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Zbigniew Jelonek

Publications and source records attributed to Zbigniew Jelonek.

32 records · Page 2Linked to original sources
arXiv2019

Quantitative properties of the non-properness set of a polynomial map

Let $f$ be a generically finite polynomial map $f: \mathbb{C}^n\to \mathbb{C}^m$ of algebraic degree $d$. Motivated by the study of the Jacobian Conjecture, we prove that the set $S_f$ of non-properness of $f$ is covered by parametric curves of degree at most $d-1$. This bound is best possible. Moreover, we prove that if $X\subset\mathbb{R}^n$ is a closed algebraic set covered by parametric curves, and $f: X\rightarrow\mathbb{R}^m$ is a generically finite polynomial map, then the set $S_f$ of non-properness of $f$ is also covered by parametric curves. Moreover, if $X$ is covered by parametric curves of degree at most $d_1$, and the map $f$ has degree $d_2$, then the set $S_f$ is covered by parametric curves of degree at most $2d_1d_2$. As an application of this result we show a real version of the Białynicki-Birula theorem: Let $G$ be a real, non-trivial, connected, unipotent group which acts effectively and polynomially on a connected smooth algebraic variety $X\subset\mathbb{R}^n$. Then the set $Fix(G)$ of fixed points has no isolated points.

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arXiv2018

On finite polynomial mappings

Let $X\subset \mathbb{C}^n$ be a smooth irreducible affine variety of dimension $k$ and let $F: X\to \mathbb{C}^m$ be a polynomial mapping. We prove that if $m\ge k$, then there is a Zariski open dense subset $U$ in the space of linear mappings ${\mathcal L}( \mathbb{C}^n, \mathbb{C}^m)$ such that for every $L\in U$ the mapping $F+L$ is a finite mapping. Moreover, we can choose $U$ in this way, that all mappings $F+L; L\in U$ are topologically equivalent.

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arXiv2018

Thom isotopy theorem for non proper maps and computation of sets of stratified generalized critical values

Let $X\subset\Bbb C^n$ be an affine variety and $f:X\to\Bbb C^m$ be the restriction to $X$ of a polynomial map $\Bbb C^n\to\Bbb C^m$. In this paper, we construct an affine Whitney stratification of $X$. The set $K(f)$ of stratified generalized critical values of $f$ can be also computed. We show that $K(f)$ is a nowhere dense subset of $\Bbb C^m$, which contains the set $B(f)$ of bifurcation values of $f$ by proving a version of the isotopy lemma for non-proper polynomial maps on singular varieties.

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arXiv2018

On real polynomial local homeomorphisms

We prove that the set of non-properness of a polynomial mapping of the three dimensional space which is a local homeomorphism cannot be homeomorphic to the real line $\R.$

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arXiv2016

Detecting asymptotic non-regular values by polar curves

We locate the Malgrange non-regular values of a given polynomial function $f:\bC^n \to \bC$ by using a series of affine polar curves. We moreover show that all non-trivial Malgrange non-regular values of $f$ are indicated by a single "super-polar curve" which we introduce here, providing also an effective algorithm of detection.

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arXiv2015

On finite regular and holomorphic mappings

Let $X, Y$ be smooth algebraic varieties of the same dimension. Let $f, g : X \to Y$ be finite polynomial mappings. We say that $f, g$ are equivalent if there exists a regular automorphism $Φ\in Aut(X)$ such that $f = g\circ Φ$. Of course if $f, g$ are equivalent, then they have the same discriminant and the same geometric degree. We show, that conversely there is only a finite number of non-equivalent proper polynomial mappings $f : X \to Y$, such that $D(f) = V$ and $μ(f) = k.$ We prove the same statement in the local holomorphic situation. In particular we show that if $f : (\Bbb C^n, 0) \to (\Bbb C^n, 0)$ is a proper and holomorphic mapping of topological degree two, then there exist biholomorphisms $P,Q : (\Bbb C^n, 0) \to (\Bbb C^n, 0)$ such that $P\circ f\circ Q (x_1, x_2,..., x_n) = (x_1^2, x_2, ..., x_n)$. Moreover, for every proper holomorphic mapping $f : (\Bbb C^n, 0) \to (\Bbb C^n, 0)$ with smooth discriminant there exist biholomorphisms $P,Q : (\Bbb C^n, 0) \to (\Bbb C^n, 0)$ such that $P\circ f\circ Q (x_1, x_2,..., x_n) = (x_1^k, x_2, ..., x_n)$, where $k = μ(f).$

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arXiv2015

The Thom Conjecture for proper polynomial mappings

Let $f,g:X \to Y$ be continuous mappings. We say that $f$ is topologically equivalent to $g$ if there exist homeomorphisms $Φ: X\to X$ and $Ψ: Y\to Y$ such that $Ψ\circ f\circ Φ=g.$ Let $X,Y$ be complex smooth irreducible affine varieties. We show that every algebraic family $F: M\times X\ni (m, x)\mapsto F(m, x)=f_m(x)\in Y$ of polynomial mappings contains only a finite number of topologically non-equivalent proper mappings. In particular there are only a finite number of topologically non-equivalent proper polynomial mappings $f: \Bbb C^n\to\Bbb C^m$ of bounded (algebraic) degree. This gives a positive answer to the Thom Conjecture in the case of proper polynomial mappings.

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arXiv2014

The set of fixed points of a unipotent group

Let $K$ be an algebraically closed field. Let $G$ be a non-trivial connected unipotent group, which acts effectively on an affine variety $X.$ Then every non-empty component $R$ of the set of fixed points of $G$ is a $K$-uniruled variety, i.e, there exists an affine cylinder $W\times K$ and a dominant, generically-finite polynomial mapping $ϕ:W\times K\rightarrow R.$ We show also that if an arbitrary infinite algebraic group $G$ acts effectively on $K^n$ and the set of fixed points contains a hypersurface $H$, then this hypersurface is $K$-uniruled.

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arXiv2014

On the set of fixed points of a polynomial automorphism

Let K be an algebraically closed field of characteristic zero. We say that a polynomial automorphism f : K^n -> K^n is special if the Jacobian of f is equal to 1. We show that every (n - 1)-dimensional component H of the set Fix(f) of fixed points of a non-trivial special polynomial automorphism f : K^n -> K^n is uniruled. Moreover, we show that if f is non-special and H is an (n-1)-dimensional component of the set Fix(f), then H is smooth, irreducible and H = Fix(f) and for K = C the Euler characteristic of H is equal to 1.

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arXiv2013

Reaching generalized critical values of a polynomial

Let f be a real or complex polynomial. We give an algorithm to compute the set of generalized critical values. The algorithm uses a finite dimensional space of rational arcs along which we can reach all generalized critical values of f.

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