Monodromy zeta-function of a polynomial on a complete intersection, and Newton polyhedra (Q2892181)

Let \(F_0, F_1, \ldots, F_k\) be polynomials in \(n\) complex variables \(z = (z_1, \ldots, z_n),\) and let \(V\subset \mathbb C^n\) be the set of common zeros of the last \(k\) polynomials. Suppose that both systems of polynomials \(\{F_0, F_1, \ldots, F_k\}\) and \(\{F_1, \ldots, F_k\}\) are non-degenerate with respect to their Newton polyhedra. The author obtains an explicit formula for the zeta function at the origin of \(F_0\) on \(V\) in terms of the integer mixed volumes of the faces of the Newton polyhedra of the polynomials. The proof is reduced to computing the zeta function of the fibration corresponding to the case \(F_0(z)=z_n,\) or, in other words, of a generic polynomial one-parameter deformation of a complete intersection, which is non-degenerate with respect to its Newton polyhedra. The author remarks that the first result can be considered as an analog of a theorem of \textit{A. Libgober} and \textit{S. Sperber} [Compos. Math. 95, No. 3, 287--307 (1995; Zbl 0968.14006)], while the latter one is a global analog of Theorem 2.2 from [\textit{G. G. Gusev}, Rev. Mat. Complut. 22, No. 2, 447--454 (2009; Zbl 1174.14003)] and Theorem 5.5 from [\textit{Y. Matsui} and \textit{K. Takeuchi}, Math. Z. 268, No. 1--2, 409--439 (2011; Zbl 1264.14005)].