Real Algebraic Number Research Articles

Polynomials with small discriminants and regular systems of real algebraic numbers

The distribution of special algebraic numbers is studied, and an optimal regular system is constructed. Bibliography: 11 titles.

Lower bounds on the projective heights of algebraic points

If $\alpha_1,\ldots,\alpha_r$ are algebraic numbers such that $$N=\sum_{i=1}^r\alpha_i \ne \sum_{i=1}^r\alpha_i^{-1}$$ for some integer $N$, then a theorem of Beukers and Zagier gives the best possible lower bound on $$\sum_{i=1}^r\log h(\alpha_i)$$ where $h$ denotes the Weil Height. We will extend this result to allow $N$ to be any totally real algebraic number. Our generalization includes a consequence of a theorem of Schinzel which bounds the height of a totally real algebraic integer.

On the approximation of quadratic algebraic numbers

In the paper we construct such second order linear recursive sequences G and H of rational integers that with their terms |a -Gn+1 /H n| < 1/ (\sqrtvDH2n) holds for every positive integer n, where a denotes a real quadratic algebraic integer of discriminant D. An approximating sequence of the form Gn+1 /Hn is also given for a if it is only a real quadratic algebraic number (not an algebraic integer), but in this case the approximating constant is not the best.

Counting real Galois covers of the projective line

For Galois covers of P1 of a given rami?cation type ? essentially, a given monodromy group G and branch locus, assumed to be de?ned over R ?we ask: How many covers are de?ned over R and how many are not? J.-P. Serre showed that the number of all Galois covers with given rami?cation type can be computed from the character table of G. We adapt Serre?s method of calculation to the more re?ned situation of Galois covers de?ned over R, for which there is a group-theoretic characterization due to P. D?s and M. Fried. We obtain explicit answers to our problem. As an application, we exhibit new families of covers not de?ned over their ?eld of moduli, the monodromy group of which can be chosen arbitrarily large. We also give examples of Galois covers de?ned over the ?eld Qtr of totally real algebraic numbers with Q-rational branch locus.

On approximation of [formula omitted]-adic numbers by [formula omitted]-adic algebraic numbers

During the last 10 years the classical Khintchine theorem on approximation of real numbers by rational numbers was generalized for integral polynomials of a fixed degree and for approximation by real algebraic numbers. In this paper we prove a complete analogue of the results for the p-adic case.

EXPRESSIONS FOR VALUES OF THE GAMMA FUNCTION

This paper presents expressions for gamma values at rational points with the denominator dividing 24 or 60. These gamma values are expressed in terms of 10 distinct gamma values and rational powers of $\pi$ and a few real algebraic numbers. Our elementary list of formulas can be conveniently used to evaluate, for example, algebraic Gauss hypergeometric functions by the Gauss identity. Also, algebraic independence of gamma values and their relation to the elliptic K-function are briefly discussed.

On the complexity of algebraic numbers, II. Continued fractions

The continued fraction expansion of an irrational number $\alpha$ is eventually periodic if and only if $\alpha$ is a quadratic irrationality. However, very little is known regarding the size of the partial quotients of algebraic real numbers of degree at least three. Because of some numerical evidence and a belief that these numbers behave like most numbers in this respect, it is often conjectured that their partial quotients form an unbounded sequence. More modestly, we may expect that if the sequence of partial quotients of an irrational number $\alpha$ is, in some sense, simple, then $\alpha$ is either quadratic or transcendental. The term can of course lead to many interpretations. It may denote real numbers whose continued fraction expansion has some regularity, or can be produced by a simple algorithm (by a simple Turing machine, for example), or arises from a simple dynamical system... The aim of this paper is to present in a unified way several new results on these different approaches of the notion of simplicity/complexity for the continued fraction expansion of algebraic real numbers of degree at least three.

On the binary expansions of algebraic numbers

Employing concepts from additive number theory, to- gether with results on binary evaluations and partial series, we establish bounds on the density of 1's in the binary expansions of real algebraic numbers. A central result is that if a real y has algebraic degree D > 1, then the number #(|y|,N) of 1-bits in the expansion of |y| through bit position N satisfies

ε-Pisot numbers in any real algebraic number field are relatively dense

An algebraic integer is called an ε-Pisot number ( ε>0) if its Galois conjugates have absolute value less then ε. Let K be any real algebraic number field. We prove that the subset of K consisting of ε-Pisot numbers which have the same degree as that of the field is relatively dense in the real line R . This has some applications to non-stationary products of random matrices involving Salem numbers.

Towards faster real algebraic numbers

This paper presents a new encoding scheme for real algebraic number manipulations which enhances current Axiom’s real closure. Algebraic manipulations are performed using different instantiations of sub-resultant-like algorithms instead of Euclidean-like algorithms. We use these algorithms to compute polynomial gcds and Bezout relations, to compute the roots and the signs of algebraic numbers. This allows us to work in the ring of real algebraic integers instead of the field of real algebraic numbers avoiding many denominators.

An application of algebraic sieve theory

Let K be a fixed totally real algebraic number field of finite degree over the rationals. The theme of this paper is the problem about the occurrence of algebraic almost-primes in a polynomial sequence generated by an irreducible polynomial of K with prime arguments. The method is based on a weighted upper and lower linear Selberg-type sieve in K and makes use of a multidimensional algebraic version of Bombieri’s theorem on primes in arithmetic progressions.

A note on a lemma of Shimura

The subject of this paper is a quaternion algebra over a quadratic extension of a totally real algebraic number field. We generalize an important technical lemma of Shimura, on which a certain period of automorphic forms depends.

Best simultaneous diophantine approximations of some cubic algebraic numbers

Let α be a real algebraic number of degree 3 over ℚ whose conjugates are not real. There exists an unit ζ of the ring of integer of K=ℚ(α) for which it is possible to describe the set of all best approximation vectors of θ=(ζ,ζ 2 ).’

Exact Algorithms for Linear Programming over Algebraic Extensions

Abstract. We study the computational complexity of linear programs with coefficients that are real algebraic numbers under a Turing machine model of computation. After reviewing a method for exact representation of algebraic numbers under the Turing model, we show that the fundamental tasks of comparison and arithmetic can be performed in polynomial time. Our technique for establishing polynomial-time algorithms for comparison and arithmetic is distinct from the usual resultant-based approaches, and has the advantage that it provides a natural framework for analysis of the complexity of computational tasks, such as Gaussian elimination, that involve a sequence of arithmetic operations. Our main contribution is to show that a variant of the ellipsoid method can be used to solve linear programming in time polynomial in the encoding size of the problem coefficients and the degree of any algebraic extension that contains those coefficients.

On fractional parts of rapidly growing functions

We study the behaviour of fractional parts of functions , where is a real algebraic number of degree and is an arbitrary positive number less than one.

Some Applications of a Bombieri-Type Theoremin Totally Real Number Fields

The primary concern of this paper is to present three further applications of a multi-dimensional version of Bombieri’s theorem on primes in arithmetic progressions in the setting of a totally real algebraic number field K. First, we deal with the order of magnitude of a greatest (relative to its norm) prime ideal factor of \(\), where the product runs over prime arguments ω of a given irreducible polynomial F which lie in a certain lattice point region. Then, we turn our attention to the problem about the occurrence of algebraic primes in a polynomial sequence generated by an irreducible polynomial of K with prime arguments. Finally, we give further contributions to the binary Goldbach problem in K.

On Arithmetic Properties of the Solutions of a Universal Differential Equation at Algebraic Points

In 1981, L. A. Rubel found an explicit algebraic differential equation (ADE) of order four such that every real continuous function on the real line can be uniformly approximated by the C∞-solutions of this ADE. It is shown that an ADE of order five exists, where the C∞-solutions additionally satisfy some algebraic properties in the sense of C. L. Siegel's results from the analytical theory of numbers. For instance, all the solutions and their derivatives are transcendental at algebraic points, and large sets of these numbers are linearly independent over the field of real algebraic numbers.

On a Transfer Theorem for the [formula omitted]≠[formula omitted][formula omitted] Conjecture

A model of computation is defined over the algebraic numbers and over number fields. This model is non-uniform, and the cost of operations depends on the height of the operands and on the degree of the extension of the rational defined by those operands. A transfer theorem for the P≠NP Conjecture is proved, namely: P≠NP in this model over the real algebraic numbers if and only if P≠NP in the classical setting.

Jacobi forms over totally real number fields

We define Jacobi forms over a totally real algebraic number field K and construct examples by first embedding the group and the space into the symplectic group and the symplectic upper half space respectively. Then symplectic modular forms are created and Jacobi forms arise by taking the appropriate Fourier coefficients. Also some known relations of Jacobi forms to vector valued modular forms over rational numbers are extended to totally real fields.

Uniform bounds on complexity and transfer of global properties of Nash functions

We show that the complexity of semialgebraic sets and mappings can be used to parametrize Nash sets and mappings by Nash families. From this we deduce uniform bounds on the complexity of Nash functions that lead to first-order descriptions of many properties of Nash functions and a good behaviour under real closed field extension (e.g. primary decomposition). As a distinguished application, we derive the solution of the extension and global equations problems over arbitrary real closed fields, in particular over the field of real algebraic numbers. This last fact and a technique of change of base are used to prove that the Artin-Mazur description holds for abstract Nash functions on the real spectrum of any commutative ring, and solve extension and global equations in that abstract setting. To complete the view, we prove the idempotency of the real spectrum and an abstract version of the separation problem. We also discuss the conditions for the rings of abstract Nash functions to be noetherian.