Related papers: Local zeta functions and Newton polyhedra

We prove a recent conjecture due to Cluckers and Veys on exponential sums modulo $p^m$ for $m \geq 2$ in the special case where the phase polynomial $f$ is sufficiently non-degenerate with respect to its Newton polyhedron at the origin. Our…

In this paper we derive an explicit expression for the normal zeta function of class two nilpotent groups whose associated Pfaffian hypersurface is smooth. In particular, we show how the local zeta function depends on counting mod p…

We give an explicit formula for the motivic zeta function in terms of a log smooth model. It generalizes the classical formulas for snc-models, but it gives rise to much fewer candidate poles, in general. As an illustration, we explain how…

We use Hodge theory to relate poles of the Archimedean zeta function $Z_f$ of a holomorphic function $f$ with several invariants of singularities. First, we prove that the largest nontrivial pole of $Z_f$ is the negative of the minimal…

The functional equation f(p(z))=g(q(z)) is studied, where p,q are polynomials and f,g are trancendental meromorphic functions in C. We find all the pairs p,q for which there exist nonconstant f,g satisfying our equation and there exist no…

We determine, up to exponentiating, the polar locus of the multivariable archimedean zeta function associated to a finite collection of polynomials F. The result is the monodromy support locus of F, a topological invariant. We give a…

We develop a practical method for computing local zeta functions of groups, algebras, and modules in fortunate cases. Using our method, we obtain a complete classification of generic local representation zeta functions associated with…

The zeta-function of a manifold is closely related to, and sometimes can be calculated completely, in terms of its periods. We report here on a practical and computationally rapid implementation of this procedure for families of Calabi-Yau…

It is known that Shintani zeta functions, which generalise multiple zeta functions, extend to meromorphic functions with poles on affine hyperplanes. We refine this result in showing that the poles lie on hyperplanes parallel to the facets…

It is known that local zeta functions associated with real analytic functions can be analytically continued as meromorphic functions to the hole complex plane. In this paper, certain cases of specific (non-real analytic) smooth functions…

For a function $f$ starlike of order $\alpha$, $0\leqslant \alpha <1$, a non-constant polynomial $Q$ of degree $n$ which is non-vanishing in the unit disc $\mathbb{D}$ and $\beta>0$, we consider the function $F:\mathbb{D}\to\mathbb{C}$…

We study the localization of the poles of the best Mobius approximations for locally univalent functions in the unit disk. Sharp geometric bounds for the pole function are established in terms of Pommerenke's linear invariant orders,…

The local normal zeta functions of a finitely generated, torsion-free nilpotent group $G$ of class 2 depend on the geometry of the Pfaffian hypersurface associated to the bilinear form induced by taking commutators in $G$. The smallest…

This paper studies algebraic and analytic structures associated with the Lerch zeta function, extending the complex variables viewpoint taken in part II. The Lerch transcendent $\Phi(s, z, c)$ is obtained from the Lerch zeta function…

Let f be a regular function on a nonsingular complex algebraic variety of dimension d. We prove a formula for the motivic zeta function of f in terms of an embedded resolution. This formula is over the Grothendieck ring itself, and…

It is known that local zeta functions associated with real analytic functions can be analytically continued as meromorphic functions to the whole complex plane. In this paper, the case of specific (non-real analytic) smooth functions is…

In this paper we present a p-adic algorithm to compute the zeta function of a nondegenerate curve over a finite field using Monsky-Washnitzer cohomology. The paper vastly generalizes previous work since all known cases, e.g. hyperelliptic,…

Let $f(t)=\sum_{n=0}^{+\infty}\frac{C_{f,n}}{n!}t^n$ be an analytic function at $0$, and let $C_{f, n}(x)=\sum_{k=0}^{n}\binom{n}{k}C_{f,k} x^{n-k}$ be the sequence of Appell polynomials, referred to as $\textit{C-polynomials associated to…

We introduce and study new versions of polylogarithms and a zeta function on a completion of $\mathbb F_q (x)$ at a finite place. The construction is based on the use of the Carlitz differential equations for $\mathbb F_q$-linear functions.

Let $f(\mathbf z)$ be an analytic function defined in the neighborhood of the origin of $\mathbb C^n$ which have some Newton degenerate faces. We generalize the Varchenko formula for the zeta function of the Milnor fibration of a Newton…