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.
This is a preview of subscription content, access via your institution.
References
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.
Author information
Authors and Affiliations
Corresponding author
Additional information
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.
Rights and permissions
About this article
Cite this article
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
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10013-014-0088-9