Determination of the poles of the topological zeta function for curves (original) (raw)

Monodromy eigenvalues are induced by poles of zeta functions: the irreducible curve case

Bulletin of the London Mathematical Society, 2010

The 'monodromy conjecture' for a hypersurface singularity f predicts that a pole of its topological (or related) zeta function induces one of its monodromy eigenvalues. However, in general only a few eigenvalues are obtained this way. The second author proposed to consider zeta functions associated with the hypersurface and with a differential form and raised the following question. Can one find a list of differential forms ω i such that any pole of the zeta function of f and an ω i induces a monodromy eigenvalue of f , and such that all monodromy eigenvalues of f are obtained this way? Here we provide an affirmative answer for an arbitrary irreducible curve singularity f .

Monodromy conjecture for some surface singularities

Annales Scientifiques de l’École Normale Supérieure, 2002

In this work we give a formula for the local Denef-Loeser zeta function of a superisolated singularity of hypersurface in terms of the local Denef-Loeser zeta function of the singularities of its tangent cone. We prove the monodromy conjecture for some surfaces singularities. These results are applied to the study of rational arrangements of plane curves whose Euler-Poincaré characteristic is three.  2002 Éditions scientifiques et médicales Elsevier SAS RÉSUMÉ.-Dans ce travail, nous donnons une formule pour la fonction zêta locale de Denef-Loeser d'une singularité superisolée d'hypersurface, en termes des fonctions zêta locales des singularités de son cône tangent. Nous démontrons la conjecture de la monodromie pour certaines singularités de surfaces. Nous appliquons ces résultats à l'étude d'arrangements, de caractéristique d'Euler trois, de courbes rationnelles.

Topological zeta functions and the monodromy conjecture for complex plane curves

arXiv (Cornell University), 2020

We study topological zeta functions of complex plane curve singularities using toric modifications and further developments. As applications of the research method, we prove that the topological zeta function is a topological invariant for complex plane curve singularities, we give a short and new proof of the monodromy conjecture for plane curves.

Topological zeta functions of complex plane curve singularities

2020

We study topological zeta functions of complex plane curve singularities using toric modifications and further developments. As applications of the research method, we prove that the topological zeta function is a topological invariant for complex plane curve singularities, we give a short and new proof of the monodromy conjecture for plane curves.

Zeta Functions for Curves and Log Canonical Models

Proceedings of the London Mathematical Society, 1997

The topological zeta function and Igusa's local zeta function are respectively a geometrical invariant associated to a complex polynomial f and an arithmetical invariant associated to a polynomial f over a p{adic eld. When f is a polynomial in two variables we prove a formula for both zeta functions in terms of the so{called log canonical model of f ?1 f0g in A 2 . This result yields moreover a conceptual explanation for a known cancellation property of candidate poles for these zeta functions. Also in the formula for Igusa's local zeta function appears a remarkable non{symmetric`q{deformation' of the intersection matrix of the minimal resolution of a Hirzebruch{Jung singularity.

On the poles of topological zeta functions

Proceedings of the American Mathematical Society

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.

On the Poles of Maximal Order of the Topological Zeta Function

Bulletin of the London Mathematical Society, 1999

The global and local topological zeta functions are singularity invariants associated to a polynomial f and its germ at 0, respectively. By de nition these zeta functions are rational functions in one variable and their poles are negative rational numbers. In this paper we study their poles of maximal possible order. When f is non degenerate with respect to its Newton polyhedron we prove that its local topological zeta function has at most one such pole, in which case it is also the largest pole; concerning the global zeta function we give a similar result. Moreover for any f we show that poles of maximal possible order are always of the form ?1=N with N a positive integer.