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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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.
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