Zeta functions and monodromy for surfaces that are general for a toric idealistic cluster

In this article we consider surfaces that are general with respect to a 3- dimensional toric idealistic cluster. In particular, this means that blowing up a toric constellation provides an embedded resolution of singularities for these surfaces. First we give a formula for the topological zeta function directly in terms of the cluster. Then we study the eigenvalues of monodromy. In particular, we derive a useful criterion to be an eigenvalue. In a third part we prove the monodromy and the holomorphy conjecture for these surfaces

On the poles of topological zeta functions

We study the topological zeta function Z_{top,f}(s) associated to a polynomial f with complex coefficients. This is a rational function in one variable and we want to determine the numbers that can occur as a pole of some topological zeta function; by definition these poles are negative rational numbers. We deal with this question in any dimension. Denote P_n := {s_0 | \exists f in C[x_1,..., x_n] : Z_{top,f}(s) has a pole in s_0}. We show that {-(n-1)/2-1/i | i in Z_{>1}} is a subset of P_n; for n=2 and n=3, the last two authors proved before that these are exactly the poles less then -(n-1)/2. As main result we prove that each rational number in the interval [-(n-1)/2,0) is contained in P_n

The holomorphy conjecture for nondegenerate surface singularities

The holomorphy conjecture states roughly that Igusa's zeta function associated to a hypersurface and a character is holomorphic on $\mathbb{C}$ whenever the order of the character does not divide the order of any eigenvalue of the local monodromy of the hypersurface. In this article we prove the holomorphy conjecture for surface singularities which are nondegenerate over $\mathbb{C}$ with respect to their Newton polyhedron. In order to provide relevant eigenvalues of monodromy, we first show a relation between the normalized volume (which appears in the formula of Varchenko for the zeta function of monodromy) of faces in a simplex in arbitrary dimension. We then study some specific character sums that show up when dealing with false poles. In contrast with the context of the trivial character, we here need to show fakeness of certain poles in addition to the candidate poles contributed by $B_1$-facets.Comment: 21 pages, 3 figure

On monodromy for a class of surfaces

International audienceIn this note we present a result on the monodromy conjecture for surfaces that are generic with respect to a toric idealistic cluster. Résumé Sur la monodromie pour une certaine classe de surfaces. On présente dans cette note un résultat sur la conjecture de monodromie pour les surfaces qui sont génériques pour un amas torique idéalistique

Monodromy conjecture for nondegenerate surface singularities

International audienceWe prove the monodromy conjecture for the topological zeta function for all non-degenerate surface singularities. Fundamental in our work is a detailed study of the formula for the zeta function of monodromy by Varchenko and the study of the candidate poles of the topological zeta function yielded by what we call 'B1-facets'. In particular, new cases among the nondegenerate surface singularities for which the monodromy conjecture is proven now, are the non-isolated singularities, the singularities giving rise to a topological zeta function with multiple candidate pole and the ones for which the Newton polyhedron contains a B1-facet

The holomorphy conjecture for ideals in dimension two

International audienceThe holomorphy conjecture predicts that the topo-logical zeta function associated to a polynomial f ∈ C[x 1 ,. .. , x n ] and an integer d > 0 is holomorphic unless d divides the order of an eigenvalue of local monodromy of f. In this note, we generalise the holomorphy conjecture to the setting of arbitrary ideals in C[x 1 ,. .. , x n ], and we prove it when n = 2