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Willem Veys

Publications and source records attributed to Willem Veys.

At least 19 recordsLinked to original sources
arXiv2026

On the poles of zeta functions for finite morphisms between normal surfaces

For a divisor representing a function and another divisor representing a differential form on a normal surface singularity, there is a notion of motivic and topological zeta function. In this paper, given a finite morphism between two normal surfaces, we prove that the set of poles of the motivic zeta function associated with the target is contained in the one associated with the source. We illustrate by examples that this inclusion is strict in general, and that on the topological level there are in general no inclusions between the sets of poles on source and target. On the other hand, when the morphism is the quotient map induced by an action of a finite abelian group on $\mathbb{C}^2$, and the divisor associated with the differential form on the source is trivial, we do show equality between the corresponding sets of poles, both on motivic and topological level. In addition, again for the quotient map induced by an action of a finite abelian group on $\mathbb{C}^2$, but now with a general divisor associated with a differential form, we provide a criterion when the topological zeta function on the target is just a multiple of the one on the source. Finally, we compare log canonical models on source and target with a view on zeta functions.

math.AG↗
arXiv2026

Koba-Nielsen local zeta functions, convex subsets, and generalized Selberg-Mehta-Macdonald and Dotsenko-Fateev-like integrals

The Koba-Nielsen local zeta functions are integrals depending on several complex parameters, used to regularize the Koba-Nielsen string amplitudes. These integrals are convergent and admit meromorphic continuations in the complex parameters. In the original case, the integration is carried out on the n-dimensional Euclidean space. In this work, the integration is over a variety of (bounded or unbounded) convex subsets; the resulting integrals also admit meromorphic continuations in the complex parameters. We describe the meromorphic continuation's polar locus explicitly, using the technique of embedded resolution. This result can be reinterpreted as saying that the meromorphic continuations are weighted sums of Gamma functions, evaluated at linear combinations of the complex parameters, where the weights are holomorphic functions. The integrals announced in the title of this paper occur as a particular case of these new Koba-Nielsen local zeta functions, or of a further generalization to arbitrary hyperplane arrangements.

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arXiv2026

The Strong Monodromy Conjecture for a class of homogeneous polynomials in three variables

We consider the class of all homogeneous, possibly non-reduced, polynomials $f$ whose associated reduced projective divisor $D_{\text{red}} \subset \mathbb{P}^{n-1}$ has (at worst) quasi-homogeneous isolated singularities. In an arbitrary number of variables $n$ and with $d$ denoting the degree of $f$, we characterize when $-n/d$ is a root of the Bernstein--Sato polynomial of $f$ in terms of elementary data involving logarithmic derivations. When we restrict to three variables, we prove the resulting class of polynomials satisfies the Strong Monodromy Conjecture, in the motivic sense.

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arXiv2026

On a Generalized Monodromy Conjecture for Curves using Differential Forms

Motivic and topological zeta functions are singularity invariants, mainly associated to a function $f$ and a top differential form $ω$ on a smooth variety. When $ω$ is the standard form $dx_1\wedge \dots \wedge dx_n$ on affine $n$-space, the monodromy conjecture states that poles of these zeta functions should induce monodromy eigenvalues of $f$. We study natural generalized statements of the monodromy conjecture for functions $f$ on complex surface germs; more precisely on singular surfaces for forms $ω$ that generalize the standard form, and on the affine plane for forms $ω$ that are intrinsically associated to $f$. For all cases, we provide counterexamples to the statement. In addition, when the intrinsically associated $ω$ is given by the generic polar of $f$, we discover a relation between the poles of the zeta functions and the intersection behaviour of the polar curve.

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arXiv2025

Birational zeta functions

We define a birational analog of the motivic zeta function of a reduced polynomial in terms of minimal models. It admits an intrinsic meaning in terms of contact loci of arcs, an analog of a result of Denef and Loeser in the motivic case. We show that for local plane curve singularities the poles of the birational zeta function essentially coincide with the poles of the motivic zeta function.

math.AG↗
arXiv2025

Dicritical divisors and hypercurvettes

Germs of rational functions~$h$ on points $p$ of smooth varieties~$S$ define germs of rational maps to the projective line. Assume that $p$ is in the indeterminacy locus of $h$. If $π:\hat{S}\to S$ is a birational map which is an isomorphism outside $p$, then $h$ lifts to a germ of a rational map on $(\hat{S}, π^{-1}(p))$. The exceptional components $E_i$ of $π^{-1}(p)$ are classified according to the restriction of (the lift of) $h$ to $E_i$; the dicritical components are those where this restriction induces a dominant map. In a series of papers, Abhyankar and the first named author studied this setting in dimension $2$, where the main result is that, for any given $π$, there is a rational function $h$ with a prescribed subset of exceptional components that are dicritical of some given degree. The concept of curvette of an exceptional component played a key role in the proof. The second named author extended previously the concept of curvette to the higher dimensional case. Here we use this concept to generalize the above result to arbitrary dimension.

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arXiv2024

Filtrations associated with singularities

We fix a complex analytic normal singularity germ $(X,o)$ of dimension $\geq 2$ and a (not necessarily irreducible) reduced Weil divisor $(S,o)\subset (X,o)$. The embedded resolution of the pair determines a multi-index filtration of the local ring $\mathcal{O}_{X,o}$, which measures the embedded geometry of the pair. Furthermore, from the (induced) resolution of $(S,o)$ we also consider a multi-index filtration associated with $(S,o)$. This latter one can be lifted to a filtration of $\mathcal{O}_{X,o}$ too. The main result proves that the second filtration of $\mathcal{O}_{X,o}$ can be realized as a `limit' filtration of the first one (if we blow up certain centers sufficiently many times).

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arXiv2024

Introduction to the monodromy conjecture

The monodromy conjecture is a mysterious open problem in singularity theory. Its original version relates arithmetic and topological/geometric properties of a multivariate polynomial $f$ over the integers, more precisely, poles of the $p$-adic Igusa zeta function of $f$ should induce monodromy eigenvalues of $f$. The case of interest is when the zero set of $f$ has singular points. We first present some history and motivation. Then we expose a proof in the case of two variables, and partial results in higher dimension, together with geometric theorems of independent interest inspired by the conjecture. We conclude with several possible generalizations.

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arXiv2021

The dlt motivic zeta function is not well-defined

In arXiv:1408.4708, Xu defines the dlt motivic zeta function associated to a regular function $f$ on a smooth variety $X$ over a field of characteristic zero. This is an adaptation of the classical motivic zeta function that was introduced by Denef and Loeser. The dlt motivic zeta function is defined on a dlt modification via a Denef-Loeser-type formula, replacing classes of strata in the Grothendieck ring of varieties by stringy motives. We provide explicit examples that show that the dlt motivic zeta function depends on the choice of dlt modification, contrary to what is claimed in arXiv:1408.4708, and that it is therefore not well-defined.

math.AG↗
arXiv2020

The motivic Igusa zeta function of a space monomial curve with a plane semigroup

In this article, we compute the motivic Igusa zeta function of a space monomial curve that appears as the special fiber of an equisingular family whose generic fiber is a complex plane branch. To this end, we determine the irreducible components of the jet schemes of such a space monomial curve. This approach does not only yield a closed formula for the motivic zeta function, but also allows to determine its poles. We show that, while the family of the jet schemes of the fibers is not flat, the number of poles of the motivic zeta function associated with the space monomial curve is equal to the number of poles of the motivic zeta function associated with a generic curve in the family.

math.AG↗
arXiv2020

The monodromy conjecture for a space monomial curve with a plane semigroup

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 every pole of the motivic, or related, Igusa zeta function induces an eigenvalue of monodromy. As the poles of the motivic zeta function associated with such a space monomial curve have been determined in earlier work, it remains to study the eigenvalues of monodromy. After reducing the problem to the curve seen as a Cartier divisor on a generic embedding surface, we construct an embedded $\mathbb Q$-resolution of this pair and use an A'Campo formula in terms of this resolution to compute the zeta function of monodromy. Combining all results, we prove the monodromy conjecture for this class of monomial curves.

math.AG↗
arXiv2020

On the motivic oscillation index and bound of exponential sums modulo $p^m$ via analytic isomorphisms

Let $f$ be a polynomial in $n$ variables over some number field and $Z$ a subscheme of affine $n$-space. The notion of motivic oscillation index of $f$ at $Z$ was initiated by Cluckers (2008) and Cluckers-Mustaţǎ-Nguyen (2019). In this paper we elaborate on this notion and raise several questions. The first one is stability under base field extension; this question is linked to a deep understanding of the density of non-archimedean local fields over which Igusa's local zeta functions of $f$ has a pole with given real part. The second one is around Igusa's conjecture for exponential sums with bounds in terms of the motivic oscillation index. Thirdly, we wonder if the above questions only depend on the analytic isomorphism class of singularities. By using various techniques as the GAGA theorem, resolution of singularities and model theory, we can answer the third question up to a base field extension. Next, by using a transfer principle between non-archimedean local fields of characteristic zero and positive characteristic, we can link all three questions with a conjecture on weights of $\ell$-adic cohomology groups of Artin-Schreier sheaves associated to jet polynomials. This way, we can answer all questions positively if $f$ is a polynomial of Thom-Sebastiani type with non-rational singularities. As a consequence, we prove Igusa's conjecture for arbitrary polynomials in three variables and polynomials with singularities of $ADE$ type. In an appendix, we answer affirmatively a recent question of Cluckers-Mustaţǎ-Nguyen (2019) on poles of twisted Igusa's local zeta functions of maximal order.

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arXiv2020

Meromorphic Continuation of Koba-Nielsen String Amplitudes

In this article, we establish in a rigorous mathematical way that Koba-Nielsen amplitudes defined on any local field of characteristic zero are bona fide integrals that admit meromorphic continuations in the kinematic parameters. Our approach allows us to study in a uniform way open and closed Koba-Nielsen amplitudes over arbitrary local fields of characteristic zero. In the regularization process we use techniques of local zeta functions and embedded resolution of singularities. As an application we present the regularization of p-adic open string amplitudes with Chan-Paton factors and constant B-field. Finally, all the local zeta functions studied here are partition functions of certain 1D log-Coulomb gases, which shows an interesting connection between Koba-Nielsen amplitudes and statistical mechanics.

math-ph↗
arXiv2020

Motivic Zeta Functions on $\mathds{Q}$-Gorenstein Varieties

We study motivic zeta functions for $\mathds{Q}$-divisors in a $\mathds{Q}$-Gorenstein variety. By using a toric partial resolution of singularities we reduce this study to the local case of two normal crossing divisors where the ambient space is an abelian quotient singularity. For the latter we provide a closed formula which is worked out directly on the quotient singular variety. As a first application we provide a family of surface singularities where the use of weighted blow-ups reduces the set of candidate poles drastically. We also present an example of a quotient singularity under the action of a nonabelian group, from which we compute some invariants of motivic nature after constructing a $\mathds{Q}$-resolution.

math.AG↗
arXiv2017

Contribution of jumping numbers by exceptional divisors, with an appendix by Karen E. Smith and Kevin Tucker

We investigate some necessary and sufficient conditions for an exceptional divisor to contribute jumping numbers of an effective divisor on a variety of arbitrary dimension, inspired by the results for curves on surfaces by Smith and Thompson and Tucker. In particular, we construct an example of an exceptional divisor that is not contracted in the log canonical model, and does not contribute any jumping numbers.

math.AG↗
arXiv2016

Power moments of Kloosterman sums

In this paper we give an essential treatment for estimating arbitrary integral power moments of Kloosterman sums over the residue class ring. For prime moduli we derive explicit estimates, and for prime-power moduli we prove concrete formulas using computations with Igusa zeta functions.

math.NT↗
arXiv2015

Zeta functions and oscillatory integrals for meromorphic functions

In the 70's Igusa developed a uniform theory for local zeta functions and oscillatory integrals attached to polynomials with coefficients in a local field of characteristic zero. In the present article this theory is extended to the case of rational functions, or, more generally, meromorphic functions f/g, with coefficients in a local field of characteristic zero. This generalization is far from being straightforward due to the fact that several new geometric phenomena appear. Also, the oscillatory integrals have two different asymptotic expansions: the 'usual' one when the norm of the parameter tends to infinity, and another one when the norm of the parameter tends to zero. The first asymptotic expansion is controlled by the poles (with negative real parts) of all the twisted local zeta functions associated to the meromorphic functions f/g - c, for certain special values c. The second expansion is controlled by the poles (with positive real parts) of all the twisted local zeta functions associated to f/g.

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