Let R m be endowed with the Euclidean metric. The covering radius of a lattice Λ ⊂ R m is the least distance r such that, given any point of R m , the distance from that point to Λ is not more than r . Lattices can occur via the unit group of the ring of integers in an algebraic number field K , by applying a logarithmic embedding K ⁎ → R m . In this paper, we examine those lattices which arise from the cyclotomic number field Q ( ζ n ) , for a given positive integer n ≥ 5 such that n ≢ 2 ( mod 4 ) . We then provide improvements to a result of de Araujo in [3] , and conclude with an upper bound on the covering radius for this lattice in terms of n and the number of its distinct prime factors. In particular, we improve [3, Lemma 2] , and show that, asymptotically, it can be improved no further.

The workshop focused on various directions of arithmetic statistics in algebra and number theory. These include statistical problems for random polynomials and varieties, probabilistic Galois theory, and counting and distribution problems for algebraic functions, algebraic number fields, elliptic curves, L -functions, as well as arithmetic problems in non-abelian settings (eg, arithmetic statistics for algebraic groups).

Ranks of matrices of logarithms of algebraic numbers II: The matrix coefficient conjecture

The study of pure number fields plays a central role in algebraic number theory due to their simple defining equations and deep arithmetic properties. In this paper, we present a detailed analysis of the discriminants and integral bases of selected pure number fields. Within this framework, the paper develops an explicit and computable expression for the discriminant of the field (K). The formula depends exclusively on the prime power factorizations of (a) and (n), thereby avoiding unnecessary complexity and allowing for direct computation. This approach clarifies the role of ramification in pure number field extensions and provides insight into the structure of the corresponding rings of integers. The results obtained not only simplify discriminant calculations but also contribute to a deeper understanding of integral bases in pure number fields, offering a useful foundation for further theoretical and computational research.

On patterns occurring in binary algebraic $$\beta $$-expansions of algebraic numbers

Physical-layer security (PLS) provides an information-theoretic framework for securing wireless communications by exploiting channel and signal-structure asymmetries, thereby avoiding reliance on computational hardness assumptions. Within this setting, lattice codes and their algebraic constructions play a central role in achieving secrecy over Gaussian and fading wiretap channels. This article offers a comprehensive survey of lattice-based wiretap coding, covering foundational concepts in algebraic number theory, Construction A over number fields, and the structure of modular and unimodular lattice families. We review key secrecy metrics, including secrecy gain, flatness factor, and equivocation, and consolidate classical and recent results to provide a unified perspective that links wireless-channel models with their underlying algebraic lattice structures. In addition, we review a newly proposed family of p-modular lattices in Khodaiemehr, H., 2018 constructed from cyclotomic fields Q(ζp) for primes p≡1(mod4) via a generalized Construction A framework. We characterize their algebraic and geometric properties and establish a non-existence theorem showing that such constructions cannot be extended to prime-power cyclotomic fields Q(ζpn) with n>1. Finally, motivated by the fact that these p-modular lattices naturally yield mixed-signature structures for which classical theta series diverge, we integrate recent advances on indefinite theta series and modular completions. Drawing on Vignéras' differential framework and generalized error functions, we outline how modularly completed indefinite theta series provide a principled analytic foundation for defining secrecy-relevant quantities in the indefinite setting. Overall, this work serves both as a survey of algebraic lattice techniques for PLS and as a source of new design insights for secure wireless communication systems.

Abstract This paper studies the relation among the number of spanning trees of intermediate graphs in a Galois cover, building on results for $(\mathbb{Z}/2\mathbb{Z})^{m}$-covers previously established by Hammer, Mattman, Sands, and Vallières. We generalize their results to arbitrary finite Galois covers. Using the Ihara zeta function and the Artin–Ihara $L$-function, we prove two formulas which are graph-theoretic analogues of Kuroda’s formula and the Brauer–Kuroda relations in algebraic number theory. Furthermore, we prove that a spanning tree formula does not exist if the Galois group is cyclic.

Let $k_{0}$ be an algebraic number field of finite degree, $S_{0}$ be a finite set of primes and $L_{S_{0}}$ be the field obtained by adjoining to $k_{0}$ all primitive $q$-th roots of unity, where $q$ runs over all primes not belonging to $S_{0}$. We shall consider, for an odd prime $l$, the maximal unramified pro-$l$ abelian extension of $L_{S_{0}}$ and investigate the structure of this Galois group with certain cyclotomic action.

When studying pure number fields, the discriminant and integral basis are cornerstone ideas that shed light on their structure and mathematical characteristics. The algebraic invariants and arithmetic behavior of a number field are affected by the discriminant, which encodes crucial information regarding the field’s ramification and the geometry of its ring of integers. To examine the field’s features, like the structure of its ideal class group and the solutions to Diophantine equations, an integral basis is a set of elements in the ring of integers that forms a basis over the integers. Class number analysis, ideal class group determination, and Hilbert symbol computing are only a few of the many important areas of number theory that benefit greatly from these ideas. Computerising discriminants for fields of high degree and discovering minimal integral bases for fields with complex ramification remain challenging tasks. We still require a deeper understanding of algebraic number theory and more advanced computational approaches to address these challenges, despite the significant progress we’ve made. In mathematical physics, coding theory, and cryptography, where the algebraic properties of number fields influence the efficacy and security of various algorithms, the study of integral bases and discriminants is fundamental for both theoretical and practical reasons.

Abstract For a complex number x , ∥ x ∥ := min ⁡ { | x - m | : m ∈ ℤ } {\|x\|:=\min\{|x-m|:m\in\mathbb{Z}\}} . Let k ≥ 1 {k\geq 1} be an integer, and let K be a number field. Let α 1 , … , α k {\alpha_{1},\dots,\alpha_{k}} be algebraic numbers with | α i | ≥ 1 {|\alpha_{i}|\geq 1} and let d i {d_{i}} denotes the degree of α i {\alpha_{i}} for 1 ≤ i ≤ k {1\leq i\leq k} . Set d = d 1 + ⋯ + d k {d=d_{1}+\cdots+d_{k}} . In this article, we show that if the inequality 0 < ∥ λ 1 ⁢ q ⁢ α 1 n + ⋯ + λ k ⁢ q ⁢ α k n ∥ < θ n q d + ε {0<\|\lambda_{1}q\alpha^{n}_{1}+\cdots+\lambda_{k}q\alpha^{n}_{k}\|<\frac{% \theta^{n}}{q^{d+\varepsilon}}} has infinitely many solutions in ( n , q , λ 1 , … , λ k ) ∈ ℕ 2 × ( K × ) k {(n,q,\lambda_{1},\dots,\lambda_{k})\in\mathbb{N}^{2}\times(K^{\times})^{k}} with absolute logarithmic Weil height of λ i {\lambda_{i}} is small compared to n and some θ ∈ ( 0 , 1 ) {\theta\in(0,1)} , then, in particular, the tuple ( λ 1 ⁢ q ⁢ α 1 n , … , λ k ⁢ q ⁢ α k n ) {(\lambda_{1}q\alpha^{n}_{1},\dots,\lambda_{k}q\alpha^{n}_{k})} is pseudo-Pisot, and at least one of α i {\alpha_{i}} is an algebraic integer. This result can be viewed as Roth-type theorem for linear combinations of powers of algebraic numbers over ℚ ¯ {\overline{\mathbb{Q}}} . The case q = 1 {q=1} was recently proved in [A. Kulkarni, N. M. Mavraki and K. D. Nguyen, Algebraic approximations to linear combinations of powers: An extension of results by Mahler and Corvaja–Zannier, Trans. Amer. Math. Soc. 371 2019, 6, 3787–3804], which is a generalization of Mahler’s question proved in [P. Corvaja and U. Zannier, On the rational approximations to the powers of an algebraic number: Solution of two problems of Mahler and Mendès France, Acta Math. 193 2004, 2, 175–191]. As a consequence of our result, we obtain the following generalization of this question: let α > 1 {\alpha>1} be an algebraic number with d = [ ℚ ( α ) : ℚ ] {d=[\mathbb{Q}(\alpha):\mathbb{Q}]} . For a given ε > 0 {\varepsilon>0} , if the inequality 0 < ∥ λ ⁢ q ⁢ α n ∥ < θ n q d + ε 0<\|\lambda q\alpha^{n}\|<\frac{\theta^{n}}{q^{d+\varepsilon}} has infinitely many solutions in the tuples ( n , q , λ ) ∈ ℕ 2 × K × {(n,q,\lambda)\in\mathbb{N}^{2}\times K^{\times}} with absolute logarithmic Weil height of λ is small compared to n and θ ∈ ( 0 , 1 ) {\theta\in(0,1)} , then some power of α is a Pisot number. As an application of this result, we deduce the transcendence of certain infinite products of algebraic numbers.

In this paper, we show that there is no irreducible polynomial \(f(x)\) of degree \(2p\) (\(p\geq5\) is a prime number) over \({\mathbb Q}\) whose three distinct roots sum up to zero. This extends some earlier results on linear relations between three algebraic numbers. In particular, let \(d\) be the smallest positive integer not a multiple of \(3\), for which there exists an irreducible polynomial \(f(x)\) of degree \(d\) whose three distinct roots add up to zero. In 2015, Dubickas and Jankauskas found that \(10\leq d \leq 20\). As a corollary, we show that it is either \(d=16\) or \(d=20\).

The Tribonacci-Lucas sequence {𝑆 𝑛 } ≥0 is defined by the linear recurrence relation 𝑆 𝑛+3 = 𝑆 𝑛+2 + 𝑆 𝑛+1 + 𝑆 𝑛 , for 𝑛 ≥ 0, with the initial conditions 𝑆 0 = 𝑆 2 = 3 and 𝑆 1 = 1. A palindromic number is a number that remains the same when its digits are reversed. This paper uses Baker’s theory for nozero lower bounds for linear forms in logarithms of algebraic numbers, and reduction methods involving the theory of continued fraction to determine all Tribonacci-Lucas numbers that are palindromic concatenations of two distinct repdigits.

The Fundamental Theorem of Arithmetic (FTA), first formalized in Euclid’s Elements around 300 BCE, established the foundation for classical number theory, composed around 300 BCE. The principle of unique factorization later became central to the rise of modern mathematics. In the mid-19th century, mathematicians such as J.W.R. Dedekind and D. Hilbert extended number-theoretic questions into quadratic fields and rings of algebraic integers, creating the foundations of algebraic number theory. Steinitz’s work in the early twentieth century of 1910 further generalized algebraic structures, marking the beginning of abstract algebra as an independent field. The purpose of this essay is to examine the Fundamental Theorem of Algebra's proof. and applies it to several representative problems in elementary number theory. It then extends to related corollaries and conceptual developments of unique factorization, including notable cases in non-unique factorization rings where the property does not hold. Finally, it introduces approaches grounded in properties of the FTA that have been applied to the ongoing study of major open problems, including the Goldbach Conjecture. Overall, through both review and mathematical analysis, the paper shows that the structural foundation provided by the FTA underlies the verification and proof of many of the most difficult results and open conjectures in mathematics, including Fermat’s Last Theorem and the Goldbach Conjecture.

Abstract In this paper we invest in the non-vanishing of the Fourier coefficients of powers of the Dedekind eta function. This is reflected in non-vanishing properties of the D’Arcais polynomials. We generalize and improve results of Heim–Luca–Neuhauser and Żmija. We apply methods from algebraic number theory.

Abstract Let $S_g$ denote the genus g closed orientable surface. A coherent filling pair of simple closed curves, $(\alpha,\beta)$ in $S_g$ , is a filling pair that has its geometric intersection number equal to the absolute value of its algebraic intersection number. A minimally intersecting filling pair, $(\alpha,\beta)$ in $S_g$ , is one whose intersection number is the minimal among all filling pairs of $S_g$ . In this paper, we give a simple geometric procedure for constructing minimally intersecting coherent filling pairs on $S_g, \ g \geq 3,$ from the starting point of a coherent filling pair of curves on a torus. Coherent filling pairs have a natural correspondence to square-tiled surfaces, or origamis , and we discuss the origami obtained from the construction.

Abstract A classical result of Fekete gives necessary conditions on a compact set in the complex plane so that it contains infinitely many sets of conjugate algebraic integers. For such sets, we demonstrate the existence of a sequence of algebraic integers such that most of their conjugates eventually lie near the set, while maintaining a bound on heights. Finally, we examine properties satisfied by the limiting distribution of a sequence of algebraic numbers.

Let p p be an odd prime and k k be an algebraically closed field with characteristic p p . Booher and Cais [Algebra Number Theory 14 (2020), pp. 587–641] showed that the a a -number of a Z / p Z \mathbb Z/p \mathbb Z -Galois cover of curves ϕ : Y → X \phi : Y \to X must be greater than a lower bound determined by the ramification of ϕ \phi . In this paper, we provide evidence that the lower bound is optimal by finding examples of Artin-Schreier curves that have a a -number equal to its lower bound for all p p . Furthermore we use formal patching to generate infinite families of Artin-Schreier curves with a a -number equal to the lower bound in any characteristic.

Abstract The purpose of this paper is to explore the enigmatic world of numbers in Diophantine D(∓2) sets, revealing fresh insights into their intricate properties and profound connections. Diophantine D(∓2) sets, which are defined by integer-based Diophantine conditions, represent a compelling domain ripe for investigation. Our study delves into these sets, disregarding their cardinalities, aiming to unveil the concealed patterns and unique characteristics they harbor. Through meticulous scrutiny of their structure, our objective is to reveal the presence of prime numbers within these sets. In our investigation, we draw upon the foundational principles of Elementary and Algebraic Number Theory, invoking the Quadratic Reciprocity Law, Diophantine equations, and the enduring contributions of eminent mathematicians such as Gauss, Dirichlet, and Fermat. These tools and insights serve as guides in our exploration, ultimately leading to a deeper comprehension of the numbers within the Diophantine D(∓2) set and their significance within the broader landscape of mathematics.

Any pair of intersecting cylinders on a translation surface is "coherent," in that the geometric and algebraic intersection numbers of their core curves are equal (up to sign).In this paper, we investigate when a pair of multicurves can be simultaneously realized as the core curves of cylinders on some translation surface.Our main tools are surface topology and the "flat grafting" deformation introduced by Ser-Wei Fu.

We investigate the size of the set of [Formula: see text] for which the orbits of 1 are nonrecurrent under the [Formula: see text]-transformation [Formula: see text]. We prove that the set of [Formula: see text] with [Formula: see text] has Lebesgue measure 0. By constructing fractal sets in a neighborhood of an algebraic number [Formula: see text], we show that the set [Formula: see text] is of full Hausdorff dimension. The result of this paper significantly extends the related known result of [Formula: see text]-transformation and that of [Formula: see text] for [Formula: see text].