相关论文: Topological zeta functions of complex plane curve …
A characterization of dynamically defined zeta functions is presented. It comprises a list of axioms, natural extension of the one which characterizes topological degree, and a uniqueness theorem. Lefschetz zeta function is the main (and…
We show some examples of topological zeta functions associated to an isolated plane curve singular point and an allowed, in the sense of N\'emethi and Veys, differential form that have several poles of order two. This is in contrast to the…
This paper deals with the study of the behaviour of the value semigroup of a curve singularity define over a global field reduced modulo a maximal ideal. We also define a global zeta function of the curve by means of motivic integration…
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…
A new, seemingly useful presentation of zeta functions on complex tori is derived by using contour integration. It is shown to agree with the one obtained by using the Chowla-Selberg series formula, for which an alternative proof is thereby…
Let $(X,x)$ be an isolated complete intersection singularity and let $f : (X,x) \to (\CC,0)$ be the germ of an analytic function with an isolated singularity at $x$. An important topological invariant in this situation is the…
This article investigates the monodromy conjecture for a space monomial curve that appears as the special fiber of an equisingular family of curves with a plane branch as generic fiber. Roughly speaking, the monodromy conjecture states that…
In this contribution we announce a complete classification and new exotic phenomena of the meromorphic structure of $\z$-functions associated to conic manifolds proved in \cite{KLP1}. In particular, we show that the meromorphic extensions…
Using analytic torsion associated to stable bundles, we introduce zeta functions for compact Riemann surfaces. To justify the well-definedness, we analyze the degenerations of analytic torsions at the boundaries of the moduli spaces, the…
We show that the motivic zeta functions of smooth, geometrically connected curves with no rational points are rational functions. This was previously known only for curves whose smooth projective models have a rational point on each…
This paper continues a series of investigations on converging representations for the Riemann Zeta function. We generalize some identities which involve Riemann's zeta function, and moreover we give new series and integrals for the zeta…
The monodromy conjecture is an umbrella term for several conjectured relationships between poles of zeta functions, monodromy eigenvalues and roots of Bernstein-Sato polynomials in arithmetic geometry and singularity theory. Even the…
We determine the zeta functions of trinomial curves in terms of Gauss sums and Jacobi sums, and we obtain an explicit formula of the genus of a trinomial curve over a finite field, then we study the conditions for a trinomial curve to be a…
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…
To investigate the topological structure of Morse functions on the projective plane we use the Reeb graphs. We describe it properties and prove that it is a complete topological invariant of simple Morse function on $\mathbb{R} P^2$. We…
The monodromy conjecture states that every pole of the topological (or related) zeta function induces an eigenvalue of monodromy. This conjecture has already been studied a lot; however, in full generality it is proven only for zeta…
Inspired by work surrounding Igusa's local zeta function, we introduce topological representation zeta functions of unipotent algebraic groups over number fields. These group-theoretic invariants capture common features of established…
We will study a monodromy preserving deformation with an irregular singular point and determine the $\tau$ function of the monodromy preserving deformation by the elliptic $\theta$ function moving the argument $z$ and the period $\Omega$.
The Poincare series of an irreducible plane curve singularity equals the zeta function of its monodromy, by a result of Campillo, Delgado and Gusein-Zade. We derive this fact from a formula of Ebeling and Gusein-Zade relating the Poincare…
In these notes, we explore possible stable properties for the zeta function of a geometric Zp-tower of curves over a finite field of characteristic p, in the spirit of Iwasawa theory. A number of fundamental questions and conjectures are…