On beta expansions for Pisot numbers

Given a number , the beta-transformation is defined for by (mod 1). The number is said to be a beta-number if the orbit is finite, hence eventually periodic. In this case is the root of a monic polynomial with integer coefficients called the characteristic polynomial of . If is the minimal polynomial of , then for some polynomial . It is the factor which concerns us here in case is a Pisot number. It is known that all Pisot numbers are beta-numbers, and it has often been asked whether must be cyclotomic in this case, particularly if . We answer this question in the negative by an examination of the regular Pisot numbers associated with the smallest 8 limit points of the Pisot numbers, by an exhaustive enumeration of the irregular Pisot numbers in (an infinite set), by a search up to degree in , to degree in , and to degree in . We find the smallest counterexample, the counterexample of smallest degree, examples where is nonreciprocal, and examples where is reciprocal but noncyclotomic. We produce infinite sequences of these two types which converge to from above, and infinite sequences of with nonreciprocal which converge to from below and to the th smallest limit point of the Pisot numbers from both sides. We conjecture that these are the only limit points of such numbers in . The Pisot numbers for which is cyclotomic are related to an interesting closed set of numbers introduced by Flatto, Lagarias and Poonen in connection with the zeta function of . Our examples show that the set of Pisot numbers is not a subset of .

     

Computation of -invariants of real quadratic fields

Let be a real quadratic field and an odd prime number which splits in . In a previous work, the author gave a sufficient condition for the Iwasawa invariant of the cyclotomic -extension of to be zero. The purpose of this paper is to study the case of this result and give new examples of with , by using information on the initial layer of the cyclotomic -extension of .

     

On the beta expansion for Salem numbers of degree 6

For a given , the beta transformation is defined for by (mod ). The number is said to be a beta number if the orbit is finite, hence eventually periodic. It is known that all Pisot numbers are beta numbers, and it is conjectured that this is true for Salem numbers, but this is known only for Salem numbers of degree . Here we consider some computational and heuristic evidence for the conjecture in the case of Salem numbers of degree , by considering the set of such numbers of trace at most . Although the orbit is small for the majority of these numbers, there are some examples for which the orbit size is shown to exceed and for which the possibility remains that the orbit is infinite. There are also some very large orbits which have been shown to be finite: an example is given for which the preperiod length is and the period length is . This is in contrast to Salem numbers of degree where the orbit size is bounded by . An heuristic probabilistic model is proposed which explains the difference between the degree- and degree- cases. The model predicts that all Salem numbers of degree and should be beta numbers but that degree- Salem numbers can have orbits which are arbitrarily large relative to the size of . Furthermore, the model predicts that a positive proportion of Salem numbers of any fixed degree will not be beta numbers. This latter prediction is not tested here.

     

-complete sequences of integers

An infinite sequence is -complete if every sufficiently large integer is the sum of such that no one divides the other. We investigate -completeness of sets of the form and with nonnegative.

     

A new algorithm for constructing large Carmichael numbers

We describe an algorithm for constructing Carmichael numbers with a large number of prime factors . This algorithm starts with a given number , representing the value of the Carmichael function . We found Carmichael numbers with up to factors.

     

The serial test for a nonlinear pseudorandom number generator

Let and A sequence is obtained by the formula The sequence is a sequence of pseudorandom numbers of the maximal period length if and only if (mod 4), (mod 4). In this note, the uniformity is investigated by the 2-dimensional serial test for the sequence. We follow closely the method of papers by Eichenauer-Herrmann and Niederreiter.

     

Results and estimates on pseudopowers

Let be a positive integer. We say looks like a power of 2 modulo a prime if there exists an integer such that . First, we provide a simple proof of the fact that a positive integer which looks like a power of modulo all but finitely many primes is in fact a power of . Next, we define an -pseudopower of the base to be a positive integer that is not a power of , but looks like a power of modulo all primes . Let denote the least such . We give an unconditional upper bound on , a conditional result (on ERH) that gives a lower bound, and a heuristic argument suggesting that is about for a certain constant . We compare our heuristic model with numerical data obtained by a sieve. Some results for bases other than are also given.

     

Asymptotic semismoothness probabilities

We call an integer semismooth with respect to and if each of its prime factors is , and all but one are . Such numbers are useful in various factoring algorithms, including the quadratic sieve. Let be the asymptotic probability that a random integer is semismooth with respect to and . We present new recurrence relations for and related functions. We then give numerical methods for computing , tables of , and estimates for the error incurred by this asymptotic approximation.

     

Spontaneous Generation of Modular Invariants

It is possible to compute and its modular equations with no perception of its related classical group structure except at . We start by taking, for prime, an unknown ``-Newtonian' polynomial equation with arbitrary coefficients (based only on Newton's polygon requirements at for and ). We then ask which choice of coefficients of leads to some consistent Laurent series solution , (where . It is conjectured that if the same Laurent series works for -Newtonian polynomials of two or more primes , then there is only a bounded number of choices for the Laurent series (to within an additive constant). These choices are essentially from the set of ``replicable functions,' which include more classical modular invariants, particularly . A demonstration for orders and is done by computation. More remarkably, if the same series works for the -Newtonian polygons of 15 special ``Fricke-Monster' values of , then is (essentially) determined uniquely. Computationally, this process stands alone, and, in a sense, modular invariants arise ``spontaneously.'

     

Turán's Pure Power Sum Problem

Let be complex numbers, and consider the power sums , . Put , where the minimum is over all possible complex numbers satisfying the above. Turán conjectured that , for some positive absolute constant. Atkinson proved this conjecture by showing . It is now known that , for . Determining whether or approaches some other limiting value as is still an open problem. Our calculations show that an upper bound for decreases for , suggesting that decreases to a limiting value less than as .

     

Two new points in the spectrum of the absolute Mahler measure of totally positive algebraic integers

For totally positive algebraic integers of degree , we consider the set of values of , where is the Mahler measure of . C. J. Smyth has found the four smallest values of and conjectured that the fifth point is . We prove that this is so and, moreover, we give the sixth point of .

     

The versions of the finite element method for problems with boundary layers

We study the uniform approximation of boundary layer functions for , , by the and versions of the finite element method. For the version (with fixed mesh), we prove super-exponential convergence in the range . We also establish, for this version, an overall convergence rate of in the energy norm error which is uniform in , and show that this rate is sharp (up to the term) when robust estimates uniform in are considered. For the version with variable mesh (i.e., the version), we show that exponential convergence, uniform in , is achieved by taking the first element at the boundary layer to be of size . Numerical experiments for a model elliptic singular perturbation problem show good agreement with our convergence estimates, even when few degrees of freedom are used and when is as small as, e.g., . They also illustrate the superiority of the approach over other methods, including a low-order version with optimal ``exponential" mesh refinement. The estimates established in this paper are also applicable in the context of corresponding spectral element methods.

     

Solvability of norm equations over cyclic number fields of prime degree

Let be an abelian number field of prime degree , and let be a nonzero rational number. We describe an algorithm which takes as input and the minimal polynomial of over , and determines if is a norm of an element of . We show that, if we ignore the time needed to obtain a complete factorization of and a complete factorization of the discriminant of , then the algorithm runs in time polynomial in the size of the input. As an application, we give an algorithm to test if a cyclic algebra over is a division algebra.

     

Explicit bounds for primes in residue classes

Let be an abelian extension of number fields, with . Let and denote the absolute discriminant and degree of . Let denote an element of the Galois group of . We prove the following theorems, assuming the Extended Riemann Hypothesis:

(1)

There is a degree- prime of such that , satisfying .

(2)

There is a degree- prime of such that generates
the same group as , satisfying .

(3)

For , there is a prime such that , satisfying
.

In (1) and (2) we can in fact take to be unramified in . A special case of this result is the following.

(4)

If , the least prime satisfies
.

It follows from our proof that (1)--(3) also hold for arbitrary Galois extensions, provided we replace by its conjugacy class . Our theorems lead to explicit versions of (1)--(4), including the following: the least prime is less than .

     

Computing the canonical height on K3 surfaces

Let be a surface in given by the intersection of a (1,1)-form and a (2,2)-form. Then is a K3 surface with two noncommuting involutions and . In 1991 the second author constructed two height functions and which behave canonically with respect to and , and in 1993 together with the first author showed in general how to decompose such canonical heights into a sum of local heights . We discuss how the geometry of the surface is related to formulas for the local heights, and we give practical algorithms for computing the involutions , , the local heights , , and the canonical heights , .

     

Some New Error Estimates for Ritz--Galerkin Methods with Minimal Regularity Assumptions

New uniform error estimates are established for finite element approximations of solutions of second-order elliptic equations using only the regularity assumption . Using an Aubin--Nitsche type duality argument we show for example that, for arbitrary (fixed) sufficiently small, there exists an such that for

Here, denotes the norm on the Sobolev space . Other related results are established.

     

Simultaneous Pell Equations

Let and be positive integers with . We shall call the simultaneous Diophantine equations

simultaneous Pell equations in and . Each such pair has the trivial solution but some pairs have nontrivial solutions too. For example, if and , then is a solution. Using theorems due to Baker, Davenport, and Waldschmidt, it is possible to show that the number of solutions is always finite, and it is possible to give a complete list of them. In this paper we report on the solutions when .

     

Convergence rate of approximate solutions to weakly coupled nonlinear systems

We study the convergence rate of approximate solutions to nonlinear hyperbolic systems which are weakly coupled through linear source terms. Such weakly coupled systems appear, for example, in the context of resonant waves in gas dynamics equations.

This work is an extension of our previous scalar analysis. This analysis asserts that a One Sided Lipschitz Condition (OSLC, or -stability) together with -consistency imply convergence to the unique entropy solution. Moreover, it provides sharp convergence rate estimates, both global (quantified in terms of the -norms) and local.

We focus our attention on the -stability of the viscosity regularization associated with such weakly coupled systems. We derive sufficient conditions, interesting for their own sake, under which the viscosity (and hence the entropy) solutions are -stable in an appropriate sense. Equipped with this, we may apply the abovementioned convergence rate analysis to approximate solutions that share this type of -stability.

     

Constructing nonresidues in finite fields and the extended Riemann hypothesis

We present a new deterministic algorithm for the problem of constructing th power nonresidues in finite fields , where is prime and is a prime divisor of . We prove under the assumption of the Extended Riemann Hypothesis (ERH), that for fixed and , our algorithm runs in polynomial time. Unlike other deterministic algorithms for this problem, this polynomial-time bound holds even if is exponentially large. More generally, assuming the ERH, in time we can construct a set of elements that generates the multiplicative group . An extended abstract of this paper appeared in Proc. 23rd Ann. ACM Symp. on Theory of Computing, 1991.