Involutions and characters of upper triangular matrix groups

We study the realizability over of representations of the group of upper-triangular matrices over . We prove that all the representations of are realizable over if , but that if , has representations not realizable over . This theorem is a variation on a result that can be obtained by combining work of J. Arregi and A. Vera-López and of the authors, but the proof of the theorem in this paper is much more natural.

     

Solving norm equations in relative number fields using -units

In this paper, we are interested in solving the so-called norm equation , where is a given arbitrary extension of number fields and a given algebraic number of . By considering -units and relative class groups, we show that if there exists at least one solution (in , but not necessarily in ), then there exists a solution for which we can describe precisely its prime ideal factorization. In fact, we prove that under some explicit conditions, the -units that are norms are norms of -units. This allows us to limit the search for rational solutions to a finite number of tests, and we give the corresponding algorithm. When is an algebraic integer, we also study the existence of an integral solution, and we can adapt the algorithm to this case.

     

Bounds for computing the tame kernel

The tame kernel of the of a number field  is the kernel of some explicit map , where the product runs over all finite primes  of  and is the residue class field at . When is a set of primes of , containing the infinite ones, we can consider the -unit group  of . Then has a natural image in . The tame kernel is contained in this image if  contains all finite primes of  up to some bound. This is a theorem due to Bass and Tate. An explicit bound for imaginary quadratic fields was given by Browkin. In this article we give a bound, valid for any number field, that is smaller than Browkin's bound in the imaginary quadratic case and has better asymptotics. A simplified version of this bound says that we only have to include in  all primes with norm up to  , where  is the discriminant of . Using this bound, one can find explicit generators for the tame kernel, and a ``long enough' search would also yield all relations. Unfortunately, we have no explicit formula to describe what ``long enough' means. However, using theorems from Keune, we can show that the tame kernel is computable.

     

A hidden number problem in small subgroups

Boneh and Venkatesan have proposed a polynomial time algorithm for recovering a hidden element , where is prime, from rather short strings of the most significant bits of the residue of modulo for several randomly chosen . González Vasco and the first author have recently extended this result to subgroups of of order at least for all and to subgroups of order at least for almost all . Here we introduce a new modification in the scheme which amplifies the uniformity of distribution of the multipliers and thus extend this result to subgroups of order at least for all primes . As in the above works, we give applications of our result to the bit security of the Diffie-Hellman secret key starting with subgroups of very small size, thus including all cryptographically interesting subgroups.

     

On iterates of Möbius transformations on fields

Let be a quadratic polynomial over a splitting field , and be the set of zeros of . We define an associative and commutative binary relation on so that every Möbius transformation with fixed point set is of the form ``plus' for some . This permits an easy proof of Aitken acceleration as well as generalizations of known results concerning Newton's method, the secant method, Halley's method, and higher order methods. If is equipped with a norm, then we give necessary and sufficient conditions for the iterates of a Möbius transformation to converge (necessarily to one of its fixed points) in the norm topology. Finally, we show that if the fixed points of are distinct and the iterates of converge, then Newton's method converges with order 2, and higher order generalizations converge accordingly.

     

Extrapolation methods and derivatives of limits of sequences

Let be an infinite sequence whose limit or antilimit can be approximated very efficiently by applying a suitable extrapolation method E to . Assume that the and hence also are differentiable functions of some parameter , being the limit or antilimit of , and that we need to approximate . A direct way of achieving this would be by applying again a suitable extrapolation method E to the sequence , and this approach has often been used efficiently in various problems of practical importance. Unfortunately, as has been observed at least in some important cases, when and have essentially different asymptotic behaviors as , the approximations to produced by this approach, despite the fact that they are good, do not converge as quickly as those obtained for , and this is puzzling. In this paper we first give a rigorous mathematical explanation of this phenomenon for the cases in which E is the Richardson extrapolation process and E is a generalization of it, thus showing that the phenomenon has very little to do with numerics. Following that, we propose a procedure that amounts to first applying the extrapolation method E to and then differentiating the resulting approximations to , and we provide a thorough convergence and stability analysis in conjunction with the Richardson extrapolation process. It follows from this analysis that the new procedure for has practically the same convergence properties as E for . We show that a very efficient way of implementing the new procedure is by actually differentiating the recursion relations satisfied by the extrapolation method used, and we derive the necessary algorithm for the Richardson extrapolation process. We demonstrate the effectiveness of the new approach with numerical examples that also support the theory. We discuss the application of this approach to numerical integration in the presence of endpoint singularities. We also discuss briefly its application in conjunction with other extrapolation methods.

     

On the modular curves

Let denote an elliptic curve over and the modular curve classifying the elliptic curves over such that the representations of in the 7-torsion points of and of are symplectically isomorphic. In case is given by a Weierstraß equation such that the invariant is a square, we exhibit here nontrivial points of . From this we deduce an infinite family of curves for which has at least four nontrivial points.

     

Algebraic integers whose conjugates all lie in an ellipse

We find all algebraic integers whose conjugates all lie in an ellipse with two of them nonreal, while the others lie in the real interval . This problem has applications to finding certain subgroups of . We use explicit auxiliary functions related to the generalized integer transfinite diameter of compact subsets of . This gives good bounds for the coefficients of the minimal polynomial of

     

Constructing representations of higher degrees of finite simple groups and covers

Let be a finite group and an irreducible character of . A simple method for constructing a representation affording can be used whenever has a subgroup such that has a linear constituent with multiplicity 1. In this paper we show that (with a few exceptions) if is a simple group or a covering group of a simple group and is an irreducible character of of degree between 32 and 100, then such a subgroup exists.

     

The convergence of harmonic Ritz values, harmonic Ritz vectors, and refined harmonic Ritz vectors

This paper concerns a harmonic projection method for computing an approximation to an eigenpair of a large matrix . Given a target point and a subspace that contains an approximation to , the harmonic projection method returns an approximation to . Three convergence results are established as the deviation of from approaches zero. First, the harmonic Ritz value converges to if a certain Rayleigh quotient matrix is uniformly nonsingular. Second, the harmonic Ritz vector converges to if the Rayleigh quotient matrix is uniformly nonsingular and remains well separated from the other harmonic Ritz values. Third, better error bounds for the convergence of are derived when converges. However, we show that the harmonic projection method can fail to find the desired eigenvalue --in other words, the method can miss if it is very close to . To this end, we propose to compute the Rayleigh quotient of with respect to and take it as a new approximate eigenvalue. is shown to converge to once tends to , no matter how is close to . Finally, we show that if the Rayleigh quotient matrix is uniformly nonsingular, then the refined harmonic Ritz vector, or more generally the refined eigenvector approximation introduced by the author, converges. We construct examples to illustrate our theory.

     

The hyperdeterminant and triangulations of the 4-cube

The hyperdeterminant of format is a polynomial of degree in unknowns which has terms. We compute the Newton polytope of this polynomial and the secondary polytope of the -cube. The regular triangulations of the -cube are classified into -equivalence classes, one for each vertex of the Newton polytope. The -cube has coarsest regular subdivisions, one for each facet of the secondary polytope, but only of them come from the hyperdeterminant.

     

Solving polynomials by radicals with roots of unity in minimum depth

Let be an algebraic number field. Let be a root of a polynomial which is solvable by radicals. Let be the splitting field of over . Let be a natural number divisible by the discriminant of the maximal abelian subextension of , as well as the exponent of , the Galois group of over . We show that an optimal nested radical with roots of unity for can be effectively constructed from the derived series of the solvable Galois group of over .

     

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.

     

Balanced multi-wavelets in

The notion of -balancing was introduced a few years ago as a condition for the construction of orthonormal scaling function vectors and multi-wavelets to ensure the property of preservation/annihilation of scalar-valued discrete polynomial data of order (or degree ), when decomposed by the corresponding matrix-valued low-pass/high-pass filters. While this condition is indeed precise, to the best of our knowledge only the proof for is known. In addition, the formulation of the -balancing condition for is so prohibitively difficult to satisfy that only a very few examples for and vector dimension 2 have been constructed in the open literature. The objective of this paper is to derive various characterizations of the -balancing condition that include the polynomial preservation property as well as equivalent formulations that facilitate the development of methods for the construction purpose. These results are established in the general multivariate and bi-orthogonal settings for any .

     

Computation of relative class numbers of CM-fields by using Hecke -functions

We develop an efficient technique for computing values at of Hecke -functions. We apply this technique to the computation of relative class numbers of non-abelian CM-fields which are abelian extensions of some totally real subfield . We note that the smaller the degree of the more efficient our technique is. In particular, our technique is very efficient whenever instead of simply choosing (the maximal totally real subfield of ) we can choose real quadratic. We finally give examples of computations of relative class numbers of several dihedral CM-fields of large degrees and of several quaternion octic CM-fields with large discriminants.

     

Even moments of generalized Rudin-Shapiro polynomials

We know from Littlewood (1968) that the moments of order of the classical Rudin-Shapiro polynomials satisfy a linear recurrence of degree . In a previous article, we developed a new approach, which enables us to compute exactly all the moments of even order for . We were also able to check a conjecture on the asymptotic behavior of , namely , where , for even and . Now for every integer there exists a sequence of generalized Rudin-Shapiro polynomials, denoted by . In this paper, we extend our earlier method to these polynomials. In particular, the moments have been completely determined for and , for and and for and . For higher values of and , we formulate a natural conjecture, which implies that , where is an explicit constant.

     

Upper bounds for the prime divisors of Wendt's determinant

Let be an even integer, . The resultant of the polynomials and is known as Wendt's determinant of order . We prove that among the prime divisors of only those which divide or can be larger than , where and is the th Lucas number, except when and . Using this estimate we derive criteria for the nonsolvability of Fermat's congruence.

     

Minimality and other properties of the width- nonadjacent form

Let be an integer and let be the set of integers that includes zero and the odd integers with absolute value less than . Every integer can be represented as a finite sum of the form , with , such that of any consecutive 's at most one is nonzero. Such representations are called width- nonadjacent forms (-NAFs). When these representations use the digits and coincide with the well-known nonadjacent forms. Width- nonadjacent forms are useful in efficiently implementing elliptic curve arithmetic for cryptographic applications. We provide some new results on the -NAF. We show that -NAFs have a minimal number of nonzero digits and we also give a new characterization of the -NAF in terms of a (right-to-left) lexicographical ordering. We also generalize a result on -NAFs and show that any base 2 representation of an integer, with digits in , that has a minimal number of nonzero digits is at most one digit longer than its binary representation.

     

The Postage Stamp Problem: An algorithm to determine the

Given an integral ``stamp" basis with and a positive integer , we define the -range as

. For given and , the extremal basis has the largest possible extremal -range

We give an algorithm to determine the -range. We prove some properties of the -range formula, and we conjecture its form for the extremal -range. We consider parameter bases , where the basis elements are given functions of . For we conjecture the extremal parameter bases for .