
147-160
EN
Let $(X,ω_X)$ and $(Y,ω_Y)$ be compact symplectic manifolds (resp. symplectic manifolds) of dimension 2n > 2. Fix 0 < s < n (resp. 0 < k ≤ n) and assume that a diffeomorphism Φ : X → Y maps all 2s-dimensional symplectic submanifolds of X to symplectic submanifolds of Y (resp. all isotropic k-dimensional tori of X to isotropic tori of Y). We prove that in both cases Φ is a conformal symplectomorphism, i.e., there is a constant c ≠0 such that $Φ*ω_{Y} = cω_{X}$.
9
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The set of points at which a morphism of affine schemes is not finite

64%
Jelonek Z., Karaś M.
Colloquium Mathematicum
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2002
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tom 92
|
nr 1
59-66
EN
Assume that X,Y are integral noetherian affine schemes. Let f:X → Y be a dominant, generically finite morphism of finite type. We show that the set of points at which the morphism f is not finite is either empty or a hypersurface. An example is given to show that this is no longer true in the non-noetherian case.
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