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

Zbigniew Jelonek

Annales Polonici Mathematici (1993)

  • Volume: 58, Issue: 3, page 259-266
  • ISSN: 0066-2216

Abstract

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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 d e g f i - μ ( f ) ) / ( m i n i = 1 , . . . , n d e g f i ) .

How to cite

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Zbigniew Jelonek. "The set of points at which a polynomial map is not proper." Annales Polonici Mathematici 58.3 (1993): 259-266. <http://eudml.org/doc/262458>.

@article{ZbigniewJelonek1993,
abstract = {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 $(∏_\{i=1\}^n deg f_i - μ (f)) / (min_\{i=1,...,n\} deg f_i)$.},
author = {Zbigniew Jelonek},
journal = {Annales Polonici Mathematici},
keywords = {polynomial mappings; proper mappings; dominant mappings; analytic covering; proper map; dominating polynomial map; uniruled hypersurface},
language = {eng},
number = {3},
pages = {259-266},
title = {The set of points at which a polynomial map is not proper},
url = {http://eudml.org/doc/262458},
volume = {58},
year = {1993},
}

TY - JOUR
AU - Zbigniew Jelonek
TI - The set of points at which a polynomial map is not proper
JO - Annales Polonici Mathematici
PY - 1993
VL - 58
IS - 3
SP - 259
EP - 266
AB - 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 $(∏_{i=1}^n deg f_i - μ (f)) / (min_{i=1,...,n} deg f_i)$.
LA - eng
KW - polynomial mappings; proper mappings; dominant mappings; analytic covering; proper map; dominating polynomial map; uniruled hypersurface
UR - http://eudml.org/doc/262458
ER -

References

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  1. [1] Z. Jelonek, The extension of regular and rational embeddings, Math. Ann. 277 (1987), 113-120. Zbl0611.14010
  2. [2] Z. Jelonek, Irreducible identity sets for polynomial automorphisms, Math. Z. 212 (1993), 601-617. Zbl0806.14011
  3. [3] Z. Jelonek, The set of points at which a polynomial map is not proper, preprint CRM no. 141, Bellaterra, March 1992. 
  4. [4] M. Kwieciński, Extending finite mappings to affine spaces, J. Pure Appl. Algebra 76 (1991), 151-154. Zbl0753.14002
  5. [5] D. Mumford, Algebraic Geometry I, Springer, Berlin, 1976. Zbl0356.14002
  6. [6] O. Perron, Algebra I (Die Grundlagen), Göschens Lehrbücherei, Berlin und Leipzig, 1932. 
  7. [7] A. Płoski, Algebraic dependence and polynomial automorphisms, Bull. Polish Acad. Sci. Math. 34 (1986), 653-659. Zbl0616.32006
  8. [8] A. Płoski, On the growth of proper polynomial mappings, Ann. Polon. Math. 45 (1985), 297-309. Zbl0584.32006
  9. [9] I. R. Shafarevich, Basic Algebraic Geometry, Springer, 1974. Zbl0284.14001
  10. [10] P. Tworzewski and T. Winiarski, Analytic sets with proper projections, J. Reine Angew. Math. 337 (1982), 68-76. Zbl0497.32024

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