Monodromy zeta-functions of deformations and Newton diagrams

Abstract

For a one-parameter deformation of an analytic complex function germ of several variables, there is defined its monodromy zeta-function. We give a Varchenko type formula for this zeta-function if the deformation is non-degenerate with respect to its Newton diagram.

Milnor fibers over singular toric varieties and nearby cycle sheaves

“…The proof of Theorem 3.12 is very simple and follows also from the functorial property (Proposition 2.9) of the nearby cycle functor. Note that on C n Gusev [7] obtained independently a similar result in a special case as a corollary of his main result. In Section 5, we extend our results to the monodromy zeta functions of T -invariant constructible sheaves.…”

“…Acknowledgements: After submitting this paper to a preprint server, we were informed by Professor Gusev that he obtained a similar result on C n . We thank him cordially for showing us his very interesting paper [7].…”

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

“…The fibration L is trivial over a neighbourhood of a point x ∈ W. Therefore, using a fixed coordinate system one can consider the family of germs at the point x of sections q σ as a deformation in the parameter σ of a function germ. We denote by ζ qσ| W 1 ,x (t) the zeta-function of the germ at the point σ = 0 of the above deformation restricted to the set W 1 (see, ex., [3]).…”

“…, F k . One can consider this result as a global analog of [3,Theorem 2.2] and an analog of [5,Theorem 5.5], where the monodromy zeta-function at infinity is calculated. In section 2 we consider the case F 0 (z) = z n , where the fibration corresponds to a polynomial deformation of a set of polynomials in n − 1 variables z 1 , z 2 , .…”