Jung's type theorem for polynomial transformations of ℂ²

Sławomir Kołodziej

Annales Polonici Mathematici (1991)

  • Volume: 55, Issue: 1, page 207-212
  • ISSN: 0066-2216

Abstract

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We prove that among counterexamples to the Jacobian Conjecture, if there are any, we can find one of lowest degree, the coordinates of which have the form x m y n + terms of degree < m+n.

How to cite

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Sławomir Kołodziej. "Jung's type theorem for polynomial transformations of ℂ²." Annales Polonici Mathematici 55.1 (1991): 207-212. <http://eudml.org/doc/262440>.

@article{SławomirKołodziej1991,
abstract = {We prove that among counterexamples to the Jacobian Conjecture, if there are any, we can find one of lowest degree, the coordinates of which have the form $x^m y^n$ + terms of degree < m+n.},
author = {Sławomir Kołodziej},
journal = {Annales Polonici Mathematici},
keywords = {Jacobian conjecture; birational automorphisms; polynomial automorphism},
language = {eng},
number = {1},
pages = {207-212},
title = {Jung's type theorem for polynomial transformations of ℂ²},
url = {http://eudml.org/doc/262440},
volume = {55},
year = {1991},
}

TY - JOUR
AU - Sławomir Kołodziej
TI - Jung's type theorem for polynomial transformations of ℂ²
JO - Annales Polonici Mathematici
PY - 1991
VL - 55
IS - 1
SP - 207
EP - 212
AB - We prove that among counterexamples to the Jacobian Conjecture, if there are any, we can find one of lowest degree, the coordinates of which have the form $x^m y^n$ + terms of degree < m+n.
LA - eng
KW - Jacobian conjecture; birational automorphisms; polynomial automorphism
UR - http://eudml.org/doc/262440
ER -

References

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  1. [1] S. S. Abhyankar, Expansion Techniques in Algebraic Geometry, Tata Inst. Fund. Research, Bombay 1977. Zbl0818.14001
  2. [2] H. Bass, E. H. Connell and D. Wright, The Jacobian Conjecture : reduction of degree and formal expansion of the inverse, Bull. Amer. Math. Soc. 7 (2) (1982), 287-330. Zbl0539.13012
  3. [3] R. C. Heitmann, On the Jacobian Conjecture, J. Pure Appl. Algebra 64 (1990), 35-72. Zbl0704.13010
  4. [4] H. W. E. Jung, Über ganze birationale Transformationen der Ebene, J. Reine Angew. Math. 184 (1942), 161-174. Zbl0027.08503
  5. [5] O.-H. Keller, Ganze Cremona-Transformationen, Monatsh. Math. Phys. 47 (1939), 299-306. 
  6. [6] L. G. Makar-Limanov, On automorphisms of the free algebra on two generators, Funktsional. Anal. i Prilozhen. 4 (3) (1970), 107-108 (in Russian). 
  7. [7] K. Rusek, Polynomial automorphisms, preprint 456, IM PAN, 1989. 
  8. [8] A. G. Vitushkin, On polynomial transformations of ℂ n , in: Manifolds, Tokyo 1973, Univ. of Tokyo Press, 1975, 415-417. 

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