Invariance of the global monodromies in families of polynomials of two complex variables
Let \(f_\alpha(x,y)\) be a family of primitive polynomials in two complex variables with coefficients which are smooth complex-valued functions of \(\alpha \in [0,1].\) For such a polynomial \(f\) the author considers the following four numerical invariants of \(f :\) the total Milnor number \(\mu,\) the degree \(d,\) the number of critical values at infinity \(\sigma,\) and certain invariant \(\lambda\) that characterizes the intersection of the degree \(d\) homogeneous part \(f_d\) of the polynomial \(f\) and the line at infinity. In particular, for any \(M\)-tame polynomial \(f\) we have \(\lambda(f) = \sigma(f) = 0\) [see \textit{Ha Huy Vui} and \textit{A. Zaharia}, Math. Ann. 304, No. 3, 481-488 (1996; Zbl 0844.32021)]. It is proved that the constancy \(\mu(f_\alpha), d(f_\alpha), \sigma(f_\alpha)\) and \(\lambda(f_\alpha)\) on the interval \([0,1]\) implies the conjugacy of the geometric global monodromies of the polynomials \(f_0\) and \(f_1.\) In authors' opinion the above result can be considered as a global analog of the Lê-Ramanujam theorem [\textit{Lê Dung Trang} and \textit{C. P. Ramanujam}, Am. J. Math. 98, 67-79 (1976; Zbl 0351.32009)].
scientific article; zbMATH DE number 1145964