Families of irreducible polynomials of Gaussian periods and matrices of cyclotomic numbers

Given an odd prime we show a way to construct large families of polynomials , , where is a set of primes of the form mod and is the irreducible polynomial of the Gaussian periods of degree in . Examples of these families when are worked in detail. We also show, given an integer and a prime mod , how to represent by matrices the Gaussian periods of degree in , and how to calculate in a simple way, with the help of a computer, irreducible polynomials for elements of .

     

Random Fibonacci sequences and the number

For the familiar Fibonacci sequence (defined by , and for ), increases exponentially with at a rate given by the golden ratio . But for a simple modification with both additions and subtractions - the random Fibonacci sequences defined by , and for , , where each sign is independent and either or - with probability - it is not even obvious if should increase with . Our main result is that

with probability . Finding the number involves the theory of random matrix products, Stern-Brocot division of the real line, a fractal measure, a computer calculation, and a rounding error analysis to validate the computer calculation.

     

Computing

Let denote the Von Mangoldt function and . We describe an elementary method for computing isolated values of . The complexity of the algorithm is time and space. A table of values of for up to is included, and some times of computation are given.

     

An algorithm for evaluation of discrete logarithms in some nonprime finite fields

In this paper we propose an algorithm for evaluation of logarithms in the finite fields , where the number has a small primitive factor . The heuristic estimate of the complexity of the algorithm is equal to
, where grows to , and is limited by a polynomial in . The evaluation of logarithms is founded on a new congruence of the kind of D. Coppersmith, , which has a great deal of solutions-pairs of polynomials of small degrees.

     

Nonexistence conditions of a solution for the congruence

We obtain nonexistence conditions of a solution for of the congruence , where , and are integers, and is a prime power. We give nonexistence conditions of the form for , , , , , and of the form for , , , . Furthermore, we complete some tables concerned with Waring's problem in -adic fields that were computed by Hardy and Littlewood.

     

Vector subdivision schemes and multiple wavelets

We consider solutions of a system of refinement equations written in the form

where the vector of functions is in and is a finitely supported sequence of matrices called the refinement mask. Associated with the mask is a linear operator defined on by . This paper is concerned with the convergence of the subdivision scheme associated with , i.e., the convergence of the sequence in the -norm.

Our main result characterizes the convergence of a subdivision scheme associated with the mask in terms of the joint spectral radius of two finite matrices derived from the mask. Along the way, properties of the joint spectral radius and its relation to the subdivision scheme are discussed. In particular, the -convergence of the subdivision scheme is characterized in terms of the spectral radius of the transition operator restricted to a certain invariant subspace. We analyze convergence of the subdivision scheme explicitly for several interesting classes of vector refinement equations.

Finally, the theory of vector subdivision schemes is used to characterize orthonormality of multiple refinable functions. This leads us to construct a class of continuous orthogonal double wavelets with symmetry.

     

Perturbing polynomials with all their roots on the unit circle

Given a monic real polynomial with all its roots on the unit circle, we ask to what extent one can perturb its middle coefficient and still have a polynomial with all its roots on the unit circle. We show that the set of possible perturbations forms a closed interval of length at most , with achieved only for polynomials of the form with in . The problem can also be formulated in terms of perturbing the constant coefficient of a polynomial having all its roots in . If we restrict to integer coefficients, then the polynomials in question are products of cyclotomics. We show that in this case there are no perturbations of length that do not arise from a perturbation of length . We also investigate the connection between slightly perturbed products of cyclotomic polynomials and polynomials with small Mahler measure. We describe an algorithm for searching for polynomials with small Mahler measure by perturbing the middle coefficients of products of cyclotomic polynomials. We show that the complexity of this algorithm is , where is the degree, and we report on the polynomials found by this algorithm through degree 64.

     

Fast algorithms for discrete polynomial transforms

Consider the Vandermonde-like matrix , where the polynomials satisfy a three-term recurrence relation. If are the Chebyshev polynomials , then coincides with . This paper presents a new fast algorithm for the computation of the matrix-vector product in arithmetical operations. The algorithm divides into a fast transform which replaces with and a subsequent fast cosine transform. The first and central part of the algorithm is realized by a straightforward cascade summation based on properties of associated polynomials and by fast polynomial multiplications. Numerical tests demonstrate that our fast polynomial transform realizes with almost the same precision as the Clenshaw algorithm, but is much faster for .

     

Factorization of the tenth Fermat number

We describe the complete factorization of the tenth Fermat number by the elliptic curve method (ECM). is a product of four prime factors with 8, 10, 40 and 252 decimal digits. The 40-digit factor was found after about 140 Mflop-years of computation. We also discuss the complete factorization of other Fermat numbers by ECM, and summarize the factorizations of .