Abstract
It is shown that a polynomial map F = (P, Q) of ℂ2 is a polynomial automorphism of ℂ2 if J(P, Q) := P x Q y − P y Q x ≡ c ≠ 0 and, in addition, both of polynomials P and Q are rational, i.e., the generic fibers of P and of Q are irreducible rational curves.
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Barth, W., Peters, C., van de Ven, A.: Compact Complex Surfaces. Ergebnisse der Mathematik und ihrer Grenzgebiete (3), vol. 4. Springer, Berlin (1984)
Artal Bartolo, E., Cassou-Nogues, P., Maugendre, H.: Quotients Jacobiens d’applications polynomiales. Ann. Inst. Fourier 53, 399–428 (2003)
Van den Essen, A.: Polynomial automorphisms and the Jacobian conjecture. Progress in Mathematics 190. Basel, Birkhauser (2000)
Friedland, Sh.: Monodromy, differential equations and the Jacobian conjecture. Ann. Polon. Math. 72, 219–249 (1999)
Griffiths, P., Harris, J.: Principles of Algebraic Geometry. Wiley, New York (1978)
Heitmann, R.: On the Jacobian conjecture. J. Pure Appl. Algebra 64, 36–72 (1990), and Corrigendum ibid. 90, 199–200 (1993)
Jelonek, Z.: The set of points at which a polynomial map is not proper. Ann. Polon. Math. 58, 259–266 (1993)
Kaliman, Sh.: On the Jacobian conjecture. Proc. Am. Math. Soc. 117, 45–51 (1993)
Keller, O.: Ganze Cremona-Transformatione. Monatsh. Math. Phys. 47, 299–306 (1939)
Le, D.T., Weber, C.: Polynômes à fibres rationnelles et conjecture Jacobienne 2 variables. C.R. Acad. Sci. Paris Sr. I Math. 320, 581–584 (1995)
Némethi, A., Sigray, I.: On the monodromy representation of polynomial maps in n variables. Stud. Sci. Math. Hung. 39, 361–367 (2002)
Neumann, W., Norbury, P.: Nontrivial rational polynomials in two variables have reducible fibres. Bull. Aust. Math. Soc. 58, 501–503 (1998)
Van Chau, N.: Non-zero constant Jacobian polynomial maps of ℂ2. Ann. Pol. Math. 71, 287–310 (1999)
Van Chau, N.: Two remarks on non-zero constant Jacobian polynomial maps of ℂ2. Ann. Pol. Math. 82, 39–44 (2003)
Van Chau, N.: Note on the Jacobian condition and the non-proper value set. Ann. Pol. Math. 84, 203–210 (2004)
Van Chau, N.: A note on the plane Jacobian conjecture. Ann. Pol. Math. 105, 13–19 (2012)
Razar, M.: Polynomial maps with constant Jacobian. Israel J. Math. 32, 97–106 (1979)
Varchenko, A.N.: Theorems on the topological equisingularity of families of algebraic varieties and families of polynomial mappings. (Russian) Izv. Akad. Nauk SSSR Ser. Mat. 36, 957–1019 (1972)
Vistoli, A.: The number of reducible hypersurfaces in a pencil. Invent. Math. l12, 247–262 (1993)
Acknowledgments
This paper is written in memory of my great friend, Professor Carlos Gutierrez, who spend to me valuable helps and encouragements in a long time when I work in ICMC, University of Sao Paulo, Sao Carlos, Sao Paulo, Brazil. We would like to thank Professor Pierrette Cassou-Nogu`es for many valuable discussions and the referees for their useful comments and corrections which helped to improve the manuscript.
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In memory of Professor Carlos Gutiérrez
The author was partially supported by Vietnam National Foundation for Science and Technology Development (NAFOSTED) grant 101.04-2014.23, VAST-JSPS and VIASM.
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Van Chau, N. Jacobian Pairs of Two Rational Polynomials are Automorphisms. Vietnam J. Math. 42, 401–406 (2014). https://doi.org/10.1007/s10013-014-0088-9
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DOI: https://doi.org/10.1007/s10013-014-0088-9