An explicit Waldspurger formula for Hilbert modular forms II

The Ramanujan Journal - We describe a construction of preimages for the Shimura map on Hilbert modular forms using generalized theta series, and give an explicit Waldspurger type formula relating...     

An explicit spine for the Picard modular group over the Gaussian integers

Let Γ\D be an arithmetic quotient of a symmetric space of non-compact type. A spine D₀ is a Γ-equivariant deformation retraction of D with dimension equal to the virtual cohomological dimension of Γ. We explicitly construct a spine for the case of Γ=SU(2,1;Z[i]). The spine is then used to compute the cohomology of Γ\D with various local coefficients.     

An improvement of the complexity bound for solving systems of polynomial equations

In 1984, the author suggested an algorithm for solving systems of polynomial equations. Now we modify it and improve the bounds on its complexity as well as the degrees and lengths of coefficients from the ground filed of the elements constructed by this algorithm. Bibliography: 4 titles.     

An explicit bound for uniform perfectness of the Julia sets of rational maps

A compact set C in the Riemann sphere is called uniformly perfect if there is a uniform upper bound on the moduli of annuli which separate C. Julia sets of rational maps of degree at least two are uniformly perfect. Proofs have been given independently by Ma né and da Rocha and by Hinkkanen, but no explicit bounds are given. In this note, we shall provide such an explicit bound and, as a result, give another proof of uniform perfectness of Julia sets of rational maps of degree at least two. As an application, we provide a lower estimate of the Hausdorff dimension of the Julia sets. We also give a concrete bound for the family of quadratic polynomials in terms of the parameter c. Received: 7 June 1999; in final form: 9 November 1999 / Published online: 17 May 2001     

Measuring the height of a polynomial

Conclusions  Mahler’s measure is alive and well in several quite diverse contexts. The differing points of view seem to generate a healthy friction. If the general level of health is measured by the quantity and quality of unsolved problems, then it may help to list these.

An upper bound for the expected number of real zeros of a random polynomial

Suppose that the n + 1 coefficients of a polynomial of degree n are independent normally distributed random variables with mean 0 and variance 1. We prove that the expected number of real zeros of this polynomial is less than $\text{(}\text{2}\text{π}\text{)}\text{log}\text{n + 1.116}$, and that the constant may be reduced to 1.113 if n is even.     

An explicit upper bound for Weil-Petersson volumes of the moduli spaces of punctured Riemann surfaces

An explicit upper bound for the Weil-Petersson volumes of punctured Riemann surfaces is obtained using Penner's combinatorial integration scheme from [4]. It is shown that for a fixed number of punctures n and for genus g increasing, while this limit is exactly equal to two for n=1. Received: 17 May 2000 / Revised version: 9 August 2000 / Published online: 23 July 2001     

An explicit formula of Atkinson type for the product of the Riemann zeta-function and a Dirichlet polynomial

We prove an explicit formula of Atkinson type for the error term in the asymptotic formula for the mean square of the product of the Riemann zeta-function and a Dirichlet polynomial. To deal with the case when coefficients of the Dirichlet polynomial are complex, we apply the idea of the first author in his study on mean values of Dirichlet L-functions.     

On Cauchy's bound for zeros of a polynomial

In this paper we have improved Cauchy's bound for zeros of a polynomialp(z)=z ⁿ+a _n+1 z ^n-1+a _n-2 z ^n-2+......+a ₁ z+a ₀. Our result is best possible and sharpens some other results also.     

Elementary explicit types and polynomial time operations

This paper studies systems of explicit mathematics as introduced by Feferman [9, 11]. In particular, we propose weak explicit type systems with a restricted form of elementary comprehension whose provably terminating operations coincide with the functions on binary words that are computable in polynomial time. The systems considered are natural extensions of the first‐order applicative theories introduced in Strahm [19, 20] (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Sphere packings give an explicit bound for the Besicovitch Covering Theorem

We show that the number of disjointed families needed in the Besicovitch Covering Theorem equals the number of unit spheres that can be packed into a ball of radius five, with one at the center, and get estimates on this number.