Topological Zeta Function Research Articles

Fractal Strings and Multifractal Zeta Functions

For a Borel measure on the unit interval and a sequence of scales that tend to zero, we define a one-parameter family of zeta functions called multifractal zeta functions. These functions are a first attempt to associate a zeta function to certain multifractal measures. However, we primarily show that they associate a new zeta function, the topological zeta function, to a fractal string in order to take into account the topology of its fractal boundary. This expands upon the geometric information garnered by the traditional geometric zeta function of a fractal string in the theory of complex dimensions. In particular, one can distinguish between a fractal string whose boundary is the classical Cantor set, and one whose boundary has a single limit point but has the same sequence of lengths as the complement of the Cantor set. Later work will address related, but somewhat different, approaches to multifractals themselves, via zeta functions, partly motivated by the present paper.

Zeta Functions and Monodromy for Surfaces that are General for a Toric Idealistic Cluster

In this article we consider surfaces that are general with respect to a 3- dimensional toric idealistic cluster. In particular, this means that blowing up a toric con- stellation provides an embedded resolution of singularities for these surfaces. First we give a formula for the topological zeta function directly in terms of the cluster. Then we study the eigenvalues of monodromy. In particular, we derive a useful criterion to be an eigenvalue. In a third part we prove the monodromy and the holomorphy conjecture for these surfaces.

Poles of the topological zeta function associated to an ideal in dimension two

To an ideal in $${\mathbb{C}[x,y]}$$ one can associate a topological zeta function. This is an extension of the topological zeta function associated to one polynomial. But in this case we use a principalization of the ideal instead of an embedded resolution of the curve. In this paper we will study two questions about the poles of this zeta function. First, we will give a criterion to determine whether or not a candidate pole is a pole. It turns out that we can know this immediately by looking at the intersection diagram of the principalization, together with the numerical data of the exceptional curves. Afterwards we will completely describe the set of rational numbers that can occur as poles of a topological zeta function associated to an ideal in dimension two. The same results are valid for related zeta functions, as for instance the motivic zeta function.

On the poles of topological zeta functions

We study the topological zeta function Z t o p , f ( s ) Z_{top,f}(s) associated to a polynomial f 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 ∣ ∃ f ∈ C [ x 1 , … , x n ] : Z t o p , f ( s ) \mathcal {P}_n := \{ s_0 \mid \exists f \in \mathbb {C}[x_1,\ldots , x_n] \, : \, Z_{top,f}(s) has a pole in s 0 } s_0 \} . We show that { − ( n − 1 ) / 2 − 1 / i ∣ i ∈ Z > 1 } \{-(n-1)/2-1/i \mid i \in \mathbb {Z}_{>1}\} is a subset of P n \mathcal {P}_n ; for n = 2 n=2 and n = 3 n=3 , the last two authors proved before that these are exactly the poles less than − ( n − 1 ) / 2 -(n-1)/2 . As the main result we prove that each rational number in the interval [ − ( n − 1 ) / 2 , 0 ) [-(n-1)/2,0) is contained in P n \mathcal {P}_n .

Modification systems and integration in their Chow groups

We introduce a notion of integration on the category of proper birational maps to a given variety $X$, with value in an associated Chow group. Applications include new birational invariants; comparison results for Chern classes and numbers of nonsingular birational varieties; `stringy' Chern classes of singular varieties; and a zeta function specializing to the topological zeta function. In its simplest manifestation, the integral gives a new expression for Chern-Schwartz-MacPherson classes of possibly singular varieties, placing them into a context in which a `change-of-variable' formula holds. v2: References added, and overly optimistic claims concerning non log-terminal singularities expunged.

Quasi-ordinary power series and their zeta functions

The main objective of this paper is to prove the monodromy conjecture for the local Igusa zeta function of a quasi-ordinary polynomial of arbitrary dimension defined over a number field. In order to do it, we compute the local Denef-Loeser motivic zeta function $Z_{\text{DL}}(h,T)$ of a quasi-ordinary power series $h$ of arbitrary dimension over an algebraically closed field of characteristic zero from its characteristic exponents without using embedded resolution of singularities. This allows us to effectively represent $Z_{\text{DL}}(h,T)=P(T)/Q(T)$ such that almost all the candidate poles given by $Q(T)$ are poles. Anyway, these candidate poles give eigenvalues of the monodromy action of the complex of nearby cycles on $h^{-1}(0).$ In particular we prove in this case the monodromy conjecture made by Denef-Loeser for the local motivic zeta function and the local topological zeta function. As a consequence, if $h$ is a quasi-ordinary polynomial defined over a number field we prove the Igusa monodromy conjecture for its local Igusa zeta function.

On the smallest poles of topological zeta functions

We study the local topological zeta function associated to a complex function that is holomorphic at the origin of C 2 (respectively C 3 ). We determine all possible poles less than −1/2 (respectively −1). On C 2 our result is a generalization of the fact that the log canonical threshold is never in ]5/6,1[. Similar statements are true for the motivic zeta function.

THE DENEF–LOESER ZETA FUNCTION IS NOT A TOPOLOGICAL INVARIANT

An example is given which shows that the Denef–Loeser zeta function (usually called the topological zeta function) associated to a germ of a complex hypersurface singularity is not a topological invariant of the singularity. The idea is the following. Consider two germs of plane curves singularities with the same integral Seifert form but with different topological type and which have different topological zeta functions. Make a double suspension of these singularities (consider them in a 4-dimensional complex space). A theorem of M. Kervaire and J. Levine states that the topological type of these new hypersurface singularities is characterized by their integral Seifert form. Moreover the Seifert form of a suspension is equal (up to sign) to the original Seifert form. Hence these new singularities have the same topological type. By means of a double suspension formula the Denef–Loeser zeta functions are computed for the two 3-dimensional singularities and it is verified that they are not equal.

Holomorphy of Igusa’s and topological zeta functions for homogeneous polynomials

Holomorphy of Igusa’s and topological zeta functions for homogeneous polynomials

Zeta Functions and ‘Kontsevich Invariants’ on Singular Varieties

AbstractLet X be a nonsingular algebraic variety in characteristic zero. To an effective divisor on X Kontsevich has associated a certain motivic integral, living in a completion of the Grothendieck ring of algebraic varieties. He used this invariant to show that birational (smooth, projective) Calabi-Yau varieties have the same Hodge numbers. Then Denef and Loeser introduced the invariant motivic (Igusa) zeta function, associated to a regular function on X, which specializes to both the classical p-adic Igusa zeta function and the topological zeta function, and also to Kontsevich's invariant.This paper treats a generalization to singular varieties. Batyrev already considered such a ‘Kontsevich invariant’ for log terminal varieties (on the level of Hodge polynomials of varieties instead of in the Grothendieck ring), and previously we introduced a motivic zeta function on normal surface germs. Here on any ℚ-Gorenstein variety X we associate a motivic zeta function and a ‘Kontsevich invariant’ to effective ℚ-Cartier divisors on X whose support contains the singular locus of X.

On the Poles of Maximal Order of the Topological Zeta Function

The global and local topological zeta functions are singularity invariants associated to a polynomial f and its germ at 0, respectively. By definition, 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; we give a similar result concerning the global zeta function. Moreover, for any f we show that poles of maximal possible order are always of the form −1/N with N a positive integer. 1991 Mathematics Subject Classification 14B05, 14E15, 32S50.

THE TOPOLOGICAL ZETA FUNCTION ASSOCIATED TO A FUNCTION ON A NORMAL SURFACE GERM

We associate to a regular function f on a normal surface germ (S, 0) an invariant, called the topological zeta function, which generalizes the same invariant for a plane curve germ; by definition it is a rational function in one variable.We study its poles and their relation with the local monodromy of f, in particular, we prove the “generalized holomorphy conjecture”. We give a formula for this topological zeta function in terms of the log canonical modelof (S, f -1,{0}) , and we also introduce a still more general invariant.

On the effect of pruning on the singularity structure of zeta functions

We investigate the topological zeta function for unimodal maps in general and dynamical zeta functions for the tent map in particular. For the generic situation, when the kneading sequence is aperiodic, it is shown that the zeta functions have a natural boundary along its radius of convergence, beyond which the function lacks analytic continuation. We make a detailed study of the function ∏n=0∞(1−z2n) associated with sequences of period doublings. It is demonstrated that this function has a dense set of singular points and zeros on the unit circle, exhibiting a rich number theoretical structure.

Zeta Functions for Curves and Log Canonical Models

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 field. 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{0} in A2. 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. 1991 Mathematics Subject Classification: 32S50 11S80 14E30 (14G20)

Determination of the poles of the topological zeta function for curves

Tof ∈ℂ[x 1…,x n ] one associates thetopological zeta function which is an invariant of (the germ of)f at 0, defined in terms of an embedded resolution of (the germ of)f −1{0} inf −1{0}. By definition the topological zeta function is a rational function in one variable, and it is related to Igusa’s local zeta function. A major problem is the study of its poles. In this paper we exactly determine all poles of the topological zeta function forn=2 and anyf ∈ℂ[x 1,x 2]. In particular there exists at most one pole of order two, and in this case it is the pole closest to the origin. Our proofs rely on a new geometrical result which makes the embedded resolution graph of the germ off into an ‘ordered tree’ with respect to the so-callednumerical data of the resolution.

On Euler characteristics associated to exceptional divisors

Let k k be an algebraically closed field and f ∈ k [ x 1 , … , x n + 1 ] f \in k[{x_1}, \ldots ,{x_{n + 1}}] . Fix an embedded resolution h : X → A n + 1 of f − 1 { 0 } h:X \to {\mathbb {A}^{n + 1}}\quad {\text {of}}\quad {f^{ - 1}}\{ 0\} and denote by E i {E_i} , i ∈ S i \in S , the irreducible components of h − 1 ( f − 1 { 0 } ) {h^{ - 1}}({f^{ - 1}}\{ 0\} ) with multiplicity N i {N_i} in the divisor of f o h f{\text {o}}h . Put also E ˚ i ≔ E i ∖ ∪ j ≠ i E j \mathring {E}_i \coloneq {E_i}\backslash { \cup _{j \ne i}}{E_j} , and denote by χ ( E i ) \chi ({E_i}) its Euler characteristic. Several conjectures concerning Igusa’s local zeta function and the topological zeta function of f f motivate the study of Euler characteristics associated to subsets ∪ i ∈ T E i { \cup _{i \in T}}{E_i} of ∪ i ∈ S E i { \cup _{i \in S}}{E_i} , which are maximal connected with respect to the property that d | N i d|{N_i} for all i ∈ T i \in T . Here d ∈ N , d > 1 d \in \mathbb {N},d > 1 . We prove that if h h maps ∪ i ∈ T E i { \cup _{i \in T}}{E_i} to a point, then \[ ( − 1 ) n ∑ i ∈ T χ ( E ˚ i ) ⩾ 0 (-1)^n \sum _{i \in T} \chi (\mathring {E}_i) \geqslant 0 \] This generalizes a well-known result for curves. We also prove some vanishing results concerning the χ ( E ˚ i ) \chi (\mathring {E}_i) for such a maximal connected subset ∪ i ∈ T E i { \cup _{i \in T}}{E_i} and give an application on the above-mentioned zeta functions, yielding some confirmation of the holomorphy conjecture for those zeta functions.