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).     

On the Poles of Maximal Order of the Topological Zeta Function

The global and local topological zeta functions are singularityinvariants associated to a polynomial f and its germ at 0, respectively.By definition, these zeta functions are rational functions inone 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 mostone such pole, in which case it is also the largest pole; wegive a similar result concerning the global zeta function. Moreover,for any f we show that poles of maximal possible order are alwaysof the form –1/N with N a positive integer. 1991 MathematicsSubject Classification 14B05, 14E15, 32S50.     

Igusa's local zeta functions of semiquasihomogeneous polynomials

In this paper, we prove the rationality of Igusa's local zeta functions of semiquasihomogeneous polynomials with coefficients in a non-archimedean local field . The proof of this result is based on Igusa's stationary phase formula and some ideas on Néron -desingularization.

     

Deformation Theory and The Computation of Zeta Functions

We present a new approach to the problem of computing the zetafunction of a hypersurface over a finite field. For a hypersurfacedefined by a polynomial of degree d in n variables over thefield of q elements, one desires an algorithm whose runningtime is a polynomial function of dⁿ log(q). (Here we assumed 2, for otherwise the problem is easy.) The case n = 1 isrelated to univariate polynomial factorisation and is comparativelystraightforward. When n = 2 one is counting points on curves,and the method of Schoof and Pila yields a complexity of , where the function C_d depends exponentiallyon d. For arbitrary n, the theorem of the author and Wan givesa complexity which is a polynomial function of (pdⁿ log(q))ⁿ,where p is the characteristic of the field. A complexity estimateof this form can also be achieved for smooth hypersurfaces usingthe method of Kedlaya, although this has only been worked outin full for curves. The new approach we present should yielda complexity which is a small polynomial function of pdⁿ log(q).In this paper, we work this out in full for Artin–Schreierhypersurfaces defined by equations of the form Z^p – Z= f, where the polynomial f has a diagonal leading form. Themethod utilises a relative p-adic cohomology theory for familiesof hypersurfaces, due in essence to Dwork. As a corollary ofour main theorem, we obtain the following curious result. Letf be a multivariate polynomial with integer coefficients whoseleading form is diagonal. There exists an explicit deterministicalgorithm which takes as input a prime p, outputs the numberof solutions to the congruence equation f = 0 op, and runs in bit operations, for any >0. This improves upon the elementary estimate of bit operations, where n is the number of variables,which can be achieved using Berlekamp's root counting algorithm.2000 Mathematics Subject Classification 11Y99, 11M38, 11T99.     

All but Finitely Many Non-Trivial Zeros of the Approximations of the Epstein Zeta Function are Simple and on the Critical Line

The Chowla–Selberg formula is applied in approximatinga given Epstein zeta function. Partial sums of the series derivefrom the Chowla–Selberg formula, and although these partialsums satisfy a functional equation, as does an Epstein zetafunction, they do not possess an Euler product. What we callpartial sums throughout this paper may be considered as specialcases concerning a more general function satisfying a functionalequation only. In this article we study the distribution ofzeros of the function. We show that in any strip containingthe critical line, all but finitely many zeros of the functionare simple and on the critical line. For many Epstein zeta functionswe show that all but finitely many non-trivial zeros of partialsums in the Chowla–Selberg formula are simple and on thecritical line. 2000 Mathematics Subject Classification 11M26.     

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.     

Definiteness of the Peano Kernel Associated with the Polyharmonic Mean Value Property

The definiteness of the Peano kernel is proved for a functionalassociated with the mean-value property of Picone and Brambleand Payne for polyharmonic functions in the ball. An importantcorollary of this is that if a function f satisfying (–1)^ppf>0vanishes on p concentric spheres centered at 0, then f(0)>0.This generalizes a well-known property of subharmonic functions(which arise in the special case p = 1).     

The Le numbers of the square of a function and their applications

Lê numbers were introduced by Massey with the purposeof numerically controlling the topological properties of familiesof non-isolated hypersurface singularities and describing thetopology associated with a function with non-isolated singularities.They are a generalization of the Milnor number for isolatedhypersurface singularities. In this note the authors investigatethe composite of an arbitrary square-free f and z². They geta formula for the Lê numbers of the composite, and considertwo applications of these numbers. The first application isconcerned with the extent to which the Lê numbers areinvariant in a family of functions which satisfy some equisingularitycondition, the second is a quick proof of a new formula forthe Euler obstruction of a hypersurface singularity. Severalexamples are computed using this formula including any X definedby a function which only has transverse D(q, p) singularitiesoff the origin.     

An Optimal Representation Formula for Carnot-Caratheodory Vector Fields

The simplest example of the sort of representation formula thatwe shall study is the following familiar inequality for a smooth,real-valued function f(x) defined on a ball B in N-dimensionalEuclidean space R^N: [formula] where f denotes the gradient of f, f_B is the average |B|^–1_Bf(y)dy, |B| is the Lebesgue measure of B, and C is a constantwhich is independent of f, x and B. This formula can be found,for example, in [4] and [12]; see also the closely related estimatesin [20, pp. 228{231]. Indeed, such a formula holds in any boundedconvex domain. 1991 Mathematics Subject Classification 31B10,46E35, 35A22.     

A p Interpolation in the Unit Ball

We study interpolating sequences in the unit ball for A^–pwith p > 0, the Banach space of holomorphic functions f with(1 – |z|²)^p |f(z)| bounded. The finite unions of A^–p-interpolatingsequences are characterized by a Carleson type condition.     

On the smallest poles of Igusa's p-adic zeta functions

Let K be a p-adic field. We explore Igusa's p-adic zeta function, which is associated to a K-analytic function on an open and compact subset of Kⁿ. First we deduce a formula for an important coefficient in the Laurent series of this meromorphic function at a candidate pole. Afterwards we use this formula to determine all values less than −1/2 for n=2 and less than −1 for n=3 which occur as the real part of a pole.     

Finsler Metrics of Constant Positive Curvature on the Lie Group S3

Guided by the Hopf fibration, a family (indexed by a positiveconstant K) of right invariant Riemannian metrics on the Liegroup S³ is singled out. Using the Yasuda–Shimada paperas an inspiration, a privileged right invariant Killing fieldof constant length is determined for each K > 1. Each suchRiemannian metric couples with the corresponding Killing fieldto produce a y-global and explicit Randers metric on S³. Employingthe machinery of spray curvature and Berwald's formula, it isproved directly that the said Randers metric has constant positiveflag curvature K, as predicted by Yasuda–Shimada. It isexplained why this family of Finslerian space forms is not projectivelyflat.     

Another discrete Fourier transform pairs associated with the Lipschitz-Lerch zeta function

It is demonstrated that the alternating Lipschitz-Lerch zeta function and the alternating Hurwitz zeta function constitute a discrete Fourier transform pair. This discrete transform pair makes it possible to deduce, as special cases and consequences, many (mainly new) transformation relations involving the values at rational arguments of alternating variants of various zeta functions, such as the Lerch and Hurwitz zeta functions and Legendre chi function.     

Evaluation of Superharmonic Functions Using Limits Along Lines

If u is a superharmonic function on R², then [formula] for all (x, y) R². This follows from the fact that a line segmentin R² is non-thin at each of its constituent points. (See Doob[1, 1.XI] or Helms [7, Chapter 10] for an account of thin setsand the fine topology.) The situation is different in higherdimensions. For example, if u is the Newtonian potential onR³ defined by [formula] then [formula] Corollary 2 below will show that, nevertheless, for nearly everyvertical line L, the value of a superharmonic function at anypoint X of L is determined by its lower limit along L at X. Throughout this paper, we let n 3. A typical point of Rⁿ willbe denoted by X or (X', x), where X'R^n–1 and xR. Givenany function f:Rⁿ [–,+] and any point X, we define thevertical cluster set of f at X by [formula] and the fine cluster set of f at X by [formula] 1991 Mathematics Subject Classification 31B05.     

Pad?-approximant Bounds and Approximate Solutions for Kirkwood-Riseman Integral Equations

Two-point Pad? approximants are used to calculate tight upperand lower bounds on the quantity <?, f> associated withKirkwood-Riseman integral equations (1+yL)?=f, which arise inthe diffusion theory of flexible macromolecules. The self-adjointoperator L is an integral operator on –1 x 1, with weaklysingular kernel |x – x'|^–?, and the two specificcases (i) f = 1, (ii) f = x² are studied. In case (i) directbounds on <?, 1> are obtained; this quantity is inverselyproportional to the translational diffusion constant. In case(ii) bounds on <?, 1 > are found by a new technique involvingcombinations of bounds for the three cases f = 1, f = x² andf = bx²?b^–1. Various types of Pade and related approximantsare compared, using the information <f, Lⁿf>, n = –2,–1, 0, 1, 2, 3 and (an upper bound on L) for severalvalues of the positive parameter y. Pad?-approximant-generating trial vectors are investigated anda convergence theorem is established. The vector consistingof an optimum linear combination of L^–1f, f and Lf isfound to be an accurate approximation to a numerical solutionin case (ii), for all values of y and x. Specific analyticalexpressions are derived for the approximate solutions.     

Funk-Hecke Formula for Orthogonal Polynomials on Spheres and on Balls

Analogues of the Funk–Hecke formula for spherical harmonicsare proved for Dunkl's h-harmonics associated to the reflectiongroups, and for orthogonal polynomials related to h-harmonicson the unit ball. In particular, an analogue and its applicationare discussed for the weight function (1–|x|²)^µ–1/2on the unit ball in R^d. 2000 Mathematics Subject Classification33C50, 33C55, 42C10.     

Type 2 Semi-Algebras of Continuous Functions

A semi-algebra of continuous functions is a cone A of continuousreal functions on a compact Hausdorff space X such that A containsthe products of its elements. A cone A is said to be of typen if fA implies fⁿ(1 + f)^–1 A. Uniformly closed semi-algebrasof types 0 and 1 have long been characterized in a manner analogousto the Stone–Weierstrass theorem, but, except for thecase when A is generated by a single function, little has beenknown about type 2. Here, progress is reported on two problems.The first is the characterization of those continuous linearfunctionals on C(X) that determine semi-algebras of type 2.The second is the determination of the type of the tensor productof two type 1 semi-algebras. 1991 Mathematics Subject Classification:46J10.     

Zeta functions of three-dimensional <Emphasis Type="Italic">p</Emphasis>-adic Lie algebras

We give an explicit formula for the subalgebra zeta function of a general three-dimensional Lie algebra over the p-adic integers . To this end, we associate to such a Lie algebra a ternary quadratic form over . The formula for the zeta function is given in terms of Igusa’s local zeta function associated to this form. We acknowledge support from the Mathematisches Forschungsinstitut Oberwolfach and the Nuffield Foundation.     

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.     

An efficient algorithm for accelerating the convergence of oscillatory series,useful for computing the polylogarithm and Hurwitz zeta functions

This paper sketches a technique for improving the rate of convergence of a general oscillatory sequence, and then applies this series acceleration algorithm to the polylogarithm and the Hurwitz zeta function. As such, it may be taken as an extension of the techniques given by Borwein’s “An efficient algorithm for computing the Riemann zeta function” by Borwein for computing the Riemann zeta function, to more general series. The algorithm provides a rapid means of evaluating Li _s (z) for general values of complex s and a kidney-shaped region of complex z values given by ∣z ²/(z–1)∣<4. By using the duplication formula and the inversion formula, the range of convergence for the polylogarithm may be extended to the entire complex z-plane, and so the algorithms described here allow for the evaluation of the polylogarithm for all complex s and z values. Alternatively, the Hurwitz zeta can be very rapidly evaluated by means of an Euler–Maclaurin series. The polylogarithm and the Hurwitz zeta are related, in that two evaluations of the one can be used to obtain a value of the other; thus, either algorithm can be used to evaluate either function. The Euler–Maclaurin series is a clear performance winner for the Hurwitz zeta, while the Borwein algorithm is superior for evaluating the polylogarithm in the kidney-shaped region. Both algorithms are superior to the simple Taylor’s series or direct summation. The primary, concrete result of this paper is an algorithm allows the exploration of the Hurwitz zeta in the critical strip, where fast algorithms are otherwise unavailable. A discussion of the monodromy group of the polylogarithm is included.