Rational-fibre polynomials and the two-variable Jacobian conjecture. (Polynômes à fibres rationnelles et conjecture jacobienne à 2 variables.) (French) Zbl 0834.14007

Lê Dũng Tráng; Weber, Claude

C. R. Acad. Sci., Paris, Sér. I 320, No. 5, 581-584 (1995).

A new geometric proof of the following particular case of the Jacobian conjecture [first proved by R. C. Heitmann, J. Pure Appl. Algebra 64, No. 1, 35-72 (1990; Zbl 0704.13010)] is given: If a polynomial mapping \((f,g) : \mathbb{C}^2 \to \mathbb{C}^2\) has a non-zero constant Jacobian and \(f\) has only rational and irreducible fibers, then \((f,g)\) is an automorphism of \(\mathbb{C}^2\). The proof is based on theorems (which have not been published so far) on properties of the exceptional divisor of the resolution of the indetermination points of the rational mapping generated by \(f\) on \(\mathbb{P}^2\).

Reviewer: T.Krasiński (Łódź)

MSC:

14H37	Automorphisms of curves
13F20	Polynomial rings and ideals; rings of integer-valued polynomials

Keywords:

rational curve; Jacobian conjecture; polynomial mapping