Algebraic Integers Research Articles

Linear independence of special values of Jacobi-theta constants

Let m≥2 be any integer and let β>11$$\\end{document}]]> be a real algebraic integer such that all its other Galois conjugates have absolute value less than or equal to 1. Let a1,a2,…,am be distinct positive integers. In this article we prove that the following infinite sums 1,∑n=1∞1βa1n2,∑n=1∞1βa2n2,…,∑n=1∞1βamn2are Q(β)-linearly independent. As a consequence, we prove the linear independence of special values of Jacobi-theta constants.

Explicit discriminant of a class of polynomial, monogenity and Galois group

Let K = Q ( θ ) where θ is a root of an irreducible polynomial f ( x ) = x n − km ( x k + a ) m + b ∈ Z [ x ] , 1 ≤ km < n and Z K denote the ring of algebraic integers of K. In this paper, we explicitly compute the discriminant of f ( x ) and provide necessary and sufficient conditions involving only a, b, m, k, n for f ( x ) to be monogenic. Moreover, we characterize all the primes dividing the index of the subgroup Z [ θ ] in Z K . As an application, we also provide a class of monogenic polynomials having non square-free discriminant and Galois group S n , the permutation group on n letters.

A lower bound to the height of algebraic integers not in the fixed field of the inertia groups

A lower bound to the height of algebraic integers not in the fixed field of the inertia groups

On algebraic degrees of certain exponential sums over finite fields

Abstract From a complex perspective, Birch and Bombieri (1985) proposed the study of an important class of n-dimensional exponential sums for n = 4 {n=4} , which was then generalized by Katz (1987) to general positive integer n. Our previous paper (2022) enhanced the preceding work from a complex point of view and expanded the subject to a p-adic point of view for any positive integer n. In this paper, we investigate the algebraic degree of the above exponential sum as an algebraic integer.

INFINITE ALGEBRAIC INDEPENDENCE OF POLYADIC SERIES WITH PERIODIC COEFFICIENTS

Consider sequences of integers 𝑎𝑛(𝑘,𝑗), 𝑘 = 1, … , 𝑇𝑗, 𝑗 = 1, … , 𝑚 such that 𝑎𝑛(𝑘,𝑗) = 𝑎𝑛(𝑘+,𝑇𝑗)𝑗 , 𝑗 = 1, … , 𝑚, 𝑘 = 1, … , 𝑇𝑗, 𝑛 = 0, 1, …, and consider the series 𝐹𝑗,𝑘(𝑧) = ∞ ∑ 𝑛=0 𝑎(𝑘,𝑗)𝑛 𝑛!𝑧𝑛, 𝑘 = 1, … , 𝑇𝑗, 𝑗 = 1, … , 𝑚. The conditions are established under which the set of series 𝐹𝑗,𝑘(𝑧), 𝑘 = 2, … , 𝑇𝑗, 𝑗 = 1, … , 𝑚 and the Euler series Φ(𝑧) = ∞ ∑ 𝑛=0 𝑛!𝑧𝑛 are algebraically independent over ℂ(𝑧) and for any algebraic integer γ ≠ 0, their values at the point γ are infinitely algebraically independent.

LOWER BOUNDS FOR THE MAHLER MEASURE AND INERTIA DEGREES OF PRIMES

Abstract We investigate the relationship between lower bounds for the Mahler measure and splitting of primes, and prove various lower bounds for the Mahler measure of algebraic integers in terms of the least common multiples of all inertia degrees of primes. The results generalise work of the second author and Kumar [‘Lehmer’s problem and splitting of rational primes in number fields’, Acta Math. Hungar.169(2) (2023), 349–358].

On the quantum security of high-dimensional RSA protocol

Abstract The idea of extending the classical RSA protocol using algebraic number fields was introduced by Takagi and Naito (Construction of RSA cryptosystem over the algebraic field using ideal theory and investigation of its security. Electron Commun Japan Part III Fund Electr Sci. 2000;83:19–29). Recently, Zheng et al. proposed the use of the ring of algebraic integers of an algebraic number field and the lattice theory to present a high-dimensional form of RSA. The authors claim that their proposal is post-quantum and is significant both from the theoretical and practical point of view. In this article, we prove that the security of Zheng et al.’s scheme is still based on the factorization problem, and we present a practical quantum attack on this proposed scheme, our attack is a quantum polynomial time algorithm that employs Shor’s algorithm as a subroutine.

Periodicity of bipartite walk on biregular graphs with conditional spectra

In this paper we study a class of discrete quantum walks, known as bipartite walks. These include the well-known Grover’s walks. A discrete quantum walk is given by the powers of a unitary matrix U indexed by arcs or edges of the underlying graph. The walk is periodic if U k = I for some positive integer k. Kubota has given a characterization of periodicity of Grover’s walk when the walk is defined on a regular bipartite graph with at most five eigenvalues. We extend Kubota’s results—if a biregular graph G has eigenvalues whose squares are algebraic integers with degree at most two, we characterize periodicity of the bipartite walk over G in terms of its spectrum. We apply periodicity results of bipartite walks to get a characterization of periodicity of Grover’s walk on regular graphs.

New lower bounds for the Schur-Siegel-Smyth trace problem

We derive and implement a new way to find lower bounds on the smallest limiting trace-to-degree ratio of totally positive algebraic integers and improve the previously best known bound to 1.80203. Our method adds new constraints to Smyth’s linear programming method to decrease the number of variables required in the new problem of interest. This allows for faster convergence recovering Schur’s bound in the simplest case and Siegel’s bound in the second simplest case of our new family of bounds. We also prove the existence of a unique optimal solution to our newly phrased problem and express the optimal solution in terms of polynomials. Lastly, we solve this new problem numerically with a gradient descent algorithm to attain the new bound 1.80203.

Algebraic integers with conjugates in a prescribed distribution

Algebraic integers with conjugates in a prescribed distribution

ON HYPERBOLIC INTEGERS

The purpose of this paper is to investigate an ℓ-ring structure of algebraic integers from an arithmetic point of view. We endow the algebra D of hyperbolic numbers with its standard f-algebra structure [7]. We introduce the ring of hyperbolic integers Zh as a sub f-ring of the ring ZD of integers of D. Next, we prove that Zh is the unique, up to ring isomorphism, Archimedean f-ring of quadratic integers. Our study focuses on arithmetic properties of Zh related to its latticeordered structure. We show that many basic properties of the ring of integers Z such as primes, unique factorization theorem and the notions of floor and ceiling functions can be extended to Zh. A surprising fact is that prime numbers seen as hyperbolic integers are semiprimes. We also obtain some properties of hyperbolic Gaussian integers. As an application, we discuss the Dirichlet divisor problem using hyperbolic intervals.

Misiurewicz polynomials and dynamical units, part II

Fix an integer d≥2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d\\ge 2$$\\end{document}. The parameters c0∈Q¯\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$c_0\\in \\overline{\\mathbb {Q}}$$\\end{document} for which the unicritical polynomial fd,c(z)=zd+c∈C[z]\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f_{d,c}(z)=z^d+c\\in \\mathbb {C}[z]$$\\end{document} has finite postcritical orbit, also known as Misiurewicz parameters, play a significant role in complex dynamics. Recent work of Buff, Epstein, and Koch proved the first known cases of a long-standing dynamical conjecture of Milnor using their arithmetic properties, about which relatively little is otherwise known. Continuing our work from a companion paper, we address further arithmetic properties of Misiurewicz parameters, especially the nature of the algebraic integers obtained by evaluating the polynomial defining one such parameter at a different Misiurewicz parameter. In the most challenging such combinations, we describe a connection between such algebraic integers and the multipliers of associated periodic points. As part of our considerations, we also introduce a new class of polynomials we call p-special, which may be of independent number theoretic interest.

Some interesting birational morphisms of smooth affine quadric 3-folds *

We study a family of birational maps of smooth affine quadric 3-folds, over the complex numbers, of the form x1x4−x2x3= constant, which seems to have some (among many others) interesting/unexpected characters: (a) they are cohomologically hyperbolic, (b) their second dynamical degree is an algebraic number but not an algebraic integer, and (c) the logarithmic growth of their periodic points is strictly smaller than their algebraic entropy. These maps are restrictions of a polynomial map on C4 preserving each of the quadrics. The study in this paper is a mixture of rigorous and experimental ones, where for the experimental study we rely on Bertini which is a reliable and fast software for expensive numerical calculations in complex algebraic geometry.

A Short Proof that Rational Algebraic Integers are Integers

Summary We present a short proof that a rational algebraic integer is an integer.

CHARACTERISATION OF PRIMES DIVIDING THE INDEX OF A CLASS OF POLYNOMIALS AND ITS APPLICATIONS

AbstractLet ${\mathbb {Z}}_{K}$ denote the ring of algebraic integers of an algebraic number field $K = {\mathbb Q}(\theta )$ , where $\theta $ is a root of a monic irreducible polynomial $f(x) = x^n + a(bx+c)^m \in {\mathbb {Z}}[x]$ , $1\leq m&lt;n$ . We say $f(x)$ is monogenic if $\{1, \theta , \ldots , \theta ^{n-1}\}$ is a basis for ${\mathbb {Z}}_K$ . We give necessary and sufficient conditions involving only $a, b, c, m, n$ for $f(x)$ to be monogenic. Moreover, we characterise all the primes dividing the index of the subgroup ${\mathbb {Z}}[\theta ]$ in ${\mathbb {Z}}_K$ . As an application, we also provide a class of monogenic polynomials having non square-free discriminant and Galois group $S_n$ , the symmetric group on n letters.

Invariant set generated by a nonreal number is everywhere dense

A set of complex numbers $S$ is called invariant if it is closed under addition and multiplication, namely, for any $x, y \in S$ we have $x+y \in S$ and $xy \in S$ . For each $s \in {\mathbb {C}}$ the smallest invariant set ${\mathbb {N}}[s]$ containing $s$ consists of all possible sums $\sum _{i \in I} a_i s^i$ , where $I$ runs over all finite nonempty subsets of the set of positive integers ${\mathbb {N}}$ and $a_i \in {\mathbb {N}}$ for each $i \in I$ . In this paper, we prove that for $s \in {\mathbb {C}}$ the set ${\mathbb {N}}[s]$ is everywhere dense in ${\mathbb {C}}$ if and only if $s \notin {\mathbb {R}}$ and $s$ is not a quadratic algebraic integer. More precisely, we show that if $s \in {\mathbb {C}} \setminus {\mathbb {R}}$ is a transcendental number, then there is a positive integer $n$ such that the sumset ${\mathbb {N}} t^n+{\mathbb {N}} t^{2n} +{\mathbb {N}} t^{3n}$ is everywhere dense in ${\mathbb {C}}$ for either $t=s$ or $t=s+s^2$ . Similarly, if $s \in {\mathbb {C}} \setminus {\mathbb {R}}$ is an algebraic number of degree $d \ne 2, 4$ , then there are positive integers $n, m$ such that the sumset ${\mathbb {N}} t^n+{\mathbb {N}} t^{2n} +{\mathbb {N}} t^{3n}$ is everywhere dense in ${\mathbb {C}}$ for $t=ms+s^2$ . For quadratic and some special quartic algebraic numbers $s$ it is shown that a similar sumset of three sets cannot be dense. In each of these two cases the density of ${\mathbb {N}}[s]$ in ${\mathbb {C}}$ is established by a different method: for those special quartic numbers, it is possible to take a sumset of four sets.

Degrees of generalized Kloosterman sums

Abstract The modern study of the exponential sums is mainly about their analytic estimates as complex numbers, which is local. In this paper, we study one global property of the exponential sums by viewing them as algebraic integers. For a kind of generalized Kloosterman sums, we present their degrees as algebraic integers.

Lower Bounds for the Canonical Height of a Unicritical Polynomial and Capacity

Abstract In a recent breakthrough, Dimitrov [Dim] solved the Schinzel–Zassenhaus conjecture. We follow his approach and adapt it to certain dynamical systems arising from polynomials of the form $T^p+c$ , where p is a prime number and where the orbit of $0$ is finite. For example, if $p=2$ and $0$ is periodic under $T^2+c$ with $c\in \mathbb {R}$ , we prove a lower bound for the local canonical height of a wandering algebraic integer that is inversely proportional to the field degree. From this, we are able to deduce a lower bound for the canonical height of a wandering point that decays like the inverse square of the field degree. For these f, our method has application to the irreducibility of polynomials. Indeed, say y is preperiodic under f but not periodic. Then any iteration of f minus y is irreducible in $\mathbb {Q}(y)[T]$ .

The density of tuples restricted by relatively r-prime conditions

In order to consider j-wise relative r-primality conditions that do not necessarily require all j-tuples of elements in a Dedekind domain to be relatively r-prime, we define the notion of j-wise relative r-primality with respect to a fixed j-uniform hypergraph H. This allows us to provide further generalisations to several results on natural densities not only for a ring of algebraic integers O, but also for the ring Fq[x].

Physical-Layer Network Coding Based on Residual Field of Algebraic Integers in OFDM-VLC Two-Way Relay Networks

Physical-Layer Network Coding Based on Residual Field of Algebraic Integers in OFDM-VLC Two-Way Relay Networks