Poles of Zeta Functions on Normal Surfaces

Let (S, 0) be a normal surface germ and Let f a non-constantregular function on Let (S, 0) with Let f(0) = 0. Using anyadditive invariant on complex algebraic varieties one can associatea zeta function to these data, where the topological and motiviczeta functions are the roughest and the finest zeta functions,respectively. In this paper we are interested in a geometricdetermination of the poles of these functions. The second authorhas already provided such a determination for the topologicalzeta function in the case of non-singular surfaces. Here wegive a complete answer for all normal surfaces, at least onthe motivic level. The topological zeta function however seemsto be too rough for this purpose, although for negative poles,which are the only ones in the non-singular case, we are ableto prove exactly the same result as for non-singular surfaces. We also give and verify a (natural) definition for when a rationalnumber is a pole of the motivic zeta function. 2000 MathematicsSubject Classification 14B05, 14E15, 14J17 (primary), 32S50(secondary).     

Zeta Functions for Curves and Log Canonical Models

The topological zeta function and Igusa's local zeta functionare respectively a geometrical invariant associated to a complexpolynomial f and an arithmetical invariant associated to a polynomialf over a p-adic field. When f is a polynomial in two variables we prove a formula forboth zeta functions in terms of the so-called log canonicalmodel of f^-1{0} in A². This result yields moreover a conceptualexplanation for a known cancellation property of candidate polesfor these zeta functions. Also in the formula for Igusa's localzeta function appears a remarkable non-symmetric ‘q-deformation’of the intersection matrix of the minimal resolution of a Hirzebruch-Jungsingularity. 1991 Mathematics Subject Classification: 32S5011S80 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. The author is a Postdoctoral Fellow of the Belgian National Fund for Scientific Research N.F.W.O.     

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).     

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

To an ideal in 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. The research was partially supported by the Fund of Scientific Research—Flanders (G.0318.06).     

Monodromy eigenvalues and zeta functions with differential forms

For a complex polynomial or analytic function f, there is a strong correspondence between poles of the so-called local zeta functions or complex powers ∫|f|^2sω, where the ω are C^∞ differential forms with compact support, and eigenvalues of the local monodromy of f. In particular Barlet showed that each monodromy eigenvalue of f is of the form , where s₀ is such a pole. We prove an analogous result for similar p-adic complex powers, called Igusa (local) zeta functions, but mainly for the related algebro-geometric topological and motivic zeta functions.     

Growth and poles of meromorphic solutions of some difference equations

This paper is devoted to studying the growth property and the pole distribution of meromorphic solutions f of some complex difference equations with all coefficients being rational functions or of growth S(r,f). We find the lower bound of the lower order, or the relation between lower order and the convergence exponent of poles of meromorphic solutions of such equations.     

Harmonic Analogues of G. R. Maclane's Universal Functions

Let E denote the space of all entire functions, equipped withthe topology of local uniform convergence (the compact-opentopology). MacLane [15] constructed an entire function f whosesequence of derivatives (f, f', f', ...) is dense in E; hisconstruction is succinctly presented in a much later note byBlair and Rubel [2], who unwittingly rederived it (see also[3]). We shall call such a function f a universal entire function.In this note we show that analogous universal functions existin the space H_N of functions harmonic on R^N, where N2. We alsostudy the permissible growth rates of universal functions inH_N and show that the set of all such functions is very large. For purposes of comparison, we first review relevant facts aboutuniversal entire functions. The function constructed by MacLaneis of exponential type 1. Duyos Ruiz [7] observed that a universalentire function cannot be of exponential type less than 1. G.Herzog [11] refined MacLane's growth estimate by proving theexistence of a universal entire function f such that |f(z)|=O(re^r)as |z|=r. Finally, Grosse–Erdmann [10] proved the followingsharp result.     

The Generation of Chisholm Rational Polynomial Approximants to Power Series in Two Variables

The generation of successive Chisholm rational polynomial approximantsf_m/_m of f(x, y), a power series in two variables, is discussed. A necessary and sufficient condition for the non-degeneracyof f_m/_m is given. It is shown that the non-degeneracy of thediagonal Pad? approximants of order m in each variable separatelyis a necessary condition for the non-degeneracy of f_m/_m. In the case of a symmetric function, it is proved that the Chisholmapproximant f_m/_m is symmetric and non-degenerate if and onlyif all the diagonal Pad? approximants of order up to m in onevariable are non-degenerate. The generation of successive Chisholmapproximants to symmetric functions is also considered. The computational scheme, called the prong method, extends tocover the computation of Chisholm approximants in N-variables(Chisholm & McEwan, 1974).     

On multiply connected wandering domains of meromorphic functions

We describe conditions under which a multiply connected wanderingdomain of a transcendental meromorphic function with a finitenumber of poles must be a Baker wandering domain, and we discussthe possible eventual connectivity of Fatou components of transcendentalmeromorphic functions. We also show that if f is meromorphic,U is a bounded component of F(f) and V is the component of F(f)such that f(U)V, then f maps each component of U onto a componentof the boundary of V in . We give examples which show that our results are sharp; for example,we prove that a multiply connected wandering domain can mapto a simply connected wandering domain, and vice versa.     

A Strong Notion of Universal Taylor Series

For a holomorphic function f in the open unit disc D, the Nthpartial sum of its Taylor series with center D is denotedby S_N(f,)(z)= . Generically, all functions f in H(D) satisfy the following. For every compactset K C with KD=Ø and K^c connected and every polynomialh, there exists a sequence of positive integers such that, for every 1 {0,1,2,...},      

On the Poles of Igusa's Local Zeta Function for Algebraic Sets

Let K be a p-adic field, let Z (s, f), sC, with Re(s) > 0,be the Igusa local zeta function associated to f(x) = (f₁(x),..., f_l(x)) [K (x₁, ..., x_n)]^l, and let be a Schwartz–Bruhatfunction. The aim of this paper is to describe explicitly thepoles of the meromorphic continuation of Z (s, f). Using resolutionof singularities it is possible to express Z (s, f) as a finitesum of p-adic monomial integrals. These monomial integrals arecomputed explicitly by using techniques of toroidal geometry.In this way, an explicit list of the candidates for poles ofZ (s, f) is obtained. 2000 Mathematics Subject Classification11S40, 14M25, 11D79.     

On the derivative of meromorphic functions with multiple zeros

Let f be a transcendental meromorphic function and let R be a rational function, R?0. We show that if all zeros and poles of f are multiple, except possibly finitely many, then f′−R has infinitely many zeros. If f has finite order and R is a polynomial, then the conclusion holds without the hypothesis that poles be multiple.     

Dimensions of Julia Sets of Hyperbolic Entire Functions

It is known that, if f is a hyperbolic rational function, thenthe Hausdorff, packing and box dimensions of the Julia set J(f)are equal. It is also known that there is a family of hyperbolictranscendental meromorphic functions with infinitely many polesfor which this result fails to be true. In this paper, new methodsare used to show that there is a family of hyperbolic transcendentalentire functions f_K, K N, such that the box and packing dimensionsof Jf_K are equal to two, even though as K the Hausdorff dimensionof Jf_K tends to one, the lowest possible value for the Hausdorffdimension of the Julia set of a transcendental entire function.2000 Mathematics Subject Classification 30D05, 37F10, 37F15,37F35, 37F50.     

On Hardy, BMO and Lipschitz spaces of invariantly harmonic functions in the unit ball

The invariantly harmonic functions in the unit ball Bⁿ in Cⁿare those annihilated by the Bergman Laplacian . The Poisson-Szegökernel P(z,) solves the Dirichlet problem for : if f C(Sⁿ),the Poisson-Szegö transform of f, where d is the normalized Lebesgue measure on Sⁿ,is the unique invariantly harmonic function u in Bⁿ, continuousup to the boundary, such that u=f on Sⁿ. The Poisson-Szegötransform establishes, loosely speaking, a one-to-one correspondencebetween function theory in Sⁿ and invariantly harmonic functiontheory in Bⁿ. When n 2, it is natural to consider on Sⁿ functionspaces related to its natural non-isotropic metric, for theseare the spaces arising from complex analysis. In the paper,different characterizations of such spaces of smooth functionsare given in terms of their invariantly harmonic extensions,using maximal functions and area integrals, as in the correspondingEuclidean theory. Particular attention is given to characterizationin terms of purely radial or purely tangential derivatives.The smoothness is measured in two different scales: that ofSobolev spaces and that of Lipschitz spaces, including BMO andBesov spaces. 1991 Mathematics Subject Classification: 32A35,32A37, 32M15, 42B25.     

Discriminant of a Germ {Phi}: (C2, 0)->(C2, 0) and Seifert Fibred Manifolds

Let f and g be two analytic function germs without common branches.We define the Jacobian quotients of (g, f), which are ‘firstorder invariants’ of the discriminant curve of (g, f),and we prove that they only depend on the topological type of(g, f). We compute them with the help of the topology of (g,f). If g is a linear form transverse to f, the Jacobian quotientsare exactly the polar quotients of f and we affirm the resultsof D. T. Lê, F. Michel and C. Weber.     

Symmetrization and norm of the Hardy-Littlewood maximal operator on Lorentz and Marcinkiewicz spaces

We prove that when a function on the real line is symmetricallyrearranged, the distribution function of its uncentered Hardy–Littlewoodmaximal function increases pointwise, while it remains unchangedonly when the function is already symmetric. Equivalently, if is the maximal operator and the symmetrization, then f(x)f(x)for every x, and equality holds for all x if and only if, upto translations, f(x) = f(x) almost everywhere. Using theseresults, we then compute the exact norms of the maximal operatoracting on Lorentz and Marcinkiewicz spaces, and we determineextremal functions that realize these norms.     

On Maximal Regularity and Semivariation of Cosine Operator Functions

It is proved that a cosine operator function C(·), withgenerator A, is locally of bounded semivariation if and onlyif u'(t) = Au(t)+f(t), t>0, u(0), u'(0)D(A), has a strongsolution for every continuous function f, if and only if thefunction , is twice continuously differentiable for every continuous function f, that is, C(·)has the maximal regularity property if and only if A is a boundedoperator. Some other characterisations of bounded generatorsof cosine operator functions are also established in terms oftheir local semivariations.     

The Second Derivative of a Meromorphic Function of Finite Order

Let f be meromorphic of finite order in the plane, such thatthe second derivative f' has finitely many zeros. Then f hasfinitely many poles. This result was conjectured by the authorin 1996, and an example shows that the theorem is sharp.