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 .

     

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.

     

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.

     

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 .

     

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.

     

Expansion and Estimation of the Range of Nonlinear Functions

Many verification algorithms use an expansion , for , where the set of matrices is usually computed as a gradient or by means of slopes. In the following, an expansion scheme is described which frequently yields sharper inclusions for . This allows also to compute sharper inclusions for the range of over a domain. Roughly speaking, has to be given by means of a computer program. The process of expanding can then be fully automatized. The function need not be differentiable. For locally convex or concave functions special improvements are described. Moreover, in contrast to other methods, may be empty without implying large overestimations for . This may be advantageous in practical applications.

     

Complete solutions of a family of quartic Thue and index form equations

Continuing the recent work of the second author, we prove that the diophantine equation

for has exactly 12 solutions except when , when it has 16 solutions. If denotes one of the zeros of , then for we also find all with .

     

A Capitulation Problem and Greenberg's Conjecture on Real Quadratic Fields

We give a sufficient condition in order that an ideal of a real quadratic field capitulates in the cyclotomic -extension of by using a unit of an intermediate field. Moreover, we give new examples of 's for which Greenberg's conjecture holds by calculating units of fields of degree 6, 18, 54 and 162.

     

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.

     

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.

     

Pure product polynomials and the Prouhet-Tarry-Escott problem

An -factor pure product is a polynomial which can be expressed in the form for some natural numbers . We define the norm of a polynomial to be the sum of the absolute values of the coefficients. It is known that every -factor pure product has norm at least . We describe three algorithms for determining the least norm an -factor pure product can have. We report results of our computations using one of these algorithms which include the result that every -factor pure product has norm strictly greater than if is , , , or .

     

Quality Local Refinement of Tetrahedral Meshes Based on 8-Subtetrahedron Subdivision

Let be a tetrahedral mesh. We present a 3-D local refinement algorithm for which is mainly based on an 8-subtetrahedron subdivision procedure, and discuss the quality of refined meshes generated by the algorithm. It is proved that any tetrahedron produces a finite number of classes of similar tetrahedra, independent of the number of refinement levels. Furthermore, , where , is a positive constant independent of and the number of refinement levels, is any refined tetrahedron of , and is a tetrahedron shape measure. It is also proved that local refinements on tetrahedra can be smoothly extended to their neighbors to maintain a conforming mesh. Experimental results show that the ratio of the number of tetrahedra actually refined to the number of tetrahedra chosen for refinement is bounded above by a small constant.

     

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 , .

     

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.

     

Efficient algorithms for computing the -discrepancy

The -discrepancy is a quantitative measure of precision for multivariate quadrature rules. It can be computed explicitly. Previously known algorithms needed operations, where is the number of nodes. In this paper we present algorithms which require operations.

     

The Diophantine equation

An effective method is derived for solving the equation of the title in positive integers and for given completely, and is carried out for all . If is of the form , then there is the solution , ; in the above range, except for with solution , , there are no other solutions.

     

On sequences without geometric progressions

Several papers have investigated sequences which have no -term arithmetic progressions, finding bounds on their density and looking at sequences generated by greedy algorithms. Rankin in 1960 suggested looking at sequences without -term geometric progressions, and constructed such sequences for each with positive density. In this paper we improve on Rankin's results, derive upper bounds, and look at sequences generated by a greedy algorithm.

     

Further investigations with the strong probable prime test

Recently, Damgård, Landrock and Pomerance described a procedure in which a -bit odd number is chosen at random and subjected to random strong probable prime tests. If the chosen number passes all tests, then the procedure will return that number; otherwise, another -bit odd integer is selected and then tested. The procedure ends when a number that passes all tests is found. Let denote the probability that such a number is composite. The authors above have shown that when and . In this paper we will show that this is in fact valid for all and .

     

An asymptotic expansion for the incomplete beta function

A new asymptotic expansion is derived for the incomplete beta function , which is suitable for large , small and . This expansion is of the form

where is the incomplete Gamma function ratio and . This form has some advantages over previous asymptotic expansions in this region in which depends on as well as on and .

     

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 .