Computing algebraic numbers of bounded height

On the number of integral quadratic polynomials with bounded heights and discriminants

In the paper we obtain the asymtotic number of integral quadratic polynomials with bounded heights and discriminants as the upper bound of heights tends to infinity.

Lower bounds on the projective heights of algebraic points

If $α_1,\ldots,α_r$ are algebraic numbers such that $$N=\sum_{i=1}^rα_i \ne \sum_{i=1}^rα_i^{-1}$$ for some integer $N$, then a theorem of Beukers and Zagier gives the best possible lower bound on $$\sum_{i=1}^r\log h(α_i)$$ where $h$ denotes the Weil Height. We will extend this result to allow $N$ to be any totally real algebraic number. Our generalization includes a consequence of a theorem of Schinzel which bounds the height of a totally real algebraic integer.

Height bounds for algebraic numbers satisfying splitting conditions

In an earlier work, the first author and Petsche solved an energy minimization problem for local fields and used the result to obtain lower bounds on the height of algebraic numbers all whose conjugates lie in various local fields, such as totally real and totally p-adic numbers. In this paper, we extend these techniques and solve the corresponding minimization programs for real intervals and p-adic discs, obtaining several new lower bounds for the height of algebraic numbers all of whose conjugates lie in such sets.

分圆域中有界高度代数整数的计数

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。

高度1到7的实代数数的Cantor列表

Cantor's List of Real Algebraic Numbers of Heights 1 to 7

该研究延续Cantor关于实代数数可数性的工作，系统给出了高度1到7、对应次数为k的整系数不可约多项式的实代数数列表，补充了Cantor仅给出前三个高度实例的研究内容。

The Sato--Tate Distribution in Families of Elliptic Curves with a Rational Parameter of Bounded Height

We obtain new results concerning the Sato--Tate conjecture on the distribution of Frobenius angles over parametric families of elliptic curves with a rational parameter of bounded height.

Algebraic $S$-integers of fixed degree and bounded height

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 $S$.

Endomorphisms of Bounded Height and Resultant

Let K be an algebraic number field. For a degree d rational morphism of projective n-space defined over K let R denote its minimal resultant ideal. For a fixed height function on the moduli space of dynamical systems this paper shows that all such morphisms of bounded height and resultant are contained in finitely many PGL(K) equivalence classes. This answers a question of Silverman in the affirmative.

Counting points of fixed degree and bounded height

We consider the set of points in projective $n$-space that generate an extension of degree $e$ over given number field $k$, and deduce an asymptotic formula for the number of such points of absolute height at most $X$, as $X$ tends to infinity. We deduce a similar such formula with instead of the absolute height, a so-called adelic-Lipschitz height.

Bounded Height in Pencils of Finitely Generated Subgroups

We prove height bounds concerning intersections of finitely generated subgroups in a torus with algebraic subvarieties, all varying in a pencil. This vastly extends the previously treated constant case and involves entirely different, and more delicate, techniques.

Rational points of bounded height on weighted projective stacks

A weighted projective stack is a stacky quotient $\mathscr P(\mathbf a)=(\mathbf A^n-\{0\})/\mathbb G_m$, where the action of $\mathbb G_m$ is with weights $\mathbf a\in\mathbb Z^n_{>0}$. Examples are: the compactified moduli stack of elliptic curves $\mathscr P(4,6)$ and the classifying stack of $μ_m$-torsors $Bμ_m=\mathscr P(m)$. We define heights on the weighted projective stacks. The heights generalize the naive height of an elliptic curve and the absolute discriminant of a torsor. We use the heights to count rational points. We find the asymptotic behaviour for the number of rational poin…

The asymptotic number of integral cubic polynomials with bounded heights and discriminants

In the paper we partially solved the problem of the distribution of the discriminants of integral polynomials in the cubic case. We proved the asymptotic formula for the number of integral cubic polynomials having bounded height and bounded discriminant. The main term of the obtained formula is the best possible for the considered range of polynomial discriminant values.

Counting degenerate polynomials of fixed degree and bounded height

In this paper, we give sharp upper and lower bounds for the number of degenerate monic (and arbitrary, not necessarily monic) polynomials with integer coefficients of fixed degree $n \ge 2$ and height bounded by $H \ge 2$. The polynomial is called degenerate if it has two distinct roots whose quotient is a root of unity. In particular, our bounds imply that non-degenerate linear recurrence sequences can be generated randomly.

On the Number of Eisenstein Polynomials of Bounded Height

We obtain a more precise version of an asymptotic formula of A. Dubickas for the number of monic Eisenstein polynomials of fixed degree $d$ and of height at most $H$, as $H\to \infty$. In particular, we give an explicit bound for the error term. We also obtain an asymptotic formula for arbitrary Eisenstein polynomials of height at most $H$.

向量群紧化上有界高度的半整点

Semi-integral points of bounded height on vector group compactifications

本文用类似[PSTVA21]的思路得到向量群等变紧化上有界高度Darmon点计数函数的渐近公式，且该公式与[Moe26a]中$\boldsymbol{\textit{M}}$-点的Manin猜想相容。

Linear Equations with Rational Fractions of Bounded Height and Stochastic Matrices

We obtain a tight, up to a logarithmic factor, upper bound on the number of solutions to the equation $$ \sum_{j=1}^n a_j \frac{s_j}{r_j} =a_0, \qquad $$ with variables $r_1,...,r_n$ in an arbitrary box at the origin and variables $s_1,..., s_n$ in an essentially arbitrary translation of this box. We apply this result to get an upper bound on the number of stochastic matrices with rational entries of bounded height.

Computing points of bounded height in projective space over a number field

We construct an algorithm for solving the following problem: given a number field $K$, a positive integer $N$, and a positive real number $B$, determine all points in $\mathbb P^N(K)$ having relative height at most $B$. A theoretical analysis of the efficiency of the algorithm is provided, as well as sample computations showing how the algorithm performs in practice. Two variants of the method are described, and examples are given to compare their running times. In the case $N=1$ we compare our method to an earlier algorithm for enumerating elements of bounded height in number fields.

Counting primitive points of bounded height

Let $k$ be a number field and $K$ a finite extension of $k$. We count points of bounded height in projective space over the field $K$ generating the extension $K/k$. As the height gets large we derive asymptotic estimates with a particularly good error term respecting the extension $K/k$. In a future paper we will use these results to get asymptotic estimates for the number of points of fixed degree over $k$. We also introduce the notion of an adelic Lipschitz height generalizing that of Masser and Vaaler. This will lead to further applications involving points of fixed degree on linear variet…

On the distribution of polynomials with bounded height

We provide an asymptotic expression for the probability that a randomly chosen polynomial with given degree, having integral coefficients bounded by some B, has a prescribed signature. We also give certain related formulas and numerical results along this line. Our theorems are closely related to earlier results of Akiyama and Pethő, and also yield extensions of recent results of Dubickas and Sha.

Totally real algebraic integers of arboreal height 2

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…

Slicing the stars: counting algebraic numbers, integers, and units by degree and height

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…

代数数高度的下界与 Dedekind zeta 函数的低位零点

Lower bound on height of algebraic numbers and low lying zeros of the Dedekind zeta-function

本文借助 Dedekind zeta 函数低位零点建立代数整数 Weil 高度下界，并在 GRH 条件下证明特定无限非 Galois 扩张中的 Lehmer 猜想。

On the number of polynomials of bounded height that satisfy Dumas's criterion

We study integer coefficient polynomials of fixed degree and maximum height $H$, that are irreducible by Dumas's criterion. We call such polynomials Dumas polynomials. We derive upper bounds on the number of Dumas polynomials, as $H$ approaches infinity. We also show that, for a fixed degree, the density of Dumas polynomials in all irreducible integer coefficient polynomials is strictly less than 1.

仿射空间中模形的循环周期点及具有有界house和高度的周期点

Cyclotomic preperiodic points for morphisms in affine spaces and preperiodic points with bounded house and height

本文研究了仿射空间中模形的周期点集在循环闭包上的稠密性问题，并探讨了具有有界house和高度的周期点的性质。

Distribution of real algebraic integers

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…

Algebraic points of small height missing a union of varieties

Let $K$ be a number field, $\overline{\mathbb Q}$, or the field of rational functions on a smooth projective curve over a perfect field, and let $V$ be a subspace of $K^N$, $N \geq 2$. Let $Z_K$ be a union of varieties defined over $K$ such that $V \nsubseteq Z_K$. We prove the existence of a point of small height in $V \setminus Z_K$, providing an explicit upper bound on the height of such a point in terms of the height of $V$ and the degree of a hypersurface containing $Z_K$, where dependence on both is optimal. This generalizes and improves upon the previous results of the author. As a part…

关于具有有界动力学典范高度点的数量

On the number of points with bounded dynamical canonical height

本文研究具有有界动力学典范高度的点的数量，利用 Hsia 的动力学高度 zeta 函数，给出特殊情形下的显式公式，并推导出点数的渐近行为。

Integral points of fixed degree and bounded height

By Northcott's Theorem there are only finitely many algebraic points in affine $n$-space of fixed degree over a given number field and of height at most $X$. For large $X$ the asymptotics of these cardinalities have been investigated by Schanuel, Schmidt, Gao, Masser and Vaaler, and the author. In this paper we study the case where the coordinates of the points are restricted to algebraic integers, and we derive the analogues of Schanuel's, Schmidt's, Gao's and the author's results.

Distribution of real algebraic integers

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 height bounds for $K$-rational points on transverse curves in powers of elliptic curves

Let $C$ be an algebraic curve embedded transversally in a power $E^N$ of an elliptic curve $E$. In this article we produce a good explicit bound for the height of all the algebraic points on $C$ contained in the union of all proper algebraic subgroups of $E^N$. The method gives a totally explicit version of the Manin-Dam'janenko Theorem in the elliptic case and it is a generalisation of previous results only proved when $E$ does not have Complex Multiplication.

由具有有限不同素因子数的有理数进行内在迪奥菲安逼近

Intrinsic Diophantine approximation by rationals of height with a bounded number of distinct prime factors

本文研究了由具有有限不同素因子数的有理数逼近的自相似集的Hausdorff维数，修正了前一版本中引理3.8的证明错误。

Lectures on height zeta functions: At the confluence of algebraic geometry, algebraic number theory, and analysis

This is a survey on the theory of height zeta functions, written on the occasion of a French-Japanese winter school, held in Miura (Kanagawa, Japan) in Jan. 2008. It does not presuppose much knowledge in algebraic geometry. The last chapter of the survey explains recent results obtained in collaboration with Yuri Tschinkel concerning asymptotics of volumes of height balls in analytic geometry over local fields, or in adelic spaces.

Integral points of bounded height on toric varieties

We establish asymptotic formulas for the number of integral points of bounded height on toric varieties.

An Arakelov-Theoretic Approach to Naïve Heights on Hyperelliptic Jacobians

We use Arakelov theory to define a height on divisors of degree zero on a hyperelliptic curve over a global field, and show that this height has computably bounded difference from the Néron-Tate height of the corresponding point on the Jacobian. We give an algorithm to compute the set of points of bounded height with respect to this new height. This provides an `in principle' solution to the problem of determining the sets of points of bounded Néron-Tate heights on the Jacobian. We give a worked example of how to compute the bound over a global function field for several curves, of genera up t…

The Bounded Height Conjecture for Semiabelian Varieties

The Bounded Height Conjecture of Bombieri, Masser, and Zannier states that for any sufficiently generic algebraic subvariety of a semiabelian $\overline{\mathbb{Q}}$-variety $G$ there is an upper bound on the Weil height of the points contained in its intersection with the union of all algebraic subgroups having (at most) complementary dimension in $G$. This conjecture has been shown by Habegger in the case where $G$ is either a multiplicative torus or an abelian variety. However, there are new obstructions to his approach if $G$ is a general semiabelian variety. In particular, the lack of Poi…

The Weil height in terms of an auxiliary polynomial

Recent theorems of Dubickas and Mossinghoff use auxiliary polynomials to give lower bounds on the Weil height of an algebraic number $α$ under certain assumptions on $α$. We prove a theorem which introduces an auxiliary polynomial for giving lower bounds on the height of any algebraic number. Our theorem contains, as corollaries, a slight generalization of the above results as well as some new lower bounds in other special cases.

多高度分布推导出Batyrev-Manin原理

The multi-height distribution implies the Batyrev-Manin principle

通过多高度分析推导出Batyrev-Manin原理，利用超椭圆法研究有理点的渐进行为。

Comments on the height reducing property II

A complex number $α$ is said to satisfy the height reducing property if there is a finite set $F\subset \mathbb{Z}$ such that $\mathbb{Z}[α]=F[α]$, where $\mathbb{Z}$ is the ring of the rational integers. It is easy to see that $α$ is an algebraic number when it satisfies the height reducing property. We prove the relation $\operatorname{Card}(F)\geq \max\{2,\left\vert M_α(0)\right\vert \},$ where $M_α$ is the minimal polynomial of $α$ over the field of the rational numbers, and discuss the related optimal cases, for some classes of algebraic numbers $α$. In addition, we show that there is an …

Points of bounded height on equivariant compactifications of vector groups, II

We prove asymptotic formulas for the number of rational points of bounded height on certain blow-ups of the projective space.

Points of bounded height on equivariant compactifications of vector groups, I

We prove asymptotic formulas for the number of rational points of bounded height on certain equivariant compactifications of the affine plane.

分圆扩张中Call-Silverman高度的一个下界

A lower bound for the Call-Silverman height in cyclotomic extensions

该研究针对数域上的二次后临界有限多项式，结合位势理论等技术，建立分圆扩张中游荡代数整数的Call-Silverman高度下界，并给出非整数情形的部分结果。

具有有界分歧的点的高度

Heights of points with bounded ramification

该研究将Baker关于分裂乘性约化椭圆曲线上有界分歧点的Néron-Tate高度下界结果有效化，证明了相关Lattès映射典范高度的类似结论，并通过改变约化类型给出反例，说明该下界存在性在有限域扩张下一般不保持。

Variation of the canonical height in a family of rational maps

Let $d\ge 2$ be an integer, let $c(t)$ be any rational map, and let $f_t(z) := (z^d+t)/z$ be a family of rational maps indexed by t. For each algebraic number $t$, we let $h_{f_t}(c(t))$ be the canonical height of $c(t)$ with respect to the rational map $f_t$. We prove that the map $H(t):=h_{f_t}(c(t))$ (as $t$ varies among the algebraic numbers) is a Weil height.

一个分裂代数群的motivic Poisson公式及其在motivic高度zeta函数中的应用

A motivic Poisson formula for split algebraic tori with an application to motivic height zeta functions

本文证明了分裂代数群adelic点的motivic版本的Poisson公式，并应用于研究分裂射影toric品种的motivic高度zeta函数，基于motivic Manin-Peyre原理。

阿贝尔簇上有界高度的Lehmer型界与有理点计数

Lehmer-Type bounds and counting rational points of bounded heights on Abelian varieties

本文研究了数域K上阿贝尔簇A的$\bar{K}$-点Néron-Tate高度的Lehmer型界，估计了高度$\leq \log B$的K-有理点数量，并针对椭圆曲线及其乘积将估计中的常数C变为显式可计算。

Integral points of bounded height on compactifications of semi-simple groups

We study the asymptotic distribution of integral points of bounded height on partial bi-equivariant compactifications of semi-simple groups of adjoint type.

Integral points of bounded height on partial equivariant compactifications of vector groups

We establish asymptotic formulas for the number of integral points of bounded height on partial equivariant compactifications of vector groups.

Lower bounds for the normalized height and non-dense subsets of varieties in an abelian variety

This work is the third part of a series of papers. In the first two we consider curves and varieties in a power of an elliptic curve. Here we deal with subvarieties of an abelian variety in general. Let V be an irreducible variety of dimension d embedded in an abelian variety A, both defined over the algebraic numbers. We say that V is weak-transverse if V is not contained in any proper algebraic subgroup of A, and transverse if it is not contained in any translate of such a subgroup. Assume a conjectural lower bound for the normalized height of V. For V transverse, we prove that the algebraic…

Height functions for motives

We define various height functions for motives over number fields. We compare these height functions with classical height functions on algebraic varieties, and also with analogous height functions for variations of Hodge structures on curves over C. These comparisons provide new questions on motives over number fields.

环面簇的某些子簇上高度有界的点

Points of bounded height on certain subvarieties of toric varieties

本文结合分裂挠子法与双曲线法，计算环面簇某些子簇上高度有界的有理点和Campana点。

整体函数域上加权射影空间中的有界高度点

Points of bounded height on weighted projective spaces over global function fields

本文研究整体函数域上加权射影叠的有界高度有理点计数问题，给出了点数的精确公式与渐近式。

Multiplicative approximation by the Weil height

Let $K/\mathbb{Q}$ be an algebraic extension of fields, and let $α\not= 0$ be contained in an algebraic closure of $K$. If $α$ can be approximated by roots of numbers in $K^{\times}$ with respect to the Weil height, we prove that some nonzero integer power of $α$ must belong to $K^{\times}$. More generally, let $K_1, K_2, \dots , K_N$, be algebraic extensions of $\mathb{Q}$ such that each pair of extensions includes one which is a (possibly infinite) Galois extension of a common subfield. If $α\not= 0$ can be approximated by a product of roots of numbers from each $K_n$ with respect to the Wei…

高度有界矩阵的特征多项式数量

Number of characteristic polynomials of matrices with bounded height

本文研究元素高度受 H 限制的整数 n×n 矩阵，给出其不同不可约特征多项式及特征值数量的上下界，并改进已有结果。

有界高度的有限群概形扭子

Torsors for finite group schemes of bounded height

本文针对整体域上有限平展驯顺群概形的扭子定义高度函数，提出有界高度扭子数量的渐近猜想，并在交换群情形证明该渐近公式，同时给出首项常数的算术解释与等分布性质。