Simple Rational Polynomials and the Jacobian Conjecture

Abstract

Consider an algebraic map Φ := (f, g) : C → C defined by the polynomial functions f and g of C. In complex dimension 2 the Jacobian conjecture asserts that, if the determinant J(f, g) of the Jacobian matrix of Φ is a nonzero constant, then the algebraic map Φ is an isomorphism. In this paper we shall show that this conjecture is true, when we assume that one of the two polynomial functions has generic rational fibers, i.e. they are diffeomorphic to a punctured 2-sphere, and furthermore is simple, i.e. the dicritical components of the polynomial in a natural compactification have degree one (see [10]). We call rational polynomial functions polynomial functions which have a general fiber diffeomorphic to a punctured 2-sphere. The corresponding polynomials are called rational polynomials. In [10] one can find a classification of simple rational polynomials up to right-left equivalence, but we shall not use this classification to establish our result.