Related papers: Motivic invariant of real polynomial functions and…

To a given real polynomial function f $\in$ R[x1, . . . , x d ], we associate real topological zeta functions Ztop,0(f\,; s) and Z $\pm$ top,0 (f\,; s) $\in$ Q(s), analogous to the topological zeta function of Denef and Loeser in the…

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 define invariants of the blow-Nash equivalence of real analytic function germs, in a similar way that the motivic zeta functions of Denef & Loeser. As a key ingredient, we extend the virtual Betti numbers, which were known for real…

We review motivic aspects of multiple zeta values, and as an application, we give an exact-numerical algorithm to decompose any (motivic) multiple zeta value of given weight into a chosen basis up to that weight.

To a given analytic function germ $f:(\mathbb{R}^d,0) \to (\mathbb{R},0)$, we associate zeta functions $Z_{f,+}$, $Z_{f,-} \in \mathbb{Z} [[T]]$, defined analogously to the motivic zeta functions of Denef and Loeser. We show that our zeta…

Given a smooth variety $X$ and a regular function $f$ on it, by considering the dlt modification, we define the dlt motivic zeta function $Z^{\rm dlt}_{\rm mot}(s)$ which does not depend on the choice of the dlt modification.

For any polynomial f with complex coefficients we find a remarkable subset of poles of the motivic zeta function. It is combinatorially determined by any log resolution and it admits an intrinsic interpretation in terms of contact loci of…

In this note, we establish an analog of the Mallows-Sloane bound for Type III formal weight enumerators. This completes the bounds for all types (Types I through IV) in synthesis of our previous results. Next we show by using the binomial…

The motivic zeta function of a smooth and proper $\mathbb{C}((t))$-variety $X$ with trivial canonical bundle is a rational function with coefficients in an appropriate Grothendieck ring of complex varieties, which measures how $X$…

This paper studies the poles of the real Archimedean zeta function for a weighted homogeneous polynomial $f \in \mathbb{R}[x, y]$ with an isolated singularity at the origin. By applying a weighted blow-up, we derive the meromorphic…

We provide the formula of motivic zeta function for semi-quasihomogeneous singularities and in dimension two, we determine the poles of zeta functions. We also give another formula for stringy E-function using embedded…

We prove an easy but interesting result about the linear independence of multiple zeta values of different weights.

In this article, we study local zeta functions over non-Archimedean locals fields of arbitrary characteristic attached to rational functions and characters $\chi$ of the units of the ring of integers $\mathcal{O}_{K}$, by using an approach…

We lift the splicing formula of N\'emethi and Veys, which deals with polynomials in two variables, to the motivic level. After defining the motivic zeta function and the monodromic motivic zeta function with respect to a differential form,…

We associate motivic zeta functions to a large class of infinite dimensional Lie algebras

Let $P$ and $Q$ be two polynomials in two variables with coefficients in an algebraic closed field of characteristic zero. We consider the rational function $f=P/Q$. For an indeterminacy point $\text{x}$ of $f$ and a value $c$, we compute…

To any Nash germ invariant under right composition with a linear action of a finite group, we associate its equivariant zeta functions, inspired from motivic zeta functions, using the equivariant virtual Poincar\'e series as a motivic…

Multiple polylogarithms are equipped with rich algebraic structures including the motivic coaction and the single-valued map which both found fruitful applications in high-energy physics. In recent work arXiv:2312.00697, the current authors…

The zeta-function of a complex variety is a power series whose nth coefficient is the nth symmetric power of the variety, viewed as an element in the Grothendieck ring of complex varieties. We prove that the zeta-function of a surface is…

In this thesis, we use logarithmic methods to study motivic objects. Let R be a complete discrete valuation ring with perfect residue field k, and denote by K its fraction field. We give in chapter 2 a new construction of the motivic Serre…