Boris Adamczewski, Yann Bugeaud
Let $b \ge 2$ be an integer. We prove that the $b$-adic expansion of every irrational algebraic number cannot have low complexity. Furthermore, we establish that irrational morphic numbers are transcendental, for a wide class of morphisms. In particular, irrational automatic numbers are transcendental. Our main tool is a new, combinatorial transcendence criterion.
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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…
Let $k$ be a number field. For $\mathcal{H}\rightarrow \infty$, we give an asymptotic formula for the number of algebraic integers of absolute Weil height bounded by $\mathcal{H}$ and fixed degree over $k$.
Algebraic degree of series of reciprocal algebraic integers
该研究针对互反代数整数相关级数的ℚ线性组合，给出了其代数次数大于任意固定整数的充分条件，涉及模快速增长的代数整数与有界正整数系数的设定。
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.
Algebraic integers in totally imaginary quartic number fields are not discrete in the complex plane under a fixed embedding, which makes it impossible to visualize all integers in the plane, unlike the quadratic imaginary algebraic integers. In this note we consider a naturally occurring discrete subset of the algebraic integers with similar properties as lattices. For the fifth cyclotomic field, we investigate those integers with absolute values under a fixed embedding in a given bound. We show that such integers form a discrete set in the complex plane. It is observed that this subset has qu…
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…
Explicit Algebraic Numbers all whose Integer Parts of Powers are Composite
该研究构造了满足除有限个n外其幂的整数部分均为合数的显式代数整数类，为数论中相关问题提供了具体实例
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.
Counting Algebraic Integers of Bounded Height in Cyclotomic Fields
本文针对素数幂阶分圆域，给出绝对高度不超过B的代数整数和单位个数的渐近上界。
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…
It has been conjectured for some time that, for any integer n\ge 2, any real number ε>0 and any transcendental real number ξ, there would exist infinitely many algebraic integers αof degree at most n with the property that |ξ-α| < H(α)^{-n+ε}, where H(α) denotes the height of α. Although this is true for n=2, we show here that, for n=3, the optimal exponent of approximation is not 3 but (3+\sqrt{5})/2 = 2.618...
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…
In 1969, H. Davenport and W. M. Schmidt studied the problem of approximation to a real number ξby algebraic integers of degree at most three. They did so, using geometry of numbers, by resorting to the dual problem of finding simultaneous approximations to ξand ξ^2 by rational numbers with the same denominator. In this paper, we show that their measure of approximation for the dual problem is optimal and that it is realized for a countable set of real numbers ξ. We give several properties of these numbers including measures of approximation by rational numbers, by quadratic real numbers and by…
Algebraic integers with conjugates in a prescribed distribution
本文研究共轭位于给定紧致实数集中的代数整数，给出其共轭分布极限的充要条件，并应用于构造全正代数整数及有限域上简单阿贝尔簇的极端点数。
On $β$-adic expansions of powers of algebraic integer omitting a digit
本文研究了代数整数幂的β进制展开中排除特定数字的性质，证明了其数量随N的变化规律。
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…
Arithmetical independence of certain uniform sets of algebraic integers
本研究针对四组次数≤3的代数整数集合，证明其具备几乎一致性与算术独立性特征，并明确了这些集合生成的代数数域对二次、三次域的覆盖程度。
Algebraic integers with continued fraction expansions containing palindromes and square roots with prescribed periods
本文刻画了连分数展开包含回文序列且具有指定周期的代数整数，特别利用 Fibonacci 多项式处理各项均为 m 的特殊情形，并推导出相应整数平方根展开及正负 Pell 方程的基本解。
Building on work of Davenport and Schmidt, we mainly prove two results. The first one is a version of Gel'fond's transcendence criterion which provides a sufficient condition for a complex or $p$-adic number $ξ$ to be algebraic in terms of the existence of polynomials of bounded degree taking small values at $ξ$ together with most of their derivatives. The second one, which follows from this criterion by an argument of duality, is a result of simultaneous approximation by conjugate algebraic integers for a fixed number $ξ$ that is either transcendental or algebraic of sufficiently large degree…
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 $c_2\cdot Q^{n-γ}<# \tilde{M}^n_φ(Q,γ, J)< c_3\cdot Q^{n-γ}$.
Let $φ(τ)=η((τ+1)/2)^2/\sqrt{2π}e^\frac{πi}{4}η(τ+1)$ where $η(τ)$ is the Dedekind eta-function. We show that if $τ_0$ is an imaginary quadratic number with $\mathrm{Im}(τ_0)>0$ and $m$ is an odd integer, then $\sqrt{m}φ(mτ_0)/φ(τ_0)$ is an algebraic integer dividing $\sqrt{m}$. This is a generalization of Theorem 4.4 given in [B. C. Berndt, H. H. Chan and L. C. Zhang, Ramanujan's remarkable product of theta-functions, Proc. Edinburgh Math. Soc. (2) 40 (1997), no. 3, 583-612]. On the other hand, let $K$ be an imaginary quadratic field and $θ_K$ be an element of $K$ with $\mathrm{Im}(θ_K)>0$ wh…
Approximation of complex algebraic numbers by algebraic numbers of bounded degree
本文研究了次数至多为 n 的代数数和次数至多为 n+1 的代数整数对复代数数的逼近问题，发现存在某些高次复代数数能被低次代数数更好地逼近，但被相应代数整数逼近的效果更差。
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…
On the structure of even $K$-groups of rings of algebraic integers
本文用数域适当扩张的类群描述了数域整数环的高阶偶 $K$-群，推广了前人关于 $K_2$ 的工作，并给出了全实域 Kummer 判据的代数 $K$-理论表述与证明。
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.
Let F and K be number fields, with F contained in K. and let O_F and O_K be their rings of integers. If there exists an elliptic curve E over F such that E(F) and E(K) have rank 1, then there exists a diophantine definition of O_F over O_K.
Let $K$ be a complete discrete valued field of characteristic zero with residue field $k_K$ of characteristic $p > 0$. Let $L/K$ be a finite Galois extension with the Galois group $G$ and suppose that the induced extension of residue fields $k_L/k_K$ is separable. In his paper, Hesselholt conjectured that $H^1(G,W(\sO_L))$ is zero, where $\sO_L$ is the ring of integers of $L$ and $W(\sO_L)$ is the Witt ring of $\sO_L$ w.r.t. the prime $p$. He partially proved this conjecture for a large class of extensions. In this paper, we prove Hesselholt's conjecture for all Galois extensions.
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.
Using a method of H. Davenport and W. M. Schmidt, we show that, for each positive integer n, the ratio 2/n is the optimal exponent of simultaneous approximation to real irrational numbers 1) by all conjugates of algebraic numbers of degree n, and 2) by all but one conjugates of algebraic integers of degree n+1.
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 $\mathcal A$.
A survey of results for Mahler measure of algebraic numbers, and one-variable polynomials with integer coefficients is presented. Related results on the maximum modulus of the conjugates (`house') of an algebraic integer are also discussed. Some generalisations are given too, though not to Mahler measure of polynomials in more than one variable.
Let $K$ be a number field of degree $n$ with ring of integers $O_K$. By means of a criterion of Gilmer for polynomially dense subsets of the ring of integers of a number field, we show that, if $h\in K[X]$ maps every element of $O_K$ of degree $n$ to an algebraic integer, then $h(X)$ is integral-valued over $O_K$, that is $h(O_K)\subset O_K$. A similar property holds if we consider the set of all algebraic integers of degree $n$ and a polynomial $f\in\mathbb{Q}[X]$: if $f(α)$ is integral over $\mathbb{Z}$ for every algebraic integer $α$ of degree $n$, then $f(β)$ is integral over $\mathbb{Z}$ …
We consider digital expansions in lattices with endomorphisms acting as base. We focus on the $w$-non-adjacent form ($w$-NAF), where each block of $w$ consecutive digits contains at most one non-zero digit. We prove that for sufficiently large $w$ and an expanding endomorphism, there is a suitable digit set such that each lattice element has an expansion as a $w$-NAF. If the eigenvalues of the endomorphism are large enough and $w$ is sufficiently large, then the $w$-NAF is shown to minimise the weight among all possible expansions of the same lattice element using the same digit system.
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.
Bicomplex Algebraic Numbers
本文研究双复数扩展的代数结构，推广原始元定理并探讨其整数环的分解性质，通过实例展示素元在双复数扩展中的变化
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…
We answer a question of Brass about vertex degrees in unit distance graphs of finitely generated additive subgroups of $\mathbb{R}^2$.
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 maximal degree over rational numbers that an n-dimensinonal Kloosterman sum defined over a finite field of characteristic p can achieve is known to be (p-1)/d where d=gcd(p-1,n+1). Wan has shown that this maximal degree is always achieved in points whose absolute trace is nonzero. By the works of Fischer, Wan we know that there exist many finite fields for which the values of the Kloosterman sums are distinct except Frobenius conjugation. For these fields we completely determine the degrees of all the Kloosterman sums. Even if the finite field does not satisfy this condition we can still o…
We develop an algebraic method of studying of Diophantine quadratic equations in three variables over the ring of Gaussian integers.
We classify all post-critically finite unicritical polynomials defined over the maximal totally real algebraic extension of ${\mathbb Q}$. Two auxiliary results used in the proof of this result may be of some independent interest. The first is a recursion formula for the $n$-diameter of an interval, which uses properties of Jacobi polynomials. The second is a numerical criterion which allows one to the give a bound on the degree of any algebraic integer having all of its complex embeddings in a real interval of length less than $4$.
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