Let K be a number field, and let a be a non-zero element of K. Fix some prime number l. We compute the density of the following set: the primes p of K such that the multiplicative order of the reduction of a modulo p is coprime to l (or, more generally, has some prescribed l-adic valuation). We evaluate the degree over K of extensions of the form K(ζ_m, \sqrt[n]{a}) with n\leq m, which are obtained by adjoining a root of unity of order l^m and the l^n-th roots of a, as this is needed for computing the above density.
The PSLQ algorithm computes integer relations for real numbers and Gaussian integer relations for complex numbers. We endeavour to extend PSLQ to find integer relations consisting of algebraic integers from some quadratic extension fields (in both the real and complex cases). We outline the algorithm, discuss the required modifications for handling algebraic integers, problems that have arisen, experimental results, and challenges to further work.
The paper assesses the top number of integer solutions for algebraic Diophantine Thue diagonal equation of the degree $n \geq 2$ and number of variables $k > 2$ and equations with explicit variable in the case when the coefficients of the equation are of the opposite signs. The author found integer conversions that maintain the asymptotic behavior of the number of integer solutions of algebraic Diophantine equation in the case of the conversion equation to diagonal form. The paper considers the estimation of the number of integer solutions for some types of algebraic Diophantine equations with…
This paper investigates the upper bound of the number of integer (natural) solutions of inhomogeneous algebraic Diophantine diagonal equations with integer coefficients without a free member via the circle method of Hardy and Littlewood. Author found the upper bound of the number of natural solutions of inhomogeneous algebraic Diophantine diagonal equations with explicit variable. He developed a method in the paper, which allows you to perform the low estimate of the number of natural (integer) solutions of algebraic Diophantine equation with integer coefficients. Author obtained a lower estim…
Let $n$ and $m$ be positive integers such that $n<m$. In this paper we compute the density of rectangular unimodular $n$ by $m$ matrices over the ring of algebraic integers of a number field.
In the paper, we study the asymptotic distribution of real algebraic integers of fixed degree as their na\"ıve height tends to infinity. For an arbitrary interval $I \subset \mathbb{R}$ and sufficiently large $Q>0$, we obtain an asymptotic formula for the number of algebraic integers $α\in I$ of fixed degree $n$ and na\"ıve height $H(α)\le Q$. In particular, we show that the real algebraic integers of degree $n$, with their height growing, tend to be distributed like the real algebraic numbers of degree $n-1$. However, we reveal two symmetric "plateaux", where the distribution of real algebrai…
The paper presents a new generalized integer orthogonal transformation which consists of a well known orthogonal transform followed by stretching the basis vectors maintaining the asymptotic behavior of the number of integer solutions for algebraic Diophantine equation. The author shows the properties of this transformation and he receives the algorithm for finding the matrix elements of a generalized integer orthogonal transformation for algebraic Diophantine equation of the second order to diagonal form. The article includes examples illustrating the reduction of algebraic equations of the s…
The author showed that any homogeneous algebraic Diophantine equation of the second order can be converted to a diagonal form using an integer non-orthogonal transformation maintaining asymptotic behavior of the number of its integer solutions. In this paper, we consider the transformation to the diagonal form of a wider class of algebraic second-order Diophantine equations, and also we consider the conditions for converting higher order algebraic Diophantine equations to this form. The author found an asymptotic estimate for the number of integer solutions of the diagonal Thue equation of odd…
We obtain results bounding the degree of the series $\sum_{n=1}^{\infty} 1/α_n$, where $\{α_n\}$ is a sequence of algebraic integers satisfying certain algebraic conditions and growth conditions. Our results extend results of Erdős, Hančl and Nair.
In the paper, we study the asymptotic distribution of real algebraic integers of fixed degree as their naive height tends to infinity. Let $I \subset \mathbb{R}$ be an arbitrary bounded interval, and $Q$ be a sufficiently large number. We obtain an asymptotic formula for the count of algebraic integers $α$ of fixed degree $n$ and naive height $H(α)\le Q$ lying in $I$. In this formula, we estimate the order of the error term from above and below. We show that algebraic integers of degree $n$ are distributed asymptotically like algebraic numbers of degree $(n-1)$ as the upper bound $Q$ of height…
Let $α$ be an algebraic integer of degree $d$, which is reciprocal. The house of $α$ is the largest modulus of its conjugates. We compute the minimum of the houses of all reciprocal algebraic integers of degree $d$ which are not roots of unity, say $\mathrm{mr}(d)$, for $d$ at most 34. We prove several lemmata and use them to avoid unnecessary calculations. The computations suggest several conjectures. The direct consequence of the last one is the conjecture of Schinzel and Zassenhaus. We demonstrate the utility of $d$-th power of the house of $α$.
By considering the norm of elements in the ring of integers in $\mathbb{Q}(\sqrt{-a})$, we give an algebraic approach to count the number of integral solutions of diophantine equations of the form $x^2+ay^2=n$ where $a$ is a Heegner number or $a=27$.
In 1970 A. Baker and W. Schmidt introduced regular systems of numbers and vectors, showing that the set of real algebraic numbers forms a regular system on any fixed interval. This fact was used to prove several important results in the metric theory of transcendental numbers. In this paper the concept of a regular system is applied to the set of algebraic integers $α$ of height $\leq Q$ in intervals of length depending on $Q$.
Given positive real numbers, we prove two inequalities involving their potential energy and their power sums. We also prove an inequality involving the energy and the discriminant and apply it to deduce a result on totally positive irreducible polynomials.
In algebraic number theory, the finiteness of the Picard group of an order in a number field is generally proved via a lattice argument: the order forms a lattice and every ideal class contains an integral ideal with a small enough non-zero element. In this work we will consider $\overline{\mathbb{Z}}$, the ring of algebraic integers, which is a lattice in a similar sense, and we will treat this lattice as intrinsically interesting. We will prove several properties of $\overline{\mathbb{Z}}$ and state some open problems. Many of these properties have connections to the indecomposable elements …
We create plots of algebraic integers in the complex plane, exploring the effect of sizing the integers according to various arithmetic invariants. We focus on Galois theoretic invariants, in particular creating plots which emphasize algebraic integers whose Galois group is not the full symmetric group--these integers we call rigid. We then give some analysis of the resulting images, suggesting avenues for future research about the geometry of so-called rigid algebraic integers.
Let $k$ be a number field and $S$ a finite set of places of $k$ containing the archimedean ones. We count the number of algebraic points of bounded height whose coordinates lie in the ring of $S$-integers of $k$. Moreover, we give an asymptotic formula for the number of $\bar{S}$-integers of bounded height and fixed degree over $k$, where $\bar{S}$ is the set of places of $\bar{k}$ lying above the ones in $S$.
We extend a result of Hančl, Kolouch and Nair on the irrationality and transcendence of continued fractions. We show that for a sequence $\{α_n\}$ of algebraic integers of bounded degree, each attaining the maximum absolute value among their conjugates and satisfying certain growth conditions, the condition $$ \limsup_{n \rightarrow \infty} \vertα_n\vert^{\frac{1}{Dd^{n-1} \prod_{i=1}^{n-2}(Dd^i + 1)}} = \infty $$ implies that the continued fraction $α= [0;α_1, α_2, \dots]$ is not an algebraic number of degree less than or equal to $D$.
Author developed a method in the paper, which, unlike the circle method of Hardy and Littlewood (CM), allows you to perform a lower estimate for the number of natural (integer) solutions of algebraic Diophantine equation with integer coefficients. It was found the lower estimate of the number of natural solutions to various types of homogeneous algebraic Diophantine equations with integer coefficients diagonal form with any number of variables using this method. Author obtained upper bound of the number of the natural solutions (using CM) of one type of homogeneous Diophantine equation for val…
Let $α$ be an algebraic integer of degree $d$, which is reciprocal. The house of $α$ is the largest modulus of its conjugates. We proved that $d$-th power of the house of reciprocal $α$ has a limit point. We presented a property of antireciprocal hexanomials. We compute the minimum of the houses of all reciprocal algebraic integers of degree $d$ having the minimal polynomial which is a factor of a $D$-th degree reciprocal or antireciprocal polynomial with at most eight monomials, say $\mathrm{mr}(d)$, for $d$ at most 180, $D\le 1.5d$ and $D\le 210$. We show that it is not necessary to take int…
We present a necessary and sufficient condition for a root greater than unity of a monic reciprocal polynomial of an even degree at least four, with integer coefficients, to be a Salem number. We determine the probability of fulfillment the condition for an arbitrary power of the root.
An algebraic integer is said large if all its real or complex embeddings have absolute value larger than $1$. An integral ideal is said \emph{large} if it admits a large generator. We investigate the notion of largeness, relating it to some arithmetic invariants of the field involved, such as the regulator and the covering radius of the lattice of units. We also study its connection with the Weil height and the Bogomolov property. We provide an algorithm for testing largeness and give some applications to the construction of floor functions arising in the theory of continued fractions.
Masser and Vaaler have given an asymptotic formula for the number of algebraic numbers of given degree $d$ and increasing height. This problem was solved by counting lattice points (which correspond to minimal polynomials over $\mathbb{Z}$) in a homogeneously expanding star body in $\mathbb{R}^{d+1}$. The volume of this star body was computed by Chern and Vaaler, who also computed the volume of the codimension-one "slice" corresponding to monic polynomials -- this led to results of Barroero on counting algebraic integers. We show how to estimate the volume of higher-codimension slices, which a…
We consider semigroup dynamical systems defined by several polynomials over a number field $\mathbb{K}$, and the orbit (tree) they generate at a given point. We obtain finiteness results for the set of preperiodic points of such systems that fall in the cyclotomic closure of $\mathbb{K}$. More generally, we consider the finiteness of initial points in the cyclotomic closure for which the orbit contains an algebraic integer of bounded house. This work extends previous results for classical obits generated by one polynomial over $\mathbb{K}$ obtained initially by Dvornicich and Zannier (for prep…
Let $φ:\mathbb{R}\rightarrow \mathbb{R}$ be a continuously differentiable function on an interval $J\subset\mathbb{R}$ and let $\boldsymbolα=(α_1,α_2)$ be a point with algebraic conjugate integer coordinates of degree $\leq n$ and of height $\leq Q$. Denote by $\tilde{M}^n_φ(Q,γ, J)$ the set of points $\boldsymbolα$ such that $|φ(α_1)-α_2|\leq c_1 Q^{-γ}$. In this paper we show that for a real $0<γ<1$ and any sufficiently large $Q$ there exist positive values $c_2<c_3$, which are independent of $Q$, such that $c_2\cdot Q^{n-γ}<# \tilde{M}^n_φ(Q,γ, J)< c_3\cdot Q^{n-γ}$.
In arXiv:1302.4423, Salez proved that every totally real algebraic integer is the eigenvalue of some tree. We define the "arboreal height" of a totally real algebraic integer $λ$ to be the minimal height of a rooted tree having $λ$ as an eigenvalue. In this paper, we prove several results about totally real algebraic integers of arboreal height $2$: We show that all real quadratic integers have arboreal height $\le 2$. We characterize the totally real cubic integers of arboreal height $2$. Finally, we prove that every totally irrational real number field is generated (as a ring over $\mathbb{Q…
Bateman and Duquette have initiated the study of Salem elements in positive characteristic. This work extends their results to 2-Salem elements whose minimal polynomials are of the type $Y^n + λ_{n-1}Y^{n-1} + \ldots+λ_1Y + λ_0 \in \mathbb{F}_q[X][Y ]$ where $n \geq2, λ_0\neq 0$ and $°λ_{n-1} < °λ_{n-2} = \max_{i\neq n-2}°(λ_i)$. This work provides an analogue of their results for 2-Salem elements whose minimal polynomials meet certain requirements.
We consider $\ell$-adic trace functions over finite fields taking values in cyclotomic integers, such as characters and exponential sums. Through ideas of Deligne and Katz, we explore probabilistic properties of the reductions modulo a prime ideal, exploiting especially the determination of their integral monodromy groups. In particular, this gives a generalization of a result of Lamzouri-Zaharescu on the distribution of short sums of the Legendre symbol reduced modulo an integer to all multiplicative characters and to hyper-Kloosterman sums.
We show that the set of algebraic extensions $F$ of $\mathbb{Q}$ in which $\mathbb{Z}$ or the ring of integers $\mathcal{O}_F$ are definable is meager in the set of all algebraic extensions.
Let $a,k\in\mathbb{N}$. For the $k-1$-th iterate of the exponential function $x\mapsto a^x$, also known as tetration, we write \[ ^k a:=a^{a^{.^{.^{.^{a}}}}}. \] In this paper, we show how an efficient algorithm for tetration modulo natural numbers $N$ may be used to compute the prime factorization of $N$. In particular, we prove that the problem of computing the squarefree part of integers is deterministically polynomial-time reducible to modular tetration.
We count various classes of algebraic integers of fixed degree by their largest absolute value. The classes of integers considered include all algebraic integers, Perron numbers, totally real integers, and totally complex integers. We give qualitative and quantitative results concerning the distribution of Perron numbers, answering in part a question of W. Thurston.
We study a linear form in the values of Euler's series $F(t)=\sum_{n=0}^\infty n!t^n$ at algebraic integer points $α_1, \ldots, α_m \in \mathbb{Z}_{\mathbb{K}}$ belonging to a number field $\mathbb{K}$. Let $v|p$ be a non-Archimedean valuation of $\mathbb{K}$. Two types of non-vanishing results for the linear form $Λ_v = λ_0 + λ_1 F_v(α_1) + \ldots + λ_m F_v(α_m)$, $λ_i \in \mathbb{Z}_{\mathbb{K}}$, are derived, the second of them containing a lower bound for the $v$-adic absolute value of $Λ_v$. The first non-vanishing result is also extended to the case of primes in residue classes. On the w…
We study periodic representations in number systems with an algebraic base $β$ (not a rational integer). We show that if $β$ has no Galois conjugate on the unit circle, then there exists a finite integer alphabet $\mathcal A$ such that every element of $\mathbb Q(β)$ admits an eventually periodic representation with base $β$ and digits in $\mathcal A$.
The $k$-dimensional generalized Euler function $φ_k(n)$ is defined to be the number of ordered $k$-tuples $(a_1,a_2,\ldots, a_k) \in \mathbb{N}^k$ with $1\leq a_1,a_2,\ldots, a_k \leq n$ such that both the product $a_1a_2\cdots a_k$ and the sum $a_1+a_2+\cdots+a_k$ are co-prime to $n$. Tóth proved that the identity \begin{equation*} \sum_{\substack{a_1,a_2,\ldots, a_k=1 \\ \gcd(a_1a_2\cdots a_k,n)=1\\ \gcd(a_1+a_2+\cdots+a_k,n)=1}}^n \gcd(a_1+a_2+\cdots+a_k-1,n) =φ_k(n)σ_0(n), \;\; \text{ where } σ_s(n) = \sum_{d\mid n}d^s \;\; \text{ holds. } \end{equation*} This identity can also be viewed a…
Elaborating on a method of Soulé, and using better estimates for the geometry of hermitian lattices, we improve the upper bounds for the torsion part of the K-theory of the rings of integers of number fields.
We provide a simple way to add, multiply, invert, and take traces and norms of algebraic integers of a number field using integral matrices. With formulas for the integral bases of the ring of integers of at least a significant proportion of numbers fields, we obtain explicit formulas for these matrices and discuss their generalization. These results are useful in proving statements about the particular number fields they work in. We give a meaningful diagonalization that is helpful in this regard and can be used to define such matrices in general. The matrix identities provided suggest that c…
