Real Algebraic Number Research Articles

On the modulo p zeros of modular forms congruent to theta series

For a prime p larger than 7, the Eisenstein series of weight p−1 has some remarkable congruence properties modulo p. Those imply, for example, that the j-invariants of its zeros (which are known to be real algebraic numbers in the interval [0,1728]), are at most quadratic over the field with p elements and are congruent modulo p to the zeros of a certain truncated hypergeometric series. In this paper we introduce “theta modular forms” of weight k≥4 for the full modular group as the modular forms for which the first dim⁡(Mk) Fourier coefficients are identical to certain theta series. We consider these theta modular forms for both the Jacobi theta series and the theta series of the hexagonal lattice. We show that the j-invariant of the zeros of the theta modular forms for the Jacobi theta series are modulo p all in the ground field with p elements. For the theta modular form of the hexagonal lattice we show that its zeros are at most quadratic over the ground field with p elements. Furthermore, we show that these zeros in both cases are congruent to the zeros of certain truncated hypergeometric functions.

On arithmetic quotients of the group SL2 over a quaternion division k-algebra

Abstract Given a totally real algebraic number field k of degree s, we consider locally symmetric spaces X G / Γ {X_{G}/\Gamma} associated with arithmetic subgroups Γ of the special linear algebraic k-group G = SL M 2 ⁢ ( D ) {G=\mathrm{SL}_{M_{2}(D)}} , attached to a quaternion division k-algebra D. The group G is k-simple, of k-rank one, and non-split over k. Using reduction theory, one can construct an open subset Y Γ ⊂ X G / Γ {Y_{\Gamma}\subset X_{G}/\Gamma} such that its closure Y ¯ Γ {\overline{Y}_{\Gamma}} is a compact manifold with boundary ∂ ⁡ Y ¯ Γ {\partial\overline{Y}_{\Gamma}} , and the inclusion Y ¯ Γ → X G / Γ {\overline{Y}_{\Gamma}\rightarrow X_{G}/\Gamma} is a homotopy equivalence. The connected components Y [ P ] {Y^{[P]}} of the boundary ∂ ⁡ Y ¯ Γ {\partial\overline{Y}_{\Gamma}} are in one-to-one correspondence with the finite set of Γ-conjugacy classes of minimal parabolic k-subgroups of G. We show that each boundary component carries the natural structure of a torus bundle. Firstly, if the quaternion division k-algebra D is totally definite, that is, D ramifies at all archimedean places of k, we prove that the basis of this bundle is homeomorphic to the torus T s - 1 {T^{s-1}} of dimension s - 1 {s-1} , has the compact fibre T 4 ⁢ s {T^{4s}} , and its structure group is SL 4 ⁢ s ⁢ ( ℤ ) {\mathrm{SL}_{4s}(\mathbb{Z})} . We determine the cohomology of Y [ P ] {Y^{[P]}} . Secondly, if the quaternion division k-algebra D is indefinite, thus, there exists at least one archimedean place v ∈ V k , ∞ {v\in V_{k,\infty}} at which D v {D_{v}} splits over ℝ {\mathbb{R}} , that is, D v ≅ M 2 ⁢ ( ℝ ) {D_{v}\cong M_{2}(\mathbb{R})} , the fibre is homeomorphic to T 4 ⁢ s {T^{4s}} , but the base space of the bundle is more complicated.

Ellis enveloping semigroups in real closed fields

We introduce the Boolean algebra of d-semialgebraic (more generally, d-definable) sets and prove that its Stone space is naturally isomorphic to the Ellis enveloping semigroup of the Stone space of the Boolean algebra of semialgebraic (definable) sets. For definably connected o-minimal groups, we prove that this family agrees with the one of externally definable sets in the one-dimensional case. Nonetheless, we prove that in general these two families differ, even in the semialgebraic case over the real algebraic numbers. On the other hand, in the semialgebraic case we characterise real semialgebraic functions representing Boolean combinations of d-semialgebraic sets.

On Transcendental Regular Continued Fractions

ABSTRACT We establish explicit constructions of real transcendental numbers that are not U-numbers with respect to Mahler’s classification by using regular continued fraction expansions of irrational real algebraic numbers.

ON JOINT DISCRETE UNIVERSALITY OF THE RIEMANN ZETA-FUNCTION IN SHORT INTERVALS

In the paper, we prove that the set of discrete shifts of the Riemann zeta-function approximating analytic nonvanishing functions f1(s),...,fr(s) defined on has a positive density in the interval [N,N + M] with with real algebraic numbers a1,...,ar linearly independent over Q. A similar result is obtained for shifts of certain absolutely convergent Dirichlet series.

On the Zeros of Exponential Polynomials

We consider the problem of deciding the existence of real roots of real-valued exponential polynomials with algebraic coefficients. Such functions arise as solutions of linear differential equations with real algebraic coefficients. We focus on two problems: theZero Problem, which asks whether an exponential polynomial has a real root, and theInfinite Zeros Problem, which asks whether such a function has infinitely many real roots. Our main result is that for differential equations of order at most 8 the Zero Problem is decidable, subject to Schanuel’s Conjecture, while the Infinite Zeros Problem is decidable unconditionally. We show moreover that a decision procedure for the Infinite Zeros Problem at order 9 would yield an algorithm for computing the Lagrange constant of any given real algebraic number to arbitrary precision, indicating that it will be very difficult to extend our decidability results to higher orders.

Every Salem number is a difference of two Pisot numbers

AbstractIn this note, we prove that every Salem number is expressible as a difference of two Pisot numbers. More precisely, we show that for each Salem number α of degree d, there are infinitely many positive integers n for which $\alpha^{2n-1}-\alpha^n+\alpha$ and $\alpha^{2n-1}-\alpha^n$ are both Pisot numbers of degree d and that the smallest such n is at most $6^{d/2-1}+1$. We also prove that every real positive algebraic number can be expressed as a quotient of two Pisot numbers. Earlier, Salem himself had proved that every Salem number can be written in this way.

On the complexity of algebraic numbers, and the bit-complexity of straight-line programs1

We investigate the complexity of languages that correspond to algebraic real numbers, and we present improved upper bounds on the complexity of these languages. Our key technical contribution is the presentation of improved uniform TC 0 circuits for division, matrix powering, and related problems, where the improvement is in terms of “majority depth” (initially studied by Maciel and Thérien). As a corollary, we obtain improved bounds on the complexity of certain problems involving arithmetic circuits, which are known to lie in the counting hierarchy, and we answer a question posed by Yap.

Различие мер Хаара цилиндров в теореме Дирихле для поля p-адических чисел

The Dirichlet box principle gives surprisingly accurate results in problems of approximation of real numbers by rational numbers, transcendental numbers by real algebraic numbers. Every polynomial taking small values at a given point $x$ also takes small values in its neighborhood. A problem of studying such neighborhoods and obtaining possible Lebesgue measure values arises frequently. In this paper we solve the problem in the p-adic case using recent results of the metric theory of Diophantine approximations.

On the generalization of the gap principle

Let \(\alpha \) be a real algebraic number of degree \(d \ge 3\) and let \(\beta \in {\mathbb {Q}}(\alpha )\) be irrational. Let \(\mu \) be a real number such that \((d/2) + 1< \mu < d\) and let \(C_0\) be a positive real number. We prove that there exist positive real numbers \(C_1\) and \(C_2\), which depend only on \(\alpha \), \(\beta \), \(\mu \) and \(C_0\), with the following property. If \(x_1/y_1\) and \(x_2/y_2\) are rational numbers in lowest terms such that $$\begin{aligned} H(x_2, y_2) \ge H(x_1, y_1) \ge C_{1} \end{aligned}$$and $$\begin{aligned} \left| \alpha - \frac{x_1}{y_1}\right|< \frac{C_0}{H(x_1, y_1)^\mu }, \quad \left| \beta - \frac{x_2}{y_2}\right| < \frac{C_0}{H(x_2, y_2)^\mu }, \end{aligned}$$then either \(H(x_2, y_2) > C_{2}^{-1} H(x_1, y_1)^{\mu - d/2}\), or there exist integers s, t, u, v, with \(sv - tu \ne ~0\), such that $$\begin{aligned} \beta = \frac{s\alpha + t}{u\alpha + v} \quad \text {and} \quad \frac{x_2}{y_2} = \frac{sx_1 + ty_1}{ux_1 + vy_1}, \end{aligned}$$or both. Here \(H(x, y) = \max (|x|, |y|)\) is the height of x/y. Since \(\mu - d/2\) exceeds 1, our result demonstrates that, unless \(\alpha \) and \(\beta \) are connected by means of a linear fractional transformation with integer coefficients, the heights of \(x_1/y_1\) and \(x_2/y_2\) have to be exponentially far apart from each other. An analogous result is established in the case when \(\alpha \) and \(\beta \) are p-adic algebraic numbers.

On Joint Universality in the Selberg–Steuding Class

The famous Selberg class is defined axiomatically and consists of Dirichlet series satisfying four axioms (Ramanujan hypothesis, analytic continuation, functional equation, multiplicativity). The Selberg–Steuding class S is a complemented Selberg class by an arithmetic hypothesis related to the distribution of prime numbers. In this paper, a joint universality theorem for the functions L from the class S on the approximation of a collection of analytic functions by shifts L(s+ia1τ),…,L(s+iarτ), where a1,…,ar are real algebraic numbers linearly independent over the field of rational numbers, is obtained. It is proved that the set of the above approximating shifts is infinite, its lower density and, with some exception, density are positive. For the proof, a probabilistic method based on weak convergence of probability measures in the space of analytic functions is applied together with the Backer theorem on linear forms of logarithms and the Mergelyan theorem on approximation of analytic functions by polynomials.

Companions of fields of rational and real algebraic numbers

Companions of the field of rational numbers and a real-closed algebraic expansion of the field of rational numbers are studied. The description of existentially closed companions of a real-closed algebraic expansion of a field of rational numbers refers to the field of study of classical algebraic structures. The general theory of companions and existentially closed companions, built on the basis of Fraisse's classes in the works of A.T. Nurtazin, is included in the classical field of existentially closed theories in model theory. The basic concept of a companion: two models of the same signature are called companions if for any finite submodel of one of them, there is an isomorphic finite submodel in the other. This approach, applied to specific classical structures and their theories, provides new tools for the study of these objects. The study of the companion class of rational and algebraic real number fields reveals companion fields containing transcendental and possibly algebraic elements with special properties of polynomials defining these elements.

Fractional parts of powers of real algebraic numbers

Let α be a real algebraic number greater than 1. We establish an effective lower bound for the distance between an integral power of α and its nearest integer.

The digit exchanges in the rotational beta expansions of algebraic numbers

In this article, we investigate the β-expansions of real algebraic numbers. In particular, we give new lower bounds for the number of digit exchanges in the case where β is a Pisot or Salem number. Moreover, we define a new class of algebraic numbers, quasi-Pisot numbers and quasi-Salem numbers, which gives a generalization of Pisot numbers and Salem numbers.Our method is applicable also to the digit expansions of complex algebraic numbers, which gives a new estimate. In particular, we investigate the digits of rotational beta expansion considered by Akiyama and Caalim [3] and zeta-expansion by Surer [20], where the base is a quasi-Pisot or quasi-Salem number.

Automorphism groups and normal forms in Normaliz

We discuss the computation of automorphism groups and normal forms of cones and polyhedra in Normaliz and indicate its implementation via nauty. The types of automorphisms include integral, rational, Euclidean and combinatorial, as well as algebraic for polytopes defined over real algebraic number fields. Examples treated in detail are the icosahedron and linear-ordering polytopes whose Euclidean automorphism groups are determined.

On Inhomogeneous Extension of Thue–Roth’s Type Inequality with Moving Targets

AbstractLet $\Gamma \subset \overline{\mathbb Q}^{\times }$ be a finitely generated multiplicative group of algebraic numbers. Let $\delta ,\linebreak\beta \in \overline{\mathbb Q}^\times $ be algebraic numbers with $\beta $ irrational. In this paper, we prove that there exist only finitely many triples $(u, q, p)\in \Gamma \times \mathbb{Z}^2$ with $d = [\mathbb{Q}(u):\mathbb{Q}]$ such that $|\delta qu|&amp;gt;1$ and $$\begin{align*} &amp; 0&amp;lt;|\delta qu+\beta-p|&amp;lt;\frac{1}{H(u)^\varepsilon q^{d+\varepsilon}}, \end{align*}$$where $H(u)$ denotes the absolute Weil height. This is an inhomogeneous analogue of the main theorem in [ 2]. As an application of our result, we also prove a transcendence result, which states as follows: let $\alpha&amp;gt;1$ be a real number. Let $\beta $ be an algebraic irrational number and $\lambda $ be a non-zero real algebraic number. For a given real number $\varepsilon&amp;gt;0$, if there are infinitely many natural numbers $n$ for which $||\lambda \alpha ^n+\beta || &amp;lt; 2^{- \varepsilon n}$ holds true, then $\alpha $ is transcendental, where $||x||$ denotes the distance from its nearest integer. When $\alpha $ and $\beta $ both are algebraic numbers satisfying same conditions, then a particular result of Kulkarni et al. [ 3] asserts that $\alpha ^d$ is a Pisot number. When $\beta $ is an algebraic irrational, our result implies that no algebraic number $\alpha $ satisfies the inequality for infinitely many natural numbers $n$. Also, our result strengthens a result of Wagner and Ziegler [ 7].

Decision problem for a class of univariate Pfaffian functions

We address the decision problem for sentences involving univariate functions constructed from a fixed Pfaffian function of order 1. We present a new symbolic procedure solving this problem with a computable complexity based on the computation of suitable Sturm sequences. For a general Pfaffian function, we assume the existence of an oracle to determine the sign that a function of the class takes at a real algebraic number. As a by-product, we obtain a new oracle-free effective algorithm solving the same problem for univariate E-polynomials based on techniques that are simpler than the previous ones, and we apply it to solve a similar decision problem in the multivariate setting. Finally, we introduce a notion of Thom encoding for zeros of an E-polynomial and describe an algorithm for their computation.

The Art Gallery Problem is ∃ℝ-complete

The Art Gallery Problem (AGP) is a classic problem in computational geometry, introduced in 1973 by Victor Klee. Given a simple polygon 풫 and an integer k , the goal is to decide if there exists a set G of k guards within 풫 such that every point p ∈ 풫 is seen by at least one guard g ∈ G . Each guard corresponds to a point in the polygon 풫, and we say that a guard g sees a point p if the line segment pg is contained in 풫. We prove that the AGP is ∃ ℝ-complete, implying that (1) any system of polynomial equations over the real numbers can be encoded as an instance of the AGP, and (2) the AGP is not in the complexity class NP unless NP = ∃ ℝ. As a corollary of our construction, we prove that for any real algebraic number α, there is an instance of the AGP where one of the coordinates of the guards equals α in any guard set of minimum cardinality. That rules out many natural geometric approaches to the problem, as it shows that any approach based on constructing a finite set of candidate points for placing guards has to include points with coordinates being roots of polynomials with arbitrary degree. As an illustration of our techniques, we show that for every compact semi-algebraic set S ⊆ [0, 1] 2 , there exists a polygon with corners at rational coordinates such that for every p ∈ [0, 1] 2 , there is a set of guards of minimum cardinality containing p if and only if p ∈ S . In the ∃ ℝ-hardness proof for the AGP, we introduce a new ∃ ℝ-complete problem ETR-INV. We believe that this problem is of independent interest, as it has already been used to obtain ∃ ℝ-hardness proofs for other problems.

A dynamical characterization for monogenity at every level of some infinite -towers

AbstractWe consider a concrete family of $2$ -towers $(\mathbb {Q}(x_n))_n$ of totally real algebraic numbers for which we prove that, for each $n$ , $\mathbb {Z}[x_n]$ is the ring of integers of $\mathbb {Q}(x_n)$ if and only if the constant term of the minimal polynomial of $x_n$ is square-free. We apply our characterization to produce new examples of monogenic number fields, which can be of arbitrary large degree under the ABC-Conjecture.

On real and pseudo-real rational maps * *Partially supported by Projects Fondecyt 1190001 and 11170129. *Partially supported by Projects Fondecyt 1190001 and 11170129.

The moduli space M d , of complex rational maps of degree d ⩾ 2, is a connected complex orbifold which carries a natural real structure, coming from usual complex conjugation. Its real points are the classes of rational maps admitting antiholomorphic automorphisms. The locus of the real points decomposes as a disjoint union of the loci , consisting of the real rational maps, and , consisting of the pseudo-real ones. We obtain that, both and , are connected and that is disconnected. We also observe that the group of holomorphic automorphisms of a pseudo-real rational map is either trivial or a cyclic group. For every n ⩾ 1, we construct pseudo-real rational maps whose group of holomorphic automorphisms is cyclic of order n. As the field of moduli of a pseudo-real rational map is contained in , these maps provide examples of rational maps which are not definable over their field of moduli. It seems that these are the only explicit examples in the literature (Silverman) of rational maps which cannot be defined over their field of moduli. We provide explicit examples of real rational maps which cannot be defined over their field of moduli. Finally, we also observe that every real rational map, which admits a model over the algebraic numbers, can be defined over the real algebraic numbers.