Topological Zeta Function Research Articles

A chaotic lattice field theory in one dimension* *Dedicated to Fritz Haake 1941–2019 [

Motivated by Gutzwiller’s semiclassical quantization, in which unstable periodic orbits of low-dimensional deterministic dynamics serve as a WKB ‘skeleton’ for chaotic quantum mechanics, we construct the corresponding deterministic skeleton for infinite-dimensional lattice-discretized scalar field theories. In the field-theoretical formulation, there is no evolution in time, and there is no ‘Lyapunov horizon’; there is only an enumeration of lattice states that contribute to the theory’s partition sum, each a global spatiotemporal solution of system’s deterministic Euler–Lagrange equations. The reformulation aligns ‘chaos theory’ with the standard solid state, field theory, and statistical mechanics. In a spatiotemporal, crystallographer formulation, the time-periodic orbits of dynamical systems theory are replaced by periodic d-dimensional Bravais cell tilings of spacetime, each weighted by the inverse of its instability, its Hill determinant. Hyperbolic shadowing of large cells by smaller ones ensures that the predictions of the theory are dominated by the smallest Bravais cells. The form of the partition function of a given field theory is determined by the group of its spatiotemporal symmetries, that is, by the space group of its lattice discretization, best studied on its reciprocal lattice. Already one-dimensional lattice discretization is of sufficient interest to be the focus of this paper. In particular, from a spatiotemporal field theory perspective, ‘time’-reversal is a purely crystallographic notion, a reflection point group, leading to a novel, symmetry quotienting perspective of time-reversible theories and associated topological zeta functions.

Bernstein-Sato ideals and hyperplane arrangements

We study the relation between zero loci of Bernstein-Sato ideals and roots of b -functions and obtain a criterion to guarantee that roots of b -functions of a reducible polynomial are determined by the zero locus of the associated Bernstein-Sato ideal. Applying the criterion together with a result of Maisonobe we prove that the set of roots of the b -function of a free hyperplane arrangement is determined by its intersection lattice. We also study the zero loci of Bernstein-Sato ideals and the associated relative characteristic cycles for arbitrary central hyperplane arrangements. We prove the multivariable n / d conjecture of Budur for complete factorizations of arbitrary hyperplane arrangements, which in turn proves the strong monodromy conjecture for the associated multivariable topological zeta functions.

Local Zeta Functions and Koba–Nielsen String Amplitudes

This article is a survey of our recent work on the connections between Koba–Nielsen amplitudes and local zeta functions (in the sense of Gel’fand, Weil, Igusa, Sato, Bernstein, Denef, Loeser, etc.). Our research program is motivated by the fact that the p-adic strings seem to be related in some interesting ways with ordinary strings. p-Adic string amplitudes share desired characteristics with their Archimedean counterparts, such as crossing symmetry and invariance under Möbius transformations. A direct connection between p-adic amplitudes and the Archimedean ones is through the limit p→1. Gerasimov and Shatashvili studied the limit p→1 of the p-adic effective action introduced by Brekke, Freund, Olson and Witten. They showed that this limit gives rise to a boundary string field theory, which was previously proposed by Witten in the context of background independent string theory. Explicit computations in the cases of 4 and 5 points show that the Feynman amplitudes at the tree level of the Gerasimov–Shatashvili Lagrangian are related to the limit p→1 of the p-adic Koba–Nielsen amplitudes. At a mathematical level, this phenomenon is deeply connected with the topological zeta functions introduced by Denef and Loeser. A Koba–Nielsen amplitude is just a new type of local zeta function, which can be studied using embedded resolution of singularities. In this way, one shows the existence of a meromorphic continuations for the Koba–Nielsen amplitudes as functions of the kinematic parameters. The Koba–Nielsen local zeta functions are algebraic-geometric integrals that can be defined over arbitrary local fields (for instance R, C, Qp, Fp((T))), and it is completely natural to expect connections between these objects. The limit p tends to one of the Koba–Nielsen amplitudes give rise to new amplitudes which we have called Denef–Loeser amplitudes. Throughout the article, we have emphasized the explicit calculations in the cases of 4 and 5 points.

The motivic zeta functions of a matroid

We introduce motivic zeta functions for matroids. These zeta functions are defined as sums over the lattice points of Bergman fans, and in the realizable case, they coincide with the motivic Igusa zeta functions of hyperplane arrangements. We show that these motivic zeta functions satisfy a functional equation arising from matroid Poincar\'{e} duality in the sense of Adiprasito-Huh-Katz. In the process, we obtain a formula for the Hilbert series of the cohomology ring of a matroid, in the sense of Feichtner-Yuzvinsky. We then show that our motivic zeta functions specialize to the topological zeta functions for matroids introduced by van der Veer, and we compute the first two coefficients in the Taylor expansion of these topological zeta functions, providing affirmative answers to two questions posed by van der Veer.

Combinatorial analogs of topological zeta functions

In this article we introduce a new matroid invariant, a combinatorial analog of the topological zeta function of a polynomial. More specifically we associate to any ranked, atomic meet-semilattice L a rational function ZL(s), in such a way that when L is the lattice of flats of a complex hyperplane arrangement we recover the usual topological zeta function. The definition is in terms of a choice of a combinatorial analog of resolutions of singularities, and the main result is that ZL(s) does not depend on this choice and depends only on L. Known properties of the topological zeta function provide a source of potential ℂ-realisability test for matroids.

On p-adic string amplitudes in the limit p approaches to one

In this article we discuss the limit p approaches to one of tree-level p-adic open string amplitudes and its connections with the topological zeta functions. There is empirical evidence that p-adic strings are related to the ordinary strings in the p → 1 limit. Previously, we established that p-adic Koba-Nielsen string amplitudes are finite sums of multivariate Igusa’s local zeta functions, consequently, they are convergent integrals that admit meromorphic continuations as rational functions. The meromorphic continuation of local zeta functions has been used for several authors to regularize parametric Feynman amplitudes in field and string theories. Denef and Loeser established that the limit p → 1 of a Igusa’s local zeta function gives rise to an object called topological zeta function. By using Denef-Loeser’s theory of topological zeta functions, we show that limit p → 1 of tree-level p-adic string amplitudes give rise to certain amplitudes, that we have named Denef-Loeser string amplitudes. Gerasimov and Shatashvili showed that in limit p → 1 the well-known non-local effective Lagrangian (reproducing the tree-level p-adic string amplitudes) gives rise to a simple Lagrangian with a logarithmic potential. We show that the Feynman amplitudes of this last Lagrangian are precisely the amplitudes introduced here. Finally, the amplitudes for four and five points are computed explicitly.

Enumerating graded ideals in graded rings associated to free nilpotent Lie rings

We compute the zeta functions enumerating graded ideals in the graded Lie rings associated with the free d-generator Lie rings $$\mathfrak {f}_{c,d}$$ of nilpotency class c for all $$c\leqslant 2$$ and for $$(c,d)\in \{(3,3),(3,2),(4,2)\}$$ . We apply our computations to obtain information about $$\mathfrak {p}$$ -adic, reduced, and topological zeta functions, in particular pertaining to their degrees and some special values.

Monodromy conjecture and the Hessian differential form

In this paper it is shown that the local topological zeta function of a germ associated with its Hessian differential form does not satisfy the monodromy conjecture, i.e., that the Hessian form is not an allowed differential 2-form.

Stability results for local zeta functions of groups algebras, and modules

AbstractVarious types of local zeta functions studied in asymptotic group theory admit two natural operations: (1) change the prime and (2) perform local base extensions. Often, the effects of both of the preceding operations can be expressed simultaneously in terms of a single formula, a statement made precise using what we call local maps of Denef type. We show that assuming the existence of such formulae, the behaviour of local zeta functions under variation of the prime in a set of density 1 in fact completely determines these functions for almost all primes and, moreover, it also determines their behaviour under local base extensions. We discuss applications to topological zeta functions, functional equations, and questions of uniformity.

Bernstein–Sato polynomials of hyperplane arrangements

We show that the Bernstein–Sato polynomial (that is, the b-function) of a hyperplane arrangement with a reduced equation is calculable by combining a generalization of Malgrange’s formula with the theory of Aomoto complexes due to Esnault, Schechtman, Terao, Varchenko, and Viehweg in certain cases. We prove in general that the roots are greater than $$-2$$ and the multiplicity of the root $$-1$$ is equal to the (effective) dimension of the ambient space. We also give an estimate of the multiplicities of the roots in terms of the multiplicities of the arrangement at the dense edges, and provide a method to calculate the Bernstein–Sato polynomial at least in the case of 3 variables with degree at most 7 and generic multiplicities at most 3. Using our argument, we can terminate the proof of a conjecture of Denef and Loeser on the relation between the topological zeta function and the Bernstein–Sato polynomial of a reduced hyperplane arrangement in the 3 variable case.

Computing topological zeta functions of groups, algebras, and modules, II

Building on our previous work, we develop the first practical algorithm for computing topological zeta functions of nilpotent groups, possibly non-associative algebras, and modules. While we previously depended upon non-degeneracy assumptions, the theory developed here allows us to overcome these restrictions in various interesting cases.

Computing topological zeta functions of groups, algebras, and modules, I

We develop techniques for computing zeta functions associated with nilpotent groups, not necessarily associative algebras, and modules, as well as Igusa-type zeta functions. At the heart of our method lies an explicit convex-geometric formula for a class of $p$-adic integrals under non-degeneracy conditions with respect to associated Newton polytopes. Our techniques prove to be especially useful for the computation of topological zeta functions associated with algebras, resulting in the first systematic investigation of their properties.

The monodromy conjecture for plane meromorphic germs

A notion of Milnor fibration for meromorphic functions and the corresponding concepts of monodromy and monodromy zeta function have been introduced by Gusein-Zade, Luengo and Melle-Hernández [‘Zeta functions for germs of meromorphic functions, and Newton diagrams’, Funct. Anal. Appl. 32 (1998)]. In this article, we define the topological zeta function for meromorphic germs and we study its poles in the plane case. We show that the poles do not behave as in the holomorphic case but still do satisfy a generalization of the monodromy conjecture.

Arcs on determinantal varieties

We study arc spaces and jet schemes of generic determinantal varieties. Using the natural group action, we decompose the arc spaces into orbits, and analyze their structure. This allows us to compute the number of irreducible components of jet schemes, log canonical thresholds, and topological zeta functions.

Zeta functions and Bernstein–Sato polynomials for ideals in dimension two

For a nonzero ideal I of C[x_1,...,x_n], with 0 in supp I, a generalization of a conjecture of Igusa - Denef - Loeser predicts that every pole of its topological zeta function is a root of its Bernstein-Sato polynomial. However, typically only a few roots are obtained this way. Following ideas of Veys, we study the following question. Is it possible to find a collection G of polynomials g in C[x_1,...,x_n], such that, for all g in G, every pole of the topological zeta function associated to I and the volume form gdx on the affine n-space, is a root of the Bernstein-Sato polynomial of I, and such that all roots are realized in this way. We obtain a negative answer to this question, providing counterexamples for monomial and principal ideals in dimension two, and give a partial positive result as well.

Generalized Monodromy Conjecture in dimension two

The aim of the article is an extension of the Monodromy Conjecture of Denef and Loeser in dimension two, incorporating zeta functions with differential forms and targeting all monodromy eigenvalues, and also considering singular ambient spaces. That is, we treat in a conceptual unity the poles of the (generalized) topological zeta function and the monodromy eigenvalues associated with a complex analytic function germ f defined on a normal surface singularity (X,0). The article targets the `right' extension in the case when the link of (X,0) is a homology sphere. As a first step, we prove a splice decomposition formula for the topological zeta function Z(f,w;s) for any f and analytic differential form w, which will play the key technical localization tool in the later definitions and proofs. Then, we define a set of `allowed' differential forms via a local restriction along each splice component. For plane curves we show the following three guiding properties: (1) if s' is any pole of Z(f,w;s) with w allowed, then exp(2\pi is') is a monodromy eigenvalue of f; (2) the `standard' form is allowed; (3) every monodromy eigenvalue of f is obtained as in (1) for some allowed w and some s'. For general (X,0) we prove (1) unconditionally, and (2)-(3) under an additional (necessary) assumption, which generalizes the semigroup condition of Neumann-Wahl. Several examples illustrate the definitions and support the basic assumptions.

The holomorphy conjecture for ideals in dimension two

The holomorphy conjecture predicts that the topological zeta function associated to a polynomial f ∈ C [ x 1 , … , x n ] f \in \mathbb {C}[x_1,\ldots ,x_n] and an integer d > 0 d > 0 is holomorphic unless d d divides the order of an eigenvalue of local monodromy of f f . In this paper, we generalise the holomorphy conjecture to the setting of arbitrary ideals in C [ x 1 , … , x n ] \mathbb {C}[x_1,\ldots ,x_n] , and we prove it when n = 2 n=2 .

Monodromy conjecture for nondegenerate surface singularities

We prove the monodromy conjecture for the topological zeta function for all non-degenerate surface singularities. Fundamental in our work is a detailed study of the formula for the zeta function of monodromy by Varchenko and the study of the candidate poles of the topological zeta function yielded by what we call 'B1-facets'. In particular, new cases among the nondegenerate surface singularities for which the monodromy conjecture is proven now, are the non-isolated singularities, the singularities giving rise to a topological zeta function with multiple candidate pole and the ones for which the Newton polyhedron contains a B1-facet.

The Monodromy Conjecture for hyperplane arrangements

The Monodromy Conjecture asserts that if c is a pole of the local topological zeta function of a hypersurface, then exp(2πic) is an eigenvalue of the monodromy on the cohomology of the Milnor fiber. A stronger version of the conjecture asserts that every such c is a root of the Bernstein-Sato polynomial of the hypersurface. In this note we prove the weak version of the conjecture for hyperplane arrangements. Furthermore, we reduce the strong version to the following conjecture: −n/d is always a root of the Bernstein-Sato polynomial of an indecomposable essential central hyperplane arrangement of d hyperplanes in C n .

Poles of the topological zeta function for plane curves and Newton polyhedra

The local topological zeta function is a rational function associated to a germ of a complex holomorphic function. This function can be computed from an embedded resolution of singularities of the germ. For functions that are nondegenerate with respect to their Newton polyhedron it is also possible to compute it from the Newton polyhedron. Both ways give rise to a set of candidate poles of the topological zeta function, containing all poles. For plane curves, W. Veys showed how to filter the actual poles out of the candidate poles induced by the resolution graph. In this Note we show how to determine from the Newton polyhedron of a nondegenerate plane curve which candidate poles are actual poles. To cite this article: A. Lemahieu, L. Van Proeyen, C. R. Acad. Sci. Paris, Ser. I 347 (2009).