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A proper subset X of vertices in the graph G is a captive dominating set if it is a total dominating set and each vertex in X dominates at least one vertex which does not belong to X ; also, the captive domination number of G , denoted by γ ca (G) , is the cardinality of a minimum captive dominating set of G. In [M. N. Al-Harere, A. A. Omran and A. T. Breesam, Captive domination in graphs, Discrete Math. Algorithms Appl. 12(6) (2020) 2050076], Al-Harere et al., presented some lower and upper bounds for the captive domination number of G , using the number of vertices and edges. Here, we present some upper bounds for γ ca (G) , using independence number and clique number. Also, the captive domination number of split graphs is studied. Among other results, some lower bounds for the captive domination number, in terms of maximum degree and the number of vertices with maximum degree, are presented. [ABSTRACT FROM AUTHOR]

Recently, Widmer introduced a new sufficient criterion for the Northcott property on the finiteness of elements of bounded height in infinite algebraic extensions of number fields. We provide a simplification of Widmer's criterion when the extension is abelian, and use this to exhibit fields with the Northcott property that do not satisfy Widmer's new criterion. We also show how the construction of pseudo algebraically closed fields with the Northcott property, carried out in previous work using the first version of Widmer's criterion, can be simplified. [ABSTRACT FROM AUTHOR]

Classical continued fractions can be introduced in the field of p-adic numbers, where p-adic continued fractions offer novel perspectives on number representation and approximation. While numerous p-adic continued fraction expansion algorithms have been proposed by the researchers, the establishment of several excellent properties, such as the Lagrange's Theorem for classic continued fractions, which indicates that every quadratic irrationals can be expanded periodically, remains elusive. In this paper, we introduce several new algorithms designed for expanding algebraic numbers in \mathbb {Q}_p for a given prime p. We give an upper bound of the number of partial quotients for the expansion of rational numbers, and prove that for small primes p, our algorithm generates periodic continued fraction expansions for all quadratic irrationals. Experimental data demonstrates that our algorithms exhibit better performance in the periodicity of expansions for quadratic irrationals compared to the existing algorithms. Furthermore, for bigger primes p, we propose a potential approach to establish a p-adic continued fraction expansion algorithm. As before, the algorithm is designed to expand algebraic numbers in \mathbb {Q}_p, while generating periodic expansions for all quadratic irrationals in \mathbb {Q}_p. [ABSTRACT FROM AUTHOR]

We prove a theorem that generalizes Schmidt's Subspace Theorem in the context of metric diophantine approximation. To do so we reformulate the Subspace Theorem in the framework of homogeneous dynamics by introducing and studying a slope formalism and the corresponding notion of semistability for diagonal flows. [ABSTRACT FROM AUTHOR]

We suggest that straight-line programs designed for algebraic computations should be accompanied by a comprehensive complexity analysis that takes into account both the number of fundamental algebraic operations needed, as well as memory requirements arising during evaluation. We introduce an approach for formalizing this idea and, as illustration, construct and analyze straight-line programs for the Bruhat decomposition of d × d matrices with determinant 1 over a finite field of order q that have length O(d2log(q)) and require storing only O(log(q)) matrices during evaluation. [ABSTRACT FROM AUTHOR]

In this paper, we prove that, for β ∈ ℂ , every α ∈ ℂ has at most finitely many (possibly none at all) representations of the form α = d n ⁢ β n + d n - 1 ⁢ β n - 1 + ... + d 0 with nonnegative integers n , d n , d n - 1 , ... , d 0 if and only if β is a transcendental number or an algebraic number which has a conjugate over ℚ (possibly β itself) in the real interval (1 , ∞) . The nontrivial part here is to show that for every algebraic number β lying with its all conjugates in ℂ ∖ (1 , ∞) , there is α ∈ ℚ ⁢ (β) with infinitely many such representations. In a particular case, when β is a quadratic algebraic number, this was recently established by Kala and Zindulka. [ABSTRACT FROM AUTHOR]

We discuss the p -adic Weierstrass zeta functions associated with elliptic curves defined over the field of algebraic numbers and linear relations for their values in the p -adic domain. These results are extensions of the p -adic analogues of results given by Wüstholz in the complex domain [see A. Baker and G. Wüstholz, Logarithmic Forms and Diophantine Geometry , New Mathematical Monographs, 9 (Cambridge University Press, Cambridge, 2007), Theorem 6.3] and also generalise a result of Bertrand to higher dimensions ['Sous-groupes à un paramètre p -adique de variétés de groupe', Invent. Math. 40 (2) (1977), 171–193]. [ABSTRACT FROM AUTHOR]

The Pólya group of an algebraic number field is the subgroup generated by the ideal classes of the products of prime ideals of equal norm inside the ideal class group. Inspired by a recent work on consecutive quadratic fields with large class numbers by Cherubini et al., we extend the notion of consecutiveness of number fields to certain parametric families of cyclic cubic fields and bi-quadratic fields and address the question of the existence of infinitely many such consecutive fields with large Pólya groups. This extends a recent result of the second author and Saikia for totally real bi-quadratic fields. [ABSTRACT FROM AUTHOR]

In this paper, we develop an explicit method to express finite algebraic numbers (in particular, certain idempotents among them) in terms of linear recurrent sequences, and give applications to the characterization of the splitting primes in a given finite Galois extension over the rational field. [ABSTRACT FROM AUTHOR]

In this paper, we show that each non-degenerate Lucas sequence contains only finitely many terms which can be written as products of double factorials or as sums or differences of double factorials. In particular, we find all the solutions to the products of double factorials which are sums or differences of two Fibonacci numbers, and Fibonacci numbers which are sums or differences of two double factorials. [ABSTRACT FROM AUTHOR]

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 ⁡ (M k) 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. [ABSTRACT FROM AUTHOR]

The primary purpose of this paper is to generalize the classical Riemann zeta function to the setting of Krull monoids with torsion class groups. We provide a first study of the same generalization by extending Euler's classical product formula to the more general scenario of Krull monoids with torsion class groups. In doing so, the Decay Theorem is fundamental as it allows us to use strong atoms instead of primes to obtain a weaker version of the Fundamental Theorem of Arithmetic in the more general setting of Krull monoids with torsion class groups. Several related examples are exhibited throughout the paper, in particular, algebraic number fields for which the generalized Riemann zeta function specializes to the Dedekind zeta function. [ABSTRACT FROM AUTHOR]

In this paper, we propose a decision procedure of reachability for a linear system ξ ′ = A ξ + u , where the matrix A ′ s eigenvalues can be arbitrary algebraic number and the input u is a vector of trigonometric-exponential polynomials. If the initial set contains only one point, the reachability problem under consideration is reduced to the decidability of the sign of trigonometric-exponential polynomial and then achieved by being reduced to verification of a series of univariate polynomial inequalities through Taylor expansions of the related exponential functions and trigonometric functions. If the initial set is open semi-algebraic, we will propose a decision procedure based on OpenCAD and an algorithm of real root isolation derived from the sign-deciding procedure for the trigonometric-exponential polynomials. The experimental results indicate the efficiency of our approach. Under the assumption of Schanuel's Conjecture, the above procedures are complete for bounded time except for several cases. [ABSTRACT FROM AUTHOR]

Let \alpha be a non-zero algebraic number. Let K be the Galois closure of \mathbb {Q}(\alpha) with Galois group G and \bar {\mathbb {Q}} be the algebraic closure of \mathbb {Q}. In this article, among the other results, we prove the following. If f\in \bar {\mathbb {Q}}[G] is a non-zero element of the group ring \bar {\mathbb {Q}}[G] and \alpha is a given algebraic number such that f(\alpha ^n) is a non-zero algebraic integer for infinitely many natural numbers n, then \alpha is an algebraic integer. This result generalises the result of Polya [Rend. Circ Mat. Palermo, 40 (1915), pp. 1–16], Corvaja and Zannier [Acta Math. 193 (2004), pp. 175–191] and Philippon and Rath [J. Number Theory 219 (2021), pp. 198–211]. We also prove the analogue of this result for rational functions with algebraic coefficients. Inspired by a result of B. de Smit [J. Number Theory 45 (1993), pp. 112–116], we prove a finite version of the Polya type result for a binary recurrence sequences of non-zero algebraic numbers. In order to prove these results, we apply the techniques of Corvaja and Zannier along with the results of Kulkarni et al. [Trans. Amer. Math. Soc. 371 (2019), pp. 3787–3804], which are applications of the Schmidt subspace theorem. [ABSTRACT FROM AUTHOR]

We prove explicit lower bounds for linear forms in two p -adic logarithms. More specifically, we establish explicit lower bounds for the p -adic distance between two integral powers of algebraic numbers, that is, | Λ | p = | α 1 b 1 − α 2 b 2 | p (and corresponding explicit upper bounds for v p (Λ)), where α 1 , α 2 are numbers that are algebraic over Q and b 1 , b 2 are positive rational integers. This work is a p -adic analogue of Gouillon's explicit lower bounds in the complex case. Our upper bound for v p (Λ) has an explicit constant of reasonable size and the dependence of the bound on B (a quantity depending on b 1 and b 2) is log ⁡ B , instead of (log ⁡ B) 2 as in the work of Bugeaud and Laurent in 1996. [ABSTRACT FROM AUTHOR]

In this paper, we propose a new algebraic winding number and prove that it computes the number of complex roots of a polynomial in a rectangle, including roots on edges or vertices with appropriate counting. The definition makes sense for the algebraic closure C = R [ i ] of a real closed field R , and the root counting result also holds in this case. We study in detail the properties of the algebraic winding number defined in [3] with respect to complex root counting in rectangles. We extend both winding numbers to rational functions, obtaining then algebraic versions of the argument principle for rectangles. [ABSTRACT FROM AUTHOR]

Let K = ℚ(ϑ) be an algebraic number field with ϑ satisfying a monic irreducible polynomial f (x) of degree n over ℚ. The polynomial f (x) is said to be monogenic if {1,ϑ,..., ϑn-1} is an integral basis of K. Deciding whether or not a monic irreducible polynomial is monogenic is an important problem in algebraic number theory. In an attempt to answer this problem for a certain family of polynomials, Jones ['A brief note on some infinite families of monogenic polynomials', Bull. Aust. Math. Soc. 100 (2019), 239-244] conjectured that if n ≥ 3, 1 ≤ m ≤ n - 1, gcd(n, mB) = 1 and A is a prime number, then the polynomial xn + A(Bx + 1)m ∈ ℤ[x] is monogenic if and only if nn + (-1)n+mBn(n - m)n-mmm A is square-free. We prove that this conjecture is true. [ABSTRACT FROM AUTHOR]

Elsner, Luca and Tachiya proved in [4] that the values of the Jacobi-theta constants θ 3 (m τ) and θ 3 (n τ) are algebraically independent over Q for distinct integers m , n under some conditions on τ . On the other hand, in [3] Elsner and Tachiya also proved that three values θ 3 (m τ) , θ 3 (n τ) and θ 3 (ℓ τ) are algebraically dependent over Q . In this article we prove the non-vanishing of linear forms in θ 3 (m τ) , θ 3 (n τ) and θ 3 (ℓ τ) under various conditions on m , n , ℓ , and τ . Among other things we prove that for odd and distinct positive integers m , n > 3 the three numbers θ 3 (τ) , θ 3 (m τ) and θ 3 (n τ) are linearly independent over Q ¯ when τ is an algebraic number of some degree greater or equal to 3. In some sense this fills the gap between the above-mentioned former results on theta constants. A theorem on the linear independence over C (τ) of the functions θ 3 (a 1 τ) , ⋯ , θ 3 (a m τ) for distinct positive rational numbers a 1 , ⋯ , a m is also established. [ABSTRACT FROM AUTHOR]

We investigate the relations between the rings E , G and D of values taken at algebraic points by arithmetic Gevrey series of order either −1 (E -functions), 0 (analytic continuations of G -functions) or 1 (renormalization of divergent series solutions at ∞ of E -operators) respectively. We prove in particular that any element of G can be written as multivariate polynomial with algebraic coefficients in elements of E and D , and is the limit at infinity of some E -function along some direction. This prompts to defining and studying the notion of mixed functions, which generalizes simultaneously E -functions and arithmetic Gevrey series of order 1. Using natural conjectures for arithmetic Gevrey series of order 1 and mixed functions (which are analogues of a theorem of André and Beukers for E -functions) and the conjecture D ∩ E = Q ‾ (but not necessarily all these conjectures at the same time), we deduce a number of interesting Diophantine results such as an analogue for mixed functions of Beukers' linear independence theorem for values of E -functions, the transcendence of the values of the Gamma function and its derivatives at all non-integral algebraic numbers, the transcendence of Gompertz constant as well as the fact that Euler's constant is not in E. [ABSTRACT FROM AUTHOR]

Let {En}n=0 and {Pn}n=0 be sequences of Perrin and Padovan numbers, respectively. We have found all repdigits that can be written as the sum or product of En and Pm in the base η, where 2 = η = 10 and m = n. In addition, we have applied the theory of linear forms in logarithms of algebraic numbers and Baker-Davenport reduction method in Diophantine approximation approaches. [ABSTRACT FROM AUTHOR]

Using Schmidt's Subspace Theorem, this paper improves and extends an existing transcendence result for sequences of algebraic numbers. The theorems thus produced correspond to a central theorem on the irrationality of sequences due to Erdős. [ABSTRACT FROM AUTHOR]

We define a semi-Brouwerian almost distributive lattice as a description of a semi-Brouwerian algebra in the category of almost distributive lattices and derive a number of algebraic properties on them. We obtain a few equivalent conditions for an almost distributive lattice to become a semi-Brouwerian almost distributive lattice. Finally we identify a class of pseudo-complemented lattices and a class of semi-Brouwerian algebras in a semi-Brouwerian almost distributive lattice. [ABSTRACT FROM AUTHOR]

Let β be a Pisot or Salem number with β > 1, and let α1 and α2 be elements in Q (β) ∩ [0, 1). In this note, we prove that α1 and α2 have either the same tail greedy expansions in a base β or independent random greedy expansions in a base β. [ABSTRACT FROM AUTHOR]

Monogenity is a classical area of algebraic number theory that continues to be actively researched. This paper collects the results obtained over the past few years in this area. Several of the listed results were presented at a series of online conferences titled "Monogenity and Power Integral Bases". We also give a collection of the most important methods used in several of these papers. A list of open problems for further research is also given. [ABSTRACT FROM AUTHOR]

We provide a technique to obtain explicit bounds for problems that can be reduced to linear forms in three complex logarithms of algebraic numbers. This technique can produce bounds significantly better than general results on lower bounds for linear forms in logarithms. We give worked examples to demonstrate both the use of our technique and the improvements it provides. Publicly shared code is also available. [ABSTRACT FROM AUTHOR]

The original article can be found online at https://doi.org/10.1007/s00222-024-01257-1The publication of this article unfortunately contained mistakes. The style of the references was not correct. The corrected reference list is given below.The original article has been corrected.Publisher's NoteSpringer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.By Yuan Liu; Melanie Matchett Wood and David Zureick-BrownReported by Author; Author; Author [Extracted from the article]

We establish the first effective improvements on the Liouville inequality for approximation to complex non-real algebraic numbers by complex algebraic numbers of degree at most 4. [ABSTRACT FROM AUTHOR]

Two algebraic number fields are called arithmetically equivalent if the Dedekind zeta functions of the fields coincide. We show that if a G-extension contains non-conjugate arithmetically equivalent fields and there is an injection from G to another group H inducing an isoclinism between G and H, then there are non-conjugate arithmetically equivalent fields inside an H-extension. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring with unity and let n ≥ 3 be an integer. Let SLn (R) and En (R) denote respectively the special linear group and elementary subgroup of the general linear group GLn (R). A result of Hurwitz says that the special linear group of size atleast three over the ring of integers of an algebraic number field is finitely generated. A celebrated theorem in group theory states that finite simple groups are two-generated. Since the special linear group of size atleast three over the ring of integers is not a finite simple group, we expect that it has more than two generators. In the special case, where R is the ring of integers of an algebraic number field which is not totally imaginary, we provide for En (R) (and hence SLn (R)) a set of Gow-Tamburini matrix generators, depending on the minimal number of generators of R as a ℤ-module. [ABSTRACT FROM AUTHOR]

Let P (x) : = a d x d + ⋯ + a 0 ∈ ℚ [ x ] , a d > 0 , be a polynomial of degree d ≥ 2. Let (x n) be a sequence of integers satisfying x n + 1 = P (x n) for all  n = 0 , 1 , 2 , ... and x n → ∞ as  n → ∞. Set α : = lim n → ∞ x n d − n . Then, under the assumption a d 1 / (d − 1) ∈ ℚ , in a recent result by [A. Dubickas, Transcendency of some constants related to integer sequences of polynomial iterations, Ramanujan J. 57 (2022) 569–581], either α is transcendental or α can be an integer or a quadratic Pisot unit with α − 1 being its conjugate over ℚ. In this paper, we study the nature of such α without the assumption that a d 1 / (d − 1) is in ℚ , and we prove that either the number α is transcendental, or α h is a Pisot number with h being the order of the torsion subgroup of the Galois closure of the number field ℚ α , a d − 1 d − 1 . Other results presented in this paper investigate the solutions of the inequality | | q 1 α 1 n + ⋯ + q k α k n + β | | < n in (n , q 1 , ... , q k) ∈ ℕ × (K ×) k , considering whether β is rational or irrational. Here, K represents a number field, and ∈ (0 , 1). The notation | | x | | denotes the distance between x and its nearest integer in ℤ. [ABSTRACT FROM AUTHOR]

For every algebraic number κ on the unit circle which is not a root of unity we prove the existence of a strict sequence of algebraic numbers whose height tends to zero, such that the averages of the evaluation of f κ (z) = log | z − κ | at the conjugates are essentially bounded from above by − h (κ). This completes a characterization on functions f κ initiated by Autissier and Baker–Masser, who cover the cases κ = 2 and | κ | ≠ 1 , respectively. Using the same ideas we also prove analogues in the p -adic setting. [ABSTRACT FROM AUTHOR]

In this paper, we give constructions for infinite sequences of finite nonlinear locally recoverable codes ⊆ ∏ i = 1 N q i over a product of finite fields arising from basis expansions in algebraic number fields. The codes in our sequences have increasing length and size, constant rate, fixed locality, and minimum distance going to infinity. [ABSTRACT FROM AUTHOR]

For an algebraic number α we consider the orders of the reductions of α in finite fields. In the case where α is an integer, it is known by the work on Artin's primitive root conjecture that the order is 'almost always almost maximal' assuming the Generalized Riemann Hypothesis (GRH), but unconditional results remain modest. We consider higher-degree variants under GRH. First, we modify an argument of Roskam to settle the case where α and the reduction have degree two. Second, we give a positive lower density result when α is of degree three and the reduction is of degree two. Third, we give higher-rank results in situations where the reductions are of degree two, three, four or six. As an application we give an almost equidistribution result for linear recurrences modulo primes. Finally, we present a general result conditional to GRH and a hypothesis on smooth values of polynomials at prime arguments. [ABSTRACT FROM AUTHOR]

In this article, we prove that the product of two algebraic numbers of degrees 4 and 6 over Q{\mathbb{Q}} cannot be of degree 8. This completes the classification of so-called product-feasible triplets (a,b,c)∈N3\left(a,b,c)\in {{\mathbb{N}}}^{3} with a≤b≤ca\le b\le c and b≤7b\le 7. The triplet (a,b,c)\left(a,b,c) is called product-feasible if there are algebraic numbers α,β\alpha ,\beta , and γ\gamma of degrees a,ba,b, and cc over Q{\mathbb{Q}}, respectively, such that αβ=γ\alpha \beta =\gamma . In the proof, we use a proposition that describes all monic quartic irreducible polynomials in Q[x]{\mathbb{Q}}\left[x] with four roots of equal moduli and is of independent interest. We also prove a more general statement, which asserts that for any integers n≥2n\ge 2 and k≥1k\ge 1, the triplet (a,b,c)=(n,(n−1)k,nk)\left(a,b,c)=\left(n,\left(n-1)k,nk) is product-feasible if and only if nn is a prime number. The choice (n,k)=(4,2)\left(n,k)=\left(4,2) recovers the case (a,b,c)=(4,6,8)\left(a,b,c)=\left(4,6,8) as well.

We provide a lower bound for the number of components of the algebraic monodromy group in the situation of a K 3 surface over a number field k. In the CM case, our bound is sharp. As an application, we describe, in the case of CM, the jump character [3, Definition 2.4.6] entirely in terms of the endomorphism field and the geometric Picard rank. [ABSTRACT FROM AUTHOR]

We fix a number field K and study statistical properties of the ring K [ γ ] ∩ K as γ varies over algebraic numbers of a fixed degree n ≥ 2. Given k ≥ 1 , we explicitly compute the density of γ for which K [ γ ] ∩ K = K [ 1 / k ] and show that this does not depend on the number field K. In particular, we show that the density of γ for which K [ γ ] ∩ K = K is ζ (n + 1) ζ (n) . In a recent paper [Singhal and Lin, Primes in denominators of algebraic numbers, Int. J. Number Theory (2023), doi:10.1142/S1793042124500167], the authors define X (K , γ) to be a certain finite subset of Spec ( K) and show that X (K , γ) determines the ring K [ γ ] ∩ K. We show that if 1 , 2 ∈ Spec ( K) satisfy 1 ∩ ℤ ≠ 2 ∩ ℤ , then the events 1 ∈ X (K , γ) and 2 ∈ X (K , γ) are independent. As t → ∞ , we study the asymptotics of the density of γ for which | X (K , γ) | = t. [ABSTRACT FROM AUTHOR]

We point out that there is an error in the proof of Theorem 1.1 in [The discriminant of compositum of algebraic number fields, Int. J. Number Theory 15 (2019) 353–360]. We also prove that the result of this theorem holds with an additional hypothesis. However, it is an open problem whether the result of the theorem is true in general or not. [ABSTRACT FROM AUTHOR]

A family of measures on a given set of real algebraic numbers of arbitrary fixed degree is considered. The paper presents a method for constructing a sequence of measures that weakly converges to the Cauchy distribution. The method is based on the theory of random polynomials. [ABSTRACT FROM AUTHOR]

We construct the Hamiltonian Feynman, Lagrangian Feynman, and Feynman–Kac formulas for the solution of the Cauchy problem with the Schrödinger operator −MgDα − V, where Dα is the Vladimirov operator and Mg is the operator of multiplication by a real-valued function g defined on the d-dimensional space A K d of finite adeles over the algebraic number field K. [ABSTRACT FROM AUTHOR]

A generalized construction procedure for algebraic number systems is hereby presented. This procedure offers an efficient representation and computation method for complex numbers, quaternions, and other algebraic structures. The construction method is then illustrated across a range of examples. In particular, the novel developments reported herein provide a generalized form of the Cayley–Dickson construction through involutive dimagmas, thereby allowing for the treatment of more general spaces other than vector spaces, which underlie the associated algebra structure. [ABSTRACT FROM AUTHOR]

The article proves the following statement: in any hyperelliptic field L defined over the field of algebraic numbers K which having non-trivial units of the ring of integer elements of the field L, there is an element for which the period length of the continued fraction is greater any pre-given number. [ABSTRACT FROM AUTHOR]

Let q be a prime number and K = Q () be an algebraic number field with a root of an irreducible polynomial x q s − a x − b having integer coefficients. In this paper, we provide some explicit conditions involving only s , q , a , b for which K is non-monogenic. As an application, in the special case of a = 0 and q = 2 , we show that if s ≥ 2 and 3 2 divides b − 1 , then K is not monogenic. We illustrate our results through examples. [ABSTRACT FROM AUTHOR]

Conway and Coxeter have shown that frieze patterns over positive rational integers are in bijection with triangulations of polygons. An investigation of frieze patterns over other subsets of the complex numbers has recently been initiated by Jørgensen and the first two authors. In this paper, we first show that a ring of algebraic numbers has finitely many units if and only if it is an order in a quadratic number field Q(d)$\mathbb {Q}(\sqrt {d})$ where d<0$d<0$. We conclude that these are exactly the rings of algebraic numbers over which there are finitely many non‐zero frieze patterns for any given height. We then show that apart from the cases d∈{−1,−2,−3,−7,−11}$d\in \lbrace -1,-2,-3,-7,-11\rbrace$ all non‐zero frieze patterns over the rings of integers Od$\mathcal {O}_d$ for d<0$d<0$ have only integral entries and hence are known as (twisted) Conway–Coxeter frieze patterns. [ABSTRACT FROM AUTHOR]

In this paper we prove that every totally real algebraic integer λ of degree d ≥ 2 occurs as an eigenvalue of some tree of height at most d (d + 1) / 2 + 3 . In order to prove this, for a given algebraic number α ≠ 0 , we investigate an additive semigroup that contains zero and is closed under the map x ↦ α / (1 - x) for x ≠ 1 . The problem of finding the smallest such semigroup seems to be of independent interest. [ABSTRACT FROM AUTHOR]

In this paper, we construct signal constellations from lattices in complex hyperbolic spaces. To construct a hyperbolic lattice, we identify an arithmetic Fuchsian group with the group of units O1 of a natural quaternion order O ⊂ V, in which V is some quaternion algebra over an algebraic number field. The arithmetic Fuchsian group, G, is isomorphic to the fundamental group of a regular hyperbolic polygon P, and an oriented compact surface arises from the pairwise identification of its opposite edges. The polygon P is the fundamental region associated with a regular tessellation {p, q}. The main contribution of this paper is to employ the Reidemeister-Schreier rewriting process to use proper decompositions of the full symmetry group of a tessellation {p, q}, which allows the matching of uniform signal constellations to quotient groups of G. In this direction, we consider the labelling of phaseshift keying (PSK) and amplitude-phase keying (APK) signal constellations diagrams via three approaches: cyclic quotient groups, a direct product of cyclic quotient groups and semi-direct product of cyclic groups (the dihedral group case). [ABSTRACT FROM AUTHOR]

For a smooth projective variety X over an algebraic number field k a conjecture of Bloch and Beilinson predicts that the kernel of the Albanese map of X is a torsion group. In this article we consider a product X = C 1 × ⋯ × C d of smooth projective curves and show that if the conjecture is true for any subproduct of two curves, then it is true for X. For a product X = C 1 × C 2 of two curves over Q with positive genus we construct many nontrivial examples that satisfy the weaker property that the image of the natural map J 1 (Q) ⊗ J 2 (Q) → ε CH 0 (C 1 × C 2) is finite, where J i is the Jacobian variety of C i . Our constructions include many new examples of non-isogenous pairs of elliptic curves E 1 , E 2 with positive rank, including the first known examples of rank greater than 1. Combining these constructions with our previous result, we obtain infinitely many nontrivial products X = C 1 × ⋯ × C d for which the analogous map ε has finite image. [ABSTRACT FROM AUTHOR]

The rational field Q is highly desired in many applications. Algorithms using the rational number field Q algebraic number fields use only integer arithmetics and are easy to implement. Therefore, studying and designing systems and expansions with coefficients in Q or algebraic number fields is particularly interesting. This paper discusses constructing quasi-tight framelets with symmetry over an algebraic field. Compared to tight framelets, quasi-tight framelets have very similar structures but much more flexibility in construction. Several recent papers have explored the structure of quasi-tight framelets. The construction of symmetric quasi-tight framelets directly applies the generalized spectral factorization of 2 × 2 matrices of Laurent polynomials with specific symmetry structures. We adequately formulate the latter problem and establish the necessary and sufficient conditions for such a factorization over a general subfield F of C , including algebraic number fields as particular cases. Our proofs of the main results are constructive and thus serve as a guideline for construction. We provide several examples to demonstrate our main results. [ABSTRACT FROM AUTHOR]

In this paper, considering L-algebras, which include a significant number of other algebraic structures, we present a definition of modules on L-algebras (Lmodules). Then we provide some examples and obtain some results on L-modules. Also, we present definitions of prime ideals of L-algebras and L-submodules (prime L-submodules) of L-modules, and investigate the relationship between them. Finally, by proving a number of theorems, we provide some conditions for having prime L-submodules. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]