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1991 | 55 | 1 | 95-101
Tytuł artykułu

A geometric approach to the Jacobian Conjecture in ℂ²

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
We consider polynomial mappings (f,g) of ℂ² with constant nontrivial jacobian. Using the Riemann-Hurwitz relation we prove among other things the following: If g - c (resp. f - c) has at most two branches at infinity for infinitely many numbers c or if f (resp. g) is proper on the level set $g^{-1}(0)$ (resp. $f^{-1}(0)$), then (f,g) is bijective.
Słowa kluczowe
Rocznik
Tom
55
Numer
1
Strony
95-101
Opis fizyczny
Daty
wydano
1991
otrzymano
1990-08-27
Twórcy
  • Institute of Mathematics, Jagiellonian University, Reymonta 4, 30-059 Kraków, Poland
Bibliografia
  • [1] S. S. Abhyankar, Expansion Techniques in Algebraic Geometry, Tata Inst. Fund. Research, Bombay 1977.
  • [2] S. S. Abhyankar and T. T. Moh, Embeddings of the line in the plane, J. Reine Angew. Math. 276 (1975), 149-166.
  • [3] H. Appelgate and H. Onishi, The Jacobian Conjecture in two variables, J. Pure Appl. Algebra 37 (1985), 215-227.
  • [4] 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.
  • [5] A. Białynicki-Birula and M. Rosenlicht, Injective morphisms of real algebraic varieties, Proc. Amer. Math. Soc. 13 (1962), 200-203.
  • [6] S. A. Broughton, Milnor numbers and the topology of polynomial hypersurfaces, Invent. Math. 92 (1988), 217-241.
  • [7] J. Chądzyński and T. Krasiński, Properness and the Jacobian Conjecture in ℂ², to appear.
  • [8] R. Ephraim, Special polars and curves with one place at infinity, in: Proc. Sympos. Pure Math. 40, Vol. I, Amer. Math. Soc., 1983, 353-360.
  • [9] H. Farkas and I. Kra, Riemann Surfaces, Springer, 1980.
  • [10] M. Furushima, Finite groups of polynomial automorphisms in the complex affine plane, Mem. Fac. Sci. Kyushu Univ. Ser. A 36 (1) (1982), 85-105.
  • [11] O.-H. Keller, Ganze Cremona-Transformationen, Monatsh. Math. Phys. 47 (1939), 299-306.
  • [12] S. Łojasiewicz, Introduction to Complex Analytic Geometry, Birkhäuser, to appear.
  • [13] T. T. Moh, On analytic irreducibility at ∞ of a pencil of curves, Proc. Amer. Math. Soc. 44 (1974), 22-24.
  • [14] T. T. Moh, On the Jacobian Conjecture and the configuration of roots, J. Reine Angew. Math. 340 (1983), 140-212.
  • [15] D. Mumford, Introduction to Algebraic Geometry, Springer, 1976.
  • [16] Y. Nakai and K. Baba, A generalization of Magnus theorem, Osaka J. Math. 14 (1977), 403-409.
  • [17] A Nowicki, On the Jacobian Conjecture in two variables, J. Pure Appl. Algebra 50 (1988), 195-207.
  • [18] K. Rusek and T. Winiarski, Criteria for regularity of holomorphic mappings, Bull. Acad. Polon. Sci. 28 (9-10) (1980), 471-475.
  • [19] K. Rusek and T. Winiarski, Polynomial automorphisms of $ℂ^n$, Univ. Iagel. Acta Math. 24 (1984), 143-149.
Typ dokumentu
Bibliografia
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Identyfikator YADDA
bwmeta1.element.bwnjournal-article-apmv55z1p95bwm
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