Proof of a conjecture of Pólya on the zeros of successive derivatives of real entire functions

We prove Pólya’s conjecture of 1943: For a real entire function of order greater than 2 with finitely many non-real zeros, the number of non-real zeros of the nth derivative tends to infinity, as . We use the saddle point method and potential theory, combined with the theory of analytic functions with positive imaginary part in the upper half-plane.     

Minimum relative error approximations for 1/t

Summary We give explicit solutions to the problem of minimizing the relative error for polynomial approximations to 1/t on arbitrary finite subintervals of (0, ). We give a simple algorithm, using synthetic division, for computing practical representations of the best approximating polynomials. The resulting polynomials also minimize the absolute error in a related functional equation. We show that, for any continuous function with no zeros on the interval of interest, the geometric convergence rates for best absolute error and best relative error approximants must be equal. The approximation polynomials for 1/t are useful for finding suitably precise initial approximations in iterative methods for computing reciprocals on computers.     

On Locating Clusters of Zeros of Analytic Functions

Given an analytic function f and a Jordan curve that does not pass through any zero of f, we consider the problem of computing all the zeros of f that lie inside , together with their respective multiplicities. Our principal means of obtaining information about the location of these zeros is a certain symmetric bilinear form that can be evaluated via numerical integration along . If f has one or several clusters of zeros, then the mapping from the ordinary moments associated with this form to the zeros and their respective multiplicities is very ill-conditioned. We present numerical methods to calculate the centre of a cluster and its weight, i.e., the arithmetic mean of the zeros that form a certain cluster and the total number of zeros in this cluster, respectively. Our approach relies on formal orthogonal polynomials and rational interpolation at roots of unity. Numerical examples illustrate the effectiveness of our techniques.     

The holomorphic flow of the Riemann zeta function

The flow of the Riemann zeta function, , is considered, and phase portraits are presented. Attention is given to the characterization of the flow lines in the neighborhood of the first 500 zeros on the critical line. All of these zeros are foci. The majority are sources, but in a small proportion of exceptional cases the zero is a sink. To produce these portraits, the zeta function was evaluated numerically to 12 decimal places, in the region of interest, using the Chebyshev method and using Mathematica.

The phase diagrams suggest new analytic properties of zeta, of which some are proved and others are given in the form of conjectures.

     

Strict estimation of the maximum of a function of one variable

We describe an algorithm for the strict estimation of max _xI f(x) whenf is analytic onI. If roundoff is neglected the error bound can be made arbitrarily small by an adaptive implementation. A majorant concept is utilized and an Algol program is given. Finally we outline the special application to the strict estimation of the range of a polynomial without any determination of the zeros of the derivative.     

Le critère de Li et l'hypothèse de Riemann pour la classe de Selberg

We generalise Li's criterion, already known for the Riemann zeta function, to a large class of Dirichlet series. We give first an explicit formula for the coefficients , for all positive integers n and ρ runs over all the non-trivial zeros of a function F in this class. To do so, we use the Weil Explicit Formula.     

Spectral Scattering Theory for Automorphic Forms

We construct a scattering process for L²-automorphic forms on the quotient of the upper half plane by a cofinite discrete subgroup Γ of . The construction is algebraic besides being analytic in the sense that we use some relations satisfied by real-analytic Eisenstein series with a complex parameter. Thanks to this feature, the construction of our operators and spaces is explicit. We show some properties of the Lax-Phillips generator on a scattering subspace carved out from this process. We prove that the spectrum of this operator consists only of eigenvalues, which correspond to the nontrivial zeros, counted with multiplicity, of the Dirichlet series appearing in the functional equation of the Eisenstein series. In particular, in the case of the (full) modular group , the Dirichlet series reduces to the Riemann zeta function ζ, thereby we obtain a spectral interpretation of the nontrivial zeros of ζ.      

The two-by-two spectral Nevanlinna-Pick problem

We give a criterion for the existence of an analytic matrix-valued function on the disc satisfying a finite set of interpolation conditions and having spectral radius bounded by . We also give a realization theorem for analytic functions from the disc to the symmetrised bidisc.

     

On efficient computation and asymptotic sharpness of Kalantari's bounds for zeros of polynomials

We study an infinite family of lower and upper bounds on the modulus of zeros of complex polynomials derived by Kalantari. We first give a simple characterization of these bounds which leads to an efficient algorithm for their computation. For a polynomial of degree our algorithm computes the first bounds in Kalantari's family in operations. We further prove that for every complex polynomial these lower and upper bounds converge to the tightest annulus containing the roots, and thus settle a problem raised in Kalantari's paper.

     

Adaptative Decomposition: The Case of the Drury–Arveson Space

The maximum selection principle allows to give expansions, in an adaptive way, of functions in the Hardy space \(\mathbf H_2\) of the disk in terms of Blaschke products. The expansion is specific to the given function. Blaschke factors and products have counterparts in the unit ball of \(\mathbb C^N\), and this fact allows us to extend in the present paper the maximum selection principle to the case of functions in the Drury–Arveson space of functions analytic in the unit ball of \(\mathbb C^N\). This will give rise to an algorithm which is a variation in this higher dimensional case of the greedy algorithm. We also introduce infinite Blaschke products in this setting and study their convergence.     

Applications of branched values to p-adic functional equations on analytic functions

Let \(\mathbb{K}\) be an algebraically closed field of characteristic 0, complete with respect to an ultrametric absolute value. Results on branched values obtained in a previous paper are used to prove that algebraic functional equations of the form g ^q = hf ^q + w have no solution among transcendental entire functions f, g or among unbounded analytic functions inside an open disk, when w is a polynomial or a bounded analytic function and h is a polynomial or an analytic function whose zeros are of order multiple of q. We also show that an analytic function whose zeros are multiple of an integer q inside a disk is the q-th power of another analytic function, provided q is a prime to the residue characteristic.     

Low-lying zeros of families of elliptic curves

There is a growing body of work giving strong evidence that zeros of families of -functions follow distribution laws of eigenvalues of random matrices. This philosophy is known as the random matrix model or the Katz-Sarnak philosophy. The random matrix model makes predictions for the average distribution of zeros near the central point for families of -functions. We study these low-lying zeros for families of elliptic curve -functions. For these -functions there is special arithmetic interest in any zeros at the central point (by the conjecture of Birch and Swinnerton-Dyer and the impressive partial results towards resolving the conjecture).

We calculate the density of the low-lying zeros for various families of elliptic curves. Our main foci are the family of all elliptic curves and a large family with positive rank. An important challenge has been to obtain results with test functions that are concentrated close to the origin since the central point is a location of great arithmetical interest. An application of our results is an improvement on the upper bound of the average rank of the family of all elliptic curves (conditional on the Generalized Riemann Hypothesis (GRH)). The upper bound obtained is less than , which shows that a positive proportion of curves in the family have algebraic rank equal to analytic rank and finite Tate-Shafarevich group. We show that there is an extra contribution to the density of the low-lying zeros from the family with positive rank (presumably from the ``extra" zero at the central point).

     

On the computation of the Galois group over the quotient field of ℂ[λ]

Summary We give an algorithm for the computation of the Galois group of the splitting field of polynomials in two variables with integer coefficients over the quotient field (), (the rational functions in ). The algorithm uses a constructive version of the Newton polygon method and analytic continuations.Supported in part by the Fonds National Suisse     

Weierstrass formula and zero-finding methods

Summary. Classical Weierstrass' formula [29] has been often the subject of investigation of many authors. In this paper we give some further applications of this formula for finding the zeros of polynomials and analytic functions. We are concerned with the problems of localization of polynomial zeros and the construction of iterative methods for the simultaneous approximation and inclusion of these zeros. Conditions for the safe convergence of Weierstrass' method, depending only on initial approximations, are given. In particular, we study polynomials with interval coefficients. Using an interval version of Weierstrass' method enclosures in the form of disks for the complex-valued set containing all zeros of a polynomial with varying coefficients are obtained. We also present Weierstrass-like algorithm for approximating, simultaneously, all zeros of a class of analytic functions in a given closed region. To demonstrate the proposed algorithms, three numerical examples are included. Received September 13, 1993     

The pluripolar hull of a graph and fine analytic continuation

We show that if the graph of an analytic function in the unit disk D is not complete pluripolar in C² then the projection of its pluripolar hull contains a fine neighborhood of a point . Moreover the projection of the pluripolar hull is always finely open. On the other hand we show that if an analytic function f in D extends to a function ℱ which is defined on a fine neighborhood of a point and is finely analytic at p then the pluripolar hull of the graph of f contains the graph of ℱ over a smaller fine neighborhood of p. We give several examples of functions with this property of fine analytic continuation. As a corollary we obtain new classes of analytic functions in the disk which have non-trivial pluripolar hulls, among them C^∞ functions on the closed unit disk which are nowhere analytically extendible and have infinitely-sheeted pluripolar hulls. Previous examples of functions with non-trivial pluripolar hull of the graph have fine analytic continuation.     

A Polynomial Cutting Surfaces Algorithm for the Convex Feasibility Problem Defined by Self-Concordant Inequalities

Consider a nonempty convex set in ^m which is defined by a finite number of smooth convex inequalities and which admits a self-concordant logarithmic barrier. We study the analytic center based column generation algorithm for the problem of finding a feasible point in this set. At each iteration the algorithm computes an approximate analytic center of the set defined by the inequalities generated in the previous iterations. If this approximate analytic center is a solution, then the algorithm terminates; otherwise either an existing inequality is shifted or a new inequality is added into the system. As the number of iterations increases, the set defined by the generated inequalities shrinks and the algorithm eventually finds a solution of the problem. The algorithm can be thought of as an extension of the classical cutting plane method. The difference is that we use analytic centers and convex cuts instead of arbitrary infeasible points and linear cuts. In contrast to the cutting plane method, the algorithm has a polynomial worst case complexity of O(Nlog 1/) on the total number of cuts to be used, where N is the number of convex inequalities in the original problem and is the maximum common slack of the original inequality system.