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Jacobian Pairs of Two Rational Polynomials are Automorphisms

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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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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Correspondence to Nguyen Van Chau.

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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

Keywords

  • Jacobian conjecture
  • Rational polynomial

Mathematics Subject Classification (2010)

  • 14R15
  • 14H20