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Nontrivial rational polynomials in two variables have reducible fibres

Published online by Cambridge University Press:  17 April 2009

Walter D. Neumann
Affiliation:
Department of Mathematics, The University of Melbourne, Parkville, Vic 3052, Australia e-mail: neumann@maths.mu.oz.au
Paul Norbury
Affiliation:
Department of Mathematics, The University of Melbourne, Parkville, Vic 3052, Australia e-mail: norbs@maths.mu.oz.au
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Abstract

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We show that every f: ℂ2 → ℂ which is a rational polynomial map with irreducible fibres is a coordinate.

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 1998

References

[1]Abhyankar, S. and Moh, T.T., ‘Embeddings of the line in the plane’, J. Reine Angew. Math. 276 (1975), 148166.Google Scholar
[2]Eisenbud, D. and Neumann, W.D., Three dimensional link theory and invariants of plane curve singularities, Ann. of Math. Stud. 101 (Princeton University Press, Princeton, NJ, 1985).Google Scholar
[3]Friedland, S., ‘On the plane jacobian conjecture’, (Preprint IHES, May 1994) (per [6]).Google Scholar
[4]Heitmann, R., ‘On the Jacobian conjecture’, J. Pure Appl. Algebra 64 (1990), 3672Google Scholar
Corrigendum J. Pure Appl. Algebra 90 (1993), 199200.CrossRefGoogle Scholar
[5]Kaliman, S., ‘Two remarks on polynomials in two variables’, Pacific J. Math. 154 (1992), 285295.Google Scholar
[6]Tráng, Lê Dung and Weber, C., ‘Polynômes á fibres rationnelles et conjecture de jacobienne á 2 variables’, C.R. Acad. Sci. Paris 320 (1995), 581584.Google Scholar
[7]Miyanishi, M. and Sugie, T., ‘Generically rational polynomials’, Osaka J. Math. 17 (1980), 339362.Google Scholar
[8]Neumann, W.D., ‘Complex algebraic plane curves via their links at infinity’, Invent. Math. 98 (1989), 445489.CrossRefGoogle Scholar
[9]Neumann, W.D., ‘Irregular links at infinity of complex affine plane curves’, Quart. J. Math, (to appear).Google Scholar
[10]Razar, M., ‘Polynomial maps with constant Jacobian’, Israel J. Math. 32 (1979), 97106.CrossRefGoogle Scholar
[11]Saito, H., ‘Fonctions entiè;res qui se reduisent à certains polynòmes. II’, Osaka J. Math. 9 (1977), 649674.Google Scholar
[12]Suzuki, M., Propriétés topologiques des polynômes de deux variables complexes, et automorphismes algébriques de l'espace ℂ2, J. Math. Soc. Japan 26 (1974), 241257.Google Scholar