On the Malgrange condition for complex polynomials of two variables
The main result of this paper states that if \(P \in {\mathbb C}[X,Y]\) is nonconstant then the set \(K_\infty(P)\) of asymptotic critical values of \(P\) coincides with the roots of a polynomial in one variable. More precisely, a polynomial \(P \in {\mathbb C}[X,Y]\) is said to satisfy the Malgrange condition at \(z \in {\mathbb C}\) if there is no sequence \((x_n,y_n)\) of points in \({\mathbb C}^2\) such that \(\lim_{n \to \infty} (|x_n|+ |y_n|) = \infty\), \(\lim_{n \to \infty} (|x_n|+ |y_n|) \nabla P(x_n,y_n) = 0\) and \(\lim_{n \to \infty} P(x_n,y_n) = z\). The set of values \(z \in {\mathbb C}\) where the Malgrange condition is not satisfied is denoted as \(K_\infty(P)\). These asymptotic critical values may be viewed as critical values without any corresponding critical point so \(K_\infty(P)\) is closely related to the set \(K_0(P)\) of classical critical values of \(P\). The main theorem says that if \(P \in {\mathbb C}[X,Y]\) is a nonconstant polynomial such that the line \(X=0\) is not a zero of \(P\) at infinity, then \(K_\infty(P)\) coincides with the set of zeros of the coefficient \(r \in {\mathbb C}[Z]\) of the highest power of \(X\) in the resultant \(R \in {\mathbb C}[X,Z]\) obtained by eliminating \(Y\) between \(P-Z\) and \(P_Y\). The polynomial \(r\) was first considered by \textit{H. V. Hà} [C. R. Acad. Sci., Paris, Sér. I, Math. 309, 231-234 (1989; Zbl 0672.32004)] in connection with the bifurcation set of \(P\). It must be noticed that the technical condition on the line \(X=0\) always holds after a linear change of variable and, what is really an important feature of this paper, that proofs are purely analytical, avoiding the algebraic and topological arguments of other approaches. The paper ends with some interesting applications of the main theorem to the real and complex Jacobian conjectures and to the reducibility of the polynomials \(P-z\), \(z \in {\mathbb C}\).
scientific article; zbMATH DE number 1889169