Effective Approximation to Complex Algebraic Numbers by Algebraic Numbers of Bounded Degree
Prajeet Bajpai, Yann Bugeaud
本文首次建立了 Liouville 不等式的有效改进，用于以次数至多为4的复代数数逼近复非实代数数。
我们首次建立了 Liouville 不等式的有效改进，用于以次数至多为4的复代数数逼近复非实代数数。
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.
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Computing algebraic numbers of bounded height
我们描述了一个算法，列出所有元素的有界高度在一个给定的数字字段。
Counting algebraic integers of fixed degree and bounded height
设$k$为数字字段。对于$\mathcal{H}\rightarrow \infty$，我们给出了$k$上固定次数且以$\mathcal{H}$为界的绝对Weil高度的代数整数个数的一个渐近公式.
The exceptional set in Vojta's conjecture for algebraic points of bounded degree
研究Vojta猜想中例外集对各种参数的依赖性。特别是，通过利用某些椭圆曲面，我们回答了否定的问题，是否Vojta的猜想持有时，扩大到所有的代数点（即，如果猜想持有没有固定的限制程度的代数点）。
Algebraic $S$-integers of fixed degree and bounded height
设$k$是数域，$S$是$k$中包含阿基米德数的有限位集.我们计数的代数点的有界高度的坐标位于环的$S$ -整数的$k$。此外，我们还给出了$k$上有界高度和固定次数的$\bar{S}$ -整数个数的一个渐近公式，其中$\bar{S}$是$\bar{k}$位于$S$上的位置的集合.
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。
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。
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…
Fields of algebraic numbers with bounded local degrees and their properties
我们给出了具有一致有界局部度的有理数的无限代数Galois扩张的一个刻画，并详细地证明了Albercoli和Zannier在一篇论文中所宣布的所有结果，并得到了相应的推广.特别地，我们证明了对于有理数的无限Galois扩张，以下三个性质是等价的：在每个素数处具有一致有界的局部度;在几乎每个素数处具有一致有界的局部度;具有有限指数的伽罗瓦群.证明这一结果启发有趣的连接与Zelmanov的工作限制伯恩赛德问题。在某些特殊情况下，我们给出了一个计算无限扩张局部度的界的公式。我们涉及的统一有界性的地方度的其他属性：作为一个子字段的合成的所有领域的有界度;被生成的元素的有界度。我们证明了上述性质是等价的阿贝尔扩展，但不是一般的，我们提供了反例的基础上群论建设与extrasspecial群及其模块，我们给出明确的实现。
In this paper we consider the problem of counting algebraic numbers $α$ of fixed degree $n$ and bounded height $Q$ such that the derivative of the minimal polynomial $P_α(x)$ of $α$ is bounded, $|P_α'(α)| < Q^{1-v}$. This problem has many applications to the problems of the metric theory of Diophantine approximation. We prove that the number of $α$ defined above on the interval $\left(-\frac12, \frac12\right)$ doesn't exceed $c_1(n)Q^{n+1-\frac{1}{7}v}$ for $Q>Q_0(n)$ and $1.4 \le v \le \frac{7}{16}(n+1)$. Our result is based on an improvement to the lemma on the order of zero approximation by…
The degree sequence of the algebraic numbers in an algebraic linear recurrence sequence is shown to be virtually periodic. This is proved using the Skolem-Mahler-Lech theorem. It has applications to the degree sequence and the minimal polynomial sequence for exponential sums over finite fields. The degree periodicity also holds for some more complicated non-linear recurrence sequences. We give one example from the iterations of a polynomial map. This depending on the dynamic Mordell-Lang conjecture which has been proved in some cases.
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。
Given smooth, projective, geometrically integral algebraic curves $X$ and $Y$ defined over a number field $K$, assuming that there is a non-constant $K$-morphism $φ\colon X \to Y$, we give an upper bound on the minimum of the degrees of such morphisms. The proof is based on isogeny estimates between abelian varieties.
Given a number field $L$, we define the degree of an algebraic number $v \in L$ with respect to a choice of a primitive element of $L$. We propose the question of computing the minimal degrees of algebraic numbers in $L$, and examine these values in degree $4$ Galois extensions over $\mathbb{Q}$ and triquadratic number fields. We show that computing minimal degrees of non-rational elements in triquadratic number fields is closely related to solving classical Diophantine problems such as congruent number problem as well as understanding various arithmetic properties of elliptic curves.
In different areas of discrete mathematics, a certain type of polynomials, having coefficients in a field K of finite characteristic, has been considered. The form and the degree of these polynomials, here called projective, are simply linked to the characteristic p of K. Roots of these projective polynomials are particular algebraic elements over K, called hyperquadratic. For a general algebraic element of degree d over K, we discuss the possibility of being hyperquadratic. Using a method of differential algebra, we obtain, for particular fields K = Fp, projective polynomials only having poly…
Counting algebraic points of bounded degree on curves
本文研究数域上光滑射影曲线中满足次数为ν、函数域与基域经gonality态射一致且高度不超过T的代数点数量的增长速率。
Let $f(x)$ be an irreducible polynomial with integer coefficients of degree at least two. Hooley proved that the roots of the congruence equation $f(x)\equiv 0\mod n$ is uniformly distributed. as a parallel of Hooley's theorem under ideal theoretical setting, we prove the uniformity of the distribution of residues of an algebraic number modulo degree one ideals. Then using this result we show that the roots of a system of polynomial congruences are uniformly distributed. Finally, the distribution of digits of n-adic expansions of an algebraic number is discussed.
Simultaneous approximation by conjugate algebraic numbers in fields of transcendence degree one
本文给出了同时逼近多个超越实数、复数或p-adic数xi_1，…，xi_t由Q上有界次数的共轭代数数表示，条件是给定的超越数xi_1，.，xi_t在Q上生成超越度为1的域。例如，当xi_1，.，xi_t形成具有非零代数差的等差数列，或者形成具有不同于单位根的非零代数比的等比数列。在这种情况下，我们还通过对偶得到了一个形式的Gel'fond超越准则，该准则用在xi_1，.，xi_t.
The number of algebraic cycles with bounded degree
设X是有限域上的射影概形。本文研究了X上有界度的有效圈数趋于无穷远的渐近性态。通过这个估计，我们可以定义一种与循环群相关的zeta函数。我们还考虑了一个类似的Arakelov几何。
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…
We prove a cohomological formula for non-critical residues of degree eight automorphic $L$-functions of $\mathrm{GSp}(4) \times \mathrm{GL}(2)$ in the spirit of Beilinson conjecture. We rely on the cohomological interpretation of an automorphic period integral and on the study of Novodvorsky's integral representation of the $L$-functions.
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…
On lattices generated by algebraic conjugates of prime degree
研究代数数在Minkowski嵌入下代数共轭像生成的欧氏格，证明素数次数大Pisot数生成良圆格，完成2维分类并推导特定Galois群下的行列式公式。
Linear relations of four conjugates of an algebraic number of degree eight
刻画了存在四个不同共轭满足线性关系α1=α2+α3+α4的八次代数数，并得到类似刻画α1+α2+α3+α4=0。特别地，当八次代数数的极小多项式非偶且恰有六个形如α_i1+α_i2+α_i3+α_i4=0的不同线性关系时，证明该数为二次与四次代数数之和。
On algebraic degrees of inverted Kloosterman sums
本文研究了反转 $n$ 维 Kloosterman 和作为代数整数的代数次数，扩展了 Katz 及作者前作中从复数与 $p$-adic 视角对该和的研究。
A generalized Schmidt's subspace theorem for algebraic points of bounded degree
该研究在次一般位置闭子概型的Schmidt子空间定理框架基础上，建立了针对有界次数代数点、关于数值等价丰富除子的Schmidt子空间定理，得到最优因子与广义加权版本，并解决了该领域长期存在的公开猜想。
Let $N_n(X)$ denote the number of degree $n$ number fields with discriminant bounded by $X$. In this note, we improve the best known upper bounds on $N_n(X)$, finding that $N_n(X) = O(X^{ c (\log n)^2})$ for an explicit constant $c$.
Valeurs algebriques de fonctions transcendantes
研究了解析超越函数取代数值的高度和次数有界的代数数的集合。
Bounding the Degree of Belyi Polynomials
Belyi定理指出，黎曼曲面X作为一条代数曲线被定义在有理数的代数闭包上，当且仅当存在一个全纯函数将X带到黎曼球面，且最多有三个临界值（传统上被认为是零、一和无穷大）。通过限制到X是黎曼球面并且我们的全纯函数是Belyi多项式的情况，我们将代数数的Belyi高度定义为将所述代数数映射到零或一的Belyi多项式的最小次数。利用牛顿多边形的组合数学方法，证明了对于p-adic赋值为非零的非零代数数，其Belyi高度必定大于或等于p.我们还给出了代数数的例子，表明我们的边界是尖锐的。
Degree 4 cohomological invariants of algebraic tori
本文通过确定代数环面分类主齐性空间的 motive，给出了描述其4次上同调不变量的正合序列，并利用已知结果导出了4次非分歧上同调群的公式。
Diophantine Approximation on varieties III: Approximation of non-algebraic points by algebraic points
对于数域上拟射影簇上的非代数点$θ$，证明了$θ$具有一系列高度和次数有界的代数点的逼近，这是本质上最好的逼近.这一结果的应用将包括证明一个稍微加强版本的Philippon标准，一些新的代数独立性标准，报表有关度量超越理论的品种任意尺寸，和一个相当准确的估计代数点的数量有界高度和程度的拟射影簇在数域。
We study the algebraic Brauer classes on open del Pezzo surfaces of degree $4$. I.e., on the complements of geometrically irreducible hyperplane sections of del Pezzo surfaces of degree $4$. We show that the $2$-torsion part is generated by classes of two different types. Moreover, there are two types of $4$-torsion classes. For each type, we discuss methods for the evaluation of such a class at a rational point over a $p$-adic field.
Hyperbolic three manifolds of bounded volume and trace field degree II
本文证明了作者在[2]中提出的有界高度猜想。作为推论，得出了只有有限个体积和迹场度有界的双曲三流形。
Integral-valued polynomials over the set of algebraic integers of bounded degree
设$K$是一个度为$n$的数域，其上有整数环$O_K$。利用Gilmer关于数域整数环的多项式稠密子集的一个判别准则，证明了若$h\in K[X]$把$n$次的$O_K$的每个元素映射到一个代数整数，则$h(X)$是$O_K$上的整值，即$h(O_K)\subset O_K$ .如果我们考虑次数为$n$的所有代数整数和多项式$f\in\mathbb{Q}[X]$的集合，一个类似的性质也成立：如果对于次数为$n$的每个代数整数$α$，$f(α)$在$\mathbb{Z}$上是积分的，那么对于次数小于$n$的每个代数整数$β$，$f(β)$在$\mathbb{Z}$上是积分的。这第二个结果是通过证明$\mathbb{Q}[X]$中的多项式环的积分闭包是矩阵$M_n(\mathbb{Z})$的集合上的整数值等于次数等于$n$的代数整数的集合上的积分值多项式环而建立的。
We establish a generalized Ihara zeta function formula for simple graphs with bounded degree. This is a generalization of the formula obtained by G. Chinta, J. Jorgenson and A. Karlsson from a vertex-transitive graph.
Finiteness of commutable maps of bounded degree
本文研究了两个动力系统（V，f）和（V，g）与f之间的关系。g = g。F.作为一个应用，我们证明了有界Pre（f）的自同态（分别为Zerkki稠密的多项式映射）只有1/2个有界次数的自同态（分别为多项式映射）与f可交换.
No three algebraic conjugates of degree sixteen sum to zero
通过分类16个顶点上6度顶点传递图，证明不存在16次代数数使得其三个共轭之和为零，结合已有结果得出最小可能的次数d=20。
Diagonals and algebraicity modulo $p$: a sharper degree bound
本文通过初等方法证明了Deligne定理，给出了多项式界下的代数次数 $d_p$ 的明确界。
Let a and b be algebraic numbers such that exactly one of a and b is an algebraic integer, and let f_t(z):=z^2+t be a family of polynomials parametrized by t. We prove that the set of all algebraic numbers t for which there exist positive integers m and n such that f_t^m(a)=f_t^n(b) has bounded Weil height. This is a special case of a more general result supporting a new bounded height conjecture in dynamics. Our results fit into the general setting of the principle of unlikely intersections in arithmetic dynamics.
Integral points of bounded degree on affine curves
将Siegel关于仿射曲线上积分点的定理推广到有界次数的积分点，给出了数域上具有无穷多个d次以下积分点的仿射曲线的一个完整刻画.推广Picard定理，我们证明了一个类似的结果，它刻画了复仿射曲线在$\mathbb{C}$的d次（或更少）解析覆盖上具有一个非常数全纯映射.
Hasse principle violation for algebraic families of del Pezzo surfaces of degree 4 and hyperelliptic curves of genus congruent to 1 modulo 4
本文构造了无限多个显式单参数代数族，包含次数为4的del Pezzo曲面和亏格模4余1的超椭圆曲线，证明其每个K成员都违反Hasse原则，并由Brauer-Manin障碍解释。
On the existence of non-special divisors of degree $g$ and $g-1$ in algebraic function fields over $\F_q$
研究了有限域$\F_q$上亏格为$g\geq 1$的代数函数域的非特殊次数因子$g$和$g-1$的存在性.特别地，我们证明了，总是存在一个有效的非特殊因子的程度$g\geq 2$，如果$q\geq 3$和，总是存在一个非特殊因子的程度$g-1\geq 1$，如果$q\geq 4$。我们使用我们的结果，以改善上和上渐近界的双线性复杂度的乘法在任何扩展$\F_{q^n}$的$\F_q$，当$q=2^r\geq 16$。
Let $D$ be a non-empty effective divisor on $\mathbb{P}^1$. We show that when ordered by height, any set of $(D,S)$-integral points on $\mathbb{P}^1$ of bounded degree has relative density zero. We then apply this to arithmetic dynamics: let $φ(z)\in \overline{\mathbb{Q}}(z)$ be a rational function of degree at least two whose second iterate $φ^2(z)$ is not a polynomial. We show that as we vary over points $P\in\mathbb{P}^1(\overline{\mathbb{Q}})$ of bounded degree, the number of algebraic integers in the forward orbit of $P$ is absolutely bounded and zero on average.
Turing Degrees of Isomorphism Types of Algebraic Objects
可数结构$\mathcal{A}$的图灵度谱是$\mathcal{A}$的同构副本的所有图灵度的集合。如果$\mathcal{A}$的同构类型存在，则它的图灵度是其度谱中的最小图灵度.我们证明了存在可数域、环和任意秩的无挠阿贝尔群，它们的同构类型具有任意的图灵度。我们还表明，在这些类的同构类型没有图灵度的结构。
We prove a uniform version of non-Archimedean Yomdin-Gromov parametrizations in a definable context with algebraic Skolem functions in the residue field. The parametrization result allows us to bound the number of F_q[t]-points of bounded degrees of algebraic varieties, uniformly in the cardinality q of the finite field F_q and the degree, generalizing work by Sedunova for fixed q. We also deduce a uniform non-Archimedean Pila-Wilkie theorem, generalizing work by Cluckers-Comte-Loeser.
We give new constructions of two classes of algebraic code families which are efficiently list decodable with small output list size from a fraction $1-R-ε$ of adversarial errors where $R$ is the rate of the code, for any desired positive constant $ε$. The alphabet size depends only $ε$ and is nearly-optimal. The first class of codes are obtained by folding algebraic-geometric codes using automorphisms of the underlying function field. The list decoding algorithm is based on a linear-algebraic approach, which pins down the candidate messages to a subspace with a nice "periodic" structure. The …
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