Bartosz Naskręcki
In this note we compute a constant $N$ that bounds the number of non--primitive divisors in elliptic divisibility sequences over function fields of any characteristic. We improve a result of Ingram--Mah{é}--Silverman--Stange--Streng, 2012, and we show that the constant can be chosen independently of the specific point and to some extent of the specific curve, as predicted in loc. cit.
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Sato--Tate, cyclicity, and divisibility statistics on average for elliptic curves of small height
我们得到了素数$p\le x$的数量的渐近公式，对于这些素数，椭圆曲线$$ \E_{a,b} : Y^2 = X^3 + aX + b $$的约化模$p$满足某些"自然“性质，平均在整数$a$和$b$上，具有$|a|\le A$和$|b| \le B$，其中$A$和$B$相对于$x$是小的。具体而言，我们调查行为相对于佐藤-泰特猜想，循环性，整除的点数由一个固定的整数$m$。
Integral points, divisibility between values of polynomials and entire curves on surfaces
我们证明了一些新的退化结果的积分点和整个曲线的表面，特别是，我们提供了第一个例子，据我们所知，一个简单的连接光滑簇的积分点集从来没有Zurkiki稠密（没有整个曲线的Zurkiki稠密图像）。我们的一些结果与整除问题，即问题的描述积分点在平面上的一些给定的多项式的值在两个变量除以其他给定的多项式的值。
p-adic properties of division polynomials and elliptic divisibility sequences
对于椭圆曲线的K-有理点E（K）群中的一个给定点P，我们考虑值序列（F_1（P），F_2（P），F_3（P），.）若K是一个有限域，我们证明了该序列是周期的.如果K/Q_p是一个局部域，我们证明（在一定的假设下）存在一个幂q=p^e，使得对所有m > 0，当k趋于无穷大时，F_{mq^{k}}（P）的极限存在于K中，并且是Q（E）上的代数极限.我们应用这个结果证明了一个类似的椭圆整除序列的p-adic极限和代数性结果。
We prove several statements about arithmetic hyperbolicity of certain blow-up varieties. As a corollary we obtain multiple examples of simply connected quasi-projective varieties that are pseudo-arithmetically hyperbolic. This generalizes results of Corvaja and Zannier obtained in dimension 2 to arbitrary dimension. The key input is an application of the Ru-Vojta's strategy. We also obtain the analogue results for function fields and Nevanlinna theory with the goal to apply them in a future paper in the context of Campana's conjectures.
Divisibility sequences are defined by the property that their elements divide each other whenever their indices do. The divisibility sequences that also satisfy a linear recurrence, like the Fibonacci numbers, are generated by polynomials that divide their compositions with every positive integer power. We completely characterize such polynomials in terms of their factorizations into cyclotomic polynomials using labeled Hasse diagrams, and construct new integer divisibility sequences based on them. We also show that, unlike the Fibonacci numbers, these non-classical sequences do not have the p…
Let $\mathbb{F}_q[t]$ denote the ring of polynomials over $\mathbb{F}_q$, the finite field of $q$ elements. We prove an estimate for fractional parts of polynomials over $\mathbb{F}_q[t]$ satisfying a certain divisibility condition analogous to that of intersective polynomials in the case of integers. We then extend our result to consider linear combinations of such polynomials as well.
It is known that the greatest common divisor of two Fibonacci numbers is again a Fibonacci number. It is called the strong divisibility property. However, this property does not hold for every second order sequence. In this paper we study the generalized Fibonacci polynomials and classify them in two types depending on their Binet formula. We give a complete characterization for those polynomials that satisfy the strong divisibility property. We also giveformulas to calculate the $\gcd$ of those polynomials that do not satisfy the strong divisibility property. Note. This paper is now published…
Let $f(x) \in \bbz[x]$ and consider the index divisibility set $D = \{n \in \bbn : n \mid f^n(0)\}$. We present a number of properties of $D$ in the case that $(f^n(0))_{n=1}^\infty$ is a rigid divisibility sequence, generalizing a number of results of Chen, Stange, and the first author. We then study the polynomial $x^d + x^e + c \in \bbz[x]$, where $d > e \ge 2$ and determine all cases where this map has a finite index divisibility set.
Divisibility criteria and coefficient formulas for cyclotomic polynomials
本文给出多项式被分圆多项式整除的充要条件，推导拉马努金和的新公式，并揭示分圆多项式系数的新性质及指数相差素因子时的递推关系。
In this paper, we generalize Bernoulli-Goss polynomials, and give a criterion on the divisibility of zeta functions of cyclotomic function fields. As an application of our criterion, for a given polynomial $f(u)$, we prove that there are infinitely many cyclotomic function fields whose zeta polynomial is divided by $f(u)$.
We investigate algebraic and arithmetic properties of a class of sequences of sparse polynomials that have binomial coefficients both as exponents and as coefficients. In addition to divisibility and irreducibility results we also consider rational roots. This leads to the study of an infinite class of integer sequences which have interesting properties and satisfy linear recurrence relations.
The multi-variable Schmidt polynomials are defined by $$ S_n^{(r)}(x_0,\ldots,x_n):=\sum_{k=0}^n {n+k \choose 2k}^{r}{2k\choose k} x_k. $$ We prove that, for any positive integers $m$, $n$, $r$, and $\varepsilon=\pm 1$, all the coefficients in the polynomial $$ \sum_{k=0}^{n-1}\varepsilon^k(2k+1) S_k^{(r)}(x_0,\ldots,x_k)^m $$ are multiples of $n$. This generalizes a recent result of Pan on the divisibility of sums of Apéry polynomials.
On divisibility of sums of Apery polynomials
对任意正整数$m$和$α$，我们证明了$$\sum_{k=0}^{n-1}ε^k(2k+1)A_k^{(α)}(x)^m\equiv0\pmod{n}, $$，其中$ε\in\{1,-1\}$和$$ A_n^{(α)}(x)=\sum_{k=0}^n\binom{n}{k}^α\binom{n+k}{k}^αx^k.$$
Divisibility of Trinomials by Irreducible Polynomials over F2
在$F_2$上给定n次的不可约三项式并不总是存在，并且在不存在n次不可约三项式的情况下，使用具有n次不可约因子的三项式可能是有效的。本文考虑了$F_2$上不可约多项式可除三项式的条件。建立了环$F_2$上自反三项式可被不可约多项式整除的条件。并且推广了Welch判别三项多项式$x^m+x^s+1$是否整除为三项多项式$x^{am}+x^{bs}+1$的判别准则。
Divisibility of the coefficients of modular polynomials
研究模多项式Φ_N(X+J, Y+J)的系数在特定代数数J（如0和奇异模）处被小素数高度整除的性质，其中J具有超奇异约化。
Fibonacci sums and divisibility properties
本文利用 Sury 多项式恒等式的变体推导有限斐波那契与卢卡斯和的新表达式，并将其推广到相关多项式及 Horadam 序列，进而用于发现和证明整除性质。
We study linear divisibility sequences of order 4, providing a characterization by means of their characteristic polynomials and finding their factorization as a product of linear divisibility sequences of order 2. Moreover, we show a new interesting connection between linear divisibility sequences and Salem numbers. Specifically, we generate linear divisibility sequences of order 4 by means of Salem numbers modulo 1.
Divisibility of the Multiplicative Order Modulo Monic Irreducible Polynomials Over Finite Fields
本文研究有限域上使给定有理函数模首一不可约多项式的乘法阶被固定整数整除的多项式集合，刻画其在三种密度概念下的密度存在性并给出公式。
We study some divisibility properties related to the factors of the discriminant of the characteristic polynomial of generalized Fibonacci sequences $(G_n)_{n\ge0}$ defined by $G_0=0$, $G_1=1$ and $G_n=pG_{n-1}+qG_{n-2}$ for $n\ge2$, where $p,q$ are given integers. As corollaries, we give some divisibility properties on some well known sequences.
In this note, we study the divisibility relation $U_m\mid U_{n+k}^s-U_n^s$, where ${\bf U}:=\{U_n\}_{n\ge 0}$ is the Lucas sequence of characteristic polynomial $x^2-ax\pm 1$ and $k,m,n,s$ are positive integers.
Let $C$ be a smooth genus one curve described by a quartic polynomial equation over the rational field $\mathbb Q$ with $P\in C(\mathbb Q)$. We give an explicit criterion for the divisibility-by-$2$ of a rational point on the elliptic curve $(C,P)$. This provides an analogue to the classical criterion of the divisibility-by-$2$ on elliptic curves described by Weierstrass equations. We employ this criterion to investigate the question of extending a rational $D(q)$-quadruple to a quintuple. We give concrete examples to which we can give an affirmative answer. One of these results implies that a…
We present a different proof of the characterization of non--degenerate recurrence sequences, which are also divisibility sequences, given by Van der Poorten, Bezevin, and Petho in their paper "A Full Characterisation of Divisibility Sequence". Our proof is based on an interesting determinant identity related to impulse sequences, arising from the evaluation of a generalized Vandermonde determinant. As a consequence of this new proof we can find a more precise form for the resultant sequence presented in their paper, in the general case of non--degenerate divisibility sequences having minimal …
Congruence Jumping in Polynomial Divisibility Systems: Arithmetic Structure and Solution Sets
该论文研究整数系数多项式整除系统 $y\mid A(x), x\mid B(y)$ 的非零整数解，提出同余跳跃方法，建立互反多项式规则并分类二次情形，构造非单位循环四周期，给出解集有限化判据。
On new divisibility properties of generalized central trinomial coefficients and Legendre polynomials
本文提出了 $m=2$ 时整除和式 $B(n,m,a,b)$ 的 $a+b$ 最高幂次新公式，并据此确定了中心 Dellanoy 数等的完整 3-adic 赋值及 Legendre 多项式的整除性质。
The alternating descent statistic on permutations was introduced by Chebikin as a variant of the descent statistic. We show that the alternating descent polynomials on permutations are unimodal via a five-term recurrence relation. We also found a quadratic recursion for the alternating major index $q$-analog of the alternating descent polynomials. As an interesting application of this quadratic recursion, we show that $(1+q)^{\lfloor n/2\rfloor}$ divides $\sum_{π\in\mathfrak{S}_n}q^{\rm{altmaj}(π)}$, where $\mathfrak{S}_n$ is the set of all permutations of $\{1,2,\ldots,n\}$ and $\rm{altmaj}(π…
Divisibility properties of polynomial expressions of random integers
研究随机整数点处多项式取值集合的可整除性，证明其最大公约数与规范化计数在温和条件下依分布收敛于几乎必然有限的随机变量，方法基于整数阿代尔与Lang-Weil界。
Arbitrarily Many Primitive Prime Divisors in an Iterate of $x^d+c$
该研究证明单临界多项式$x^d+c$的临界轨道第$n$项的本原素因子个数无一致界，可构造参数使该项含任意多个本原素因子，还应用于得到分裂域的最大次数。
In this note we shall give conditions which guarantee that $\frac{1-q^b}{1-q^a}{n\brack m}\in\mathbb{Z}[q]$ holds. We shall provide a full characterisation for $\frac{1-q^b}{1-q^a}{ka\brack m}\in\mathbb{Z}[q]$. This unifies a variety of results already known in literature. We shall prove new divisibility properties for the binomial coefficients and a new divisibility result for a certain finite sum involving the roots of the unity.
Schönemann-Eisenstein-Dumas-type irreducibility conditions that use arbitrarily many prime numbers
著名的不可约性标准的舍内曼-爱森斯坦和大仲马依赖于信息的整除系数的多项式由一个单一的素数。本文给出了整系数多项式的Schönemann-Eisenstein-Dumas型不可约准则，这些准则是由其系数关于任意多个素数的可除性条件给出的.一个特别的注意将支付给那些不可约的标准，需要信息的整除系数由两个不同的素数。
Common Divisors of Elliptic Divisibility Sequences over Function Fields
设E/k（T）是定义在特征为零的有理函数域上的椭圆曲线。修正E的维尔斯特拉斯方程。对于E（k（T））中的点R，写作x_R = A_R/D_R^2，其中互素多项式A_R（T）和D_R（T）在k [T]中。对于n \ge 1的序列{D_{nR}}称为R的"椭圆整除序列".设E（k（T））中的P，Q为独立点.我们猜想deg（gcd（D_{nP}，D_{mQ}））对于m，n \ge 1是有界的，并且对于无穷多个n \ge 1，gcd（D_{nP}，D_{nQ}）= gcd（D_{P}，D_{Q}）.我们在j（E）在k中的情况下证明了这些定理。更一般地说，我们证明类似的声明与k（T）替换为任何曲线的功能领域和P和Q允许躺在不同的椭圆曲线。如果k是特征为p的有限域，并再次假设j（E）在k中，我们证明了对于满足gcd（n，p）= 1的无穷多个n，deg（gcd（D_{nP}，D_{nQ}））> n + O（sqrt {n}）.
We define the $\textit{Divisor Divisibility Sequence}$ associated to a Laurent polynomial $f\in\mathbb{Z}[X_1^{\pm1},\ldots,X_N^{\pm1}]$ to be the sequence $W_n(f)=\prod f(ζ_1,\ldots,ζ_N)$, where $ζ_1,\ldots,ζ_N$ range over all $n$'th roots of unity with $f(ζ_1,\ldots,ζ_N)\ne0$. More generally, we define $W_Λ(f)$ analogously for any finite subgroup $Λ\subset(\mathbb C^*)^N$. We investigate divisibility, factorization, and growth properties of $W_Λ(f)$ as a function of $Λ$. In particular, we give the complete factorization of $W_Λ(f)$ when $f$ has generic coefficients, and we prove an analytic …
Canonical height functions on the affine plane associated with polynomial automorphisms
设$f: \mathbb{A}^2 \to \mathbb{A}^2$是数域$K$上动力次数为$δ\geq 2$的多项式自同构. (This等价于说$f$是不可三角化的多项式自同构。）然后，我们构造了与$f$相关的定义在$\mathbb{A}^2(\bar{K})$上的正则高度函数。这些函数满足Northcott有限性，并且$\mathbb{A}^2(\bar{K})$上的$\bar{K}$值点是$f$-周期的当且仅当它的高度为零.作为正则高度函数的一个应用，我们给出了无限$f$-轨道上有界高度点的个数的一个估计。
Let $p$ denote an odd prime. In this paper, we are concerned with the $p$-divisibility of additive exponential sums associated to one variable polynomials over a finite field of characteristic $p$, and with (the very close question of) determining the Newton polygons of some families of Artin-Schreier curves, i.e. $p$-cyclic coverings of the projective line in characteristic $p$. We first give a lower bound on the $p$-divisibility of exponential sums associated to polynomials of fixed degree. Then we show that an Artin-Schreier curve defined over a finite field of characteristic $p$ cannot be …
We investigate some aspects of the $m$-division field $K({\mathcal{E}}[m])$, where $\mathcal{E}$ is an elliptic curve defined over a field $K$ with ${\textrm{char}}(K)\neq 2,3$ and $m$ is a positive integer. When $m=p^r$, with $p\geq 5$ a prime and $r$ a positive integer, we prove $K(\mathcal{E}[p^r])=K(x_1,ζ_p,y_2)$, where $\{(x_1, y_1),(x_2,y_2)\}$ is a generating system of ${\mathcal{E}}[p^r]$ and $ζ_p$ is a primitive $p$-th root of the unity. If $\mathcal{E}$ has a $K$-rational point of order $p$, then $K(\mathcal{E}[p^r])=K(ζ_{p^r},\sqrt[m_1]{a})$, with $a\in K(ζ_{p^r})$ and $m_1|p^r$. In…
We provide an elementary proof to a conjecture by Robinson that multiples of (powers of) primes in the Somos-4 sequence are equally spaced. We also show, almost as a corollary, for the generalised Somos-4 sequence defined by $τ_{n+2}τ_{n-2}=ατ_{n+1}τ_{n-1}+βτ_n^2$ and initial values $τ_1=τ_2=τ_3=τ_4=1$, that the polynomial $τ_n(α,β)$ is a divisor of $τ_{n+k(2n-5)}(α,β)$ for all $n,k\in\mathbb Z$ and establish a similar result for the generalized Somos-5 sequence.
The notion of block divisibility naturally leads one to introduce unitary cyclotomic polynomials. We formulate some basic properties of unitary cyclotomic polynomials and study how they are connected with cyclotomic, inclusion-exclusion and Kronecker polynomials. Further, we derive some related arithmetic function identities involving the unitary analog of the Dirichlet convolution.
Divisibility of class numbers of imaginary quadratic function fields by a fixed odd number
本文给出了一类群元素为固定奇数阶的函数域$\mathbb{F}_{q}(x)$的虚二次扩张个数的一个新下界。更确切地说，对于$q$，一个幂的奇素数，和$g$一个固定的奇正整数$\ge 3$，我们表明，对于每个$ε>0$，有$\gg q^{L(1/2+\frac{3}{2(g+1)}-ε)}$多项式$f \in \mathbb{F}_{q}[x]$与$°f=L$，其中的二次扩展$\mathbb{F}_{q}(x, \sqrt{f})$的类群有一个元素的顺序$g$。这锐化了Ram Murty先前的下限$q^{L(1/2+\frac{1}{g})}$。我们的结果是一个功能领域的类似结果Soundararajan数域。
On the number of integral quadratic polynomials with bounded heights and discriminants
本文给出了当高度的上界趋于无穷大时，具有有界高度的整二次多项式及其判别式的渐近数。
Periods modulo $p$ of integer sequences associated with division polynomials of genus $2$ curves
研究2阶曲线积分点相关除法多项式整数序列的模p周期性，揭示序列模p周期与积分点在Jacobian变元上阶的关系，扩展Ward关于椭圆曲线除法序列的研究。
Divisibility by 2 of partial Stirling numbers
这里使用的部分Stirling数T_n（k）被定义为（n选i）i^k中i的奇数值之和。它们的2-指数nu（T_n（k））在代数拓扑学中是重要的.我们给出了许多适用于n的所有值的具体结果，即对某个模2 ^t的同余类中的所有k，nu（T_n（k））= nu（k-k 0）+c 0，其中k 0是2-adic整数，c 0是正整数.我们的分析涉及到nu（sum（n choose 2 i +1）i^j）的几个新的一般性结果，其证明涉及到一个新的多项式族。在Clarke之后，我们将T_n解释为关于2-adic整数的函数，并且上述2-adic整数k 0作为这些函数的零。
On the number of polynomials of bounded height that satisfy Dumas's criterion
本文研究了不可约的定次数、最大高度$H$的整系数多项式。我们称这种多项式为杜马斯多项式。当$H$趋于无穷大时，我们推导出了杜马斯多项式个数的上界。我们还证明了，对于一个固定的次数，杜马斯多项式的密度在所有不可约整系数多项式严格小于1。
On the Canonical Height Gap for Polynomial Maps and Portraits of Preperiodic Points of Polynomials with Height $0$
研究\(\mathbb{P}^{1}\)上多项式映射的典范高度与朴素高度差，建立与次数\(d\)无关的精确界，以此确定\(\mathbb{Q}\)上高度为0的多项式映射的有理预周期点集并分类其画像。
We define the interpolated polynomial multiple zeta values as a generalization of all of multiple zeta values, multiple zeta-star values, interpolated multiple zeta values, symmetric multiple zeta values, and polynomial multiple zeta values. We then compute the generating function of the sum of interpolated polynomial multiple zeta values of fixed weight, depth, and height.
On the Number of Eisenstein Polynomials of Bounded Height
我们得到了A的一个更精确的渐近公式。Dubickas的数量monic艾森斯坦多项式的固定程度$d$和高度最$H$，作为$H\to \infty$。特别是，我们给出了一个明确的边界的误差项。我们还得到了任意高度至多为$H$的艾森斯坦多项式的一个渐近公式。
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.
Let $ϕ(x) = x^d + c$ be an integral polynomial of degree at least 2, and consider the sequence $(ϕ^n(0))_{n=0}^\infty$, which is the orbit of $0$ under iteration by $ϕ$. Let $D_{d,c}$ denote the set of positive integers $n$ for which $n \mid ϕ^n(0)$. We give a characterization of $D_{d,c}$ in terms of a directed graph and describe a number of its properties, including its cardinality and the primes contained therein. In particular, we study the question of which primes $p$ have the property that the orbit of $0$ is a single $p$-cycle modulo $p$. We show that the set of such primes is finite wh…
The asymptotic number of integral cubic polynomials with bounded heights and discriminants
本文部分地解决了三次整多项式判别式的分布问题。证明了具有有界高度和有界判别式的整三次多项式个数的渐近公式。所得到的公式的主项是最好的可能的多项式判别值的考虑范围。
We obtain upper bounds for the number of monic irreducible polynomials over $\mathbb Z$ of a fixed degree $n$ and a growing height $H$ for which the field generated by one of its roots has a given discriminant. We approach it via counting square-free parts of polynomial discriminants via two complementing approaches. In turn, this leads to a lower bound on the number of distinct discriminants of fields generated by roots of polynomials of degree $n$ and height at most $H$. We also give an upper bound for the number of trinomials of bounded height with given square-free part of the discriminant…
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.
Explicit height bounds on modular polynomials for the elliptic $j$-invariant, cube root of $j$, and Weber modular function $\mathfrak f$
研究改进椭圆\(j\)不变量等相关模多项式显式上界，通过改进Breuer等人给出的界，还证明了\(j\)立方根及韦伯模函数\(\mathfrak f\)的模多项式高度新显式上界。
On the height of cyclotomic polynomials
设$A_n$表示分圆多项式$Φ_n$的高度，其中$n$是$k$不同奇素数的乘积。我们证明了$A_n \le ε_kϕ(n)^{k^{-1}2^{k-1}-1}$与$-\logε_k\sim c2^k$，$c>0$ .对于逆分圆多项式$Ψ_n$的高度$C_n$也是如此。此外，我们改进了Kaplan关于$x^n-1$的因子的最大高度的一个界，记为$B_n$ .我们表明，$B_n<η_k n^{(3^k-1)/(2k)-1}$，与$-\log η_k \sim c3^k$和相同的$c$。
A Note on Heights of Cyclotomic Polynomials
本文证明了对于任意正整数 $h$，$h$ 或 $h+1$ 必为某个由三个不同素数乘积决定的分圆多项式 $Φ_n$ 的高度。
Let K be a number field, X/K a curve, and f/X a family of endomorphisms of projective N-space. It follows from a result of Call and Silverman that the canonical height associated to the family f, evaluated along a section, differs from a Weil height on the base by little-o of a Weil height. In the case where f is a family with an invariant hyperplane, whose restriction to this invariant hyperplane is isotrivial, we improve this by showing that the canonical height along a section differs from a Weil height on the base by a bounded amount.
Let $F$ be a univariate polynomial or rational fraction of degree $d$ defined over a number field. We give bounds from above on the absolute logarithmic Weil height of $F$ in terms of the heights of its values at small integers: we review well-known bounds obtained from interpolation algorithms given values at $d+1$ (resp. $2d+1$) points, and obtain tighter results when considering a larger number of evaluation points.
We prove a lower bound on the canonical height associated to polynomials over number fields evaluated at points with infinite forward orbit. The lower bound depends only on the degree of the polynomial, the degree of the number field, and the number of places of bad reduction.
Several recent papers construct auxiliary polynomials to bound the Weil height of certain classes of algebraic numbers from below. Following these techniques, the author gave a general method for introducing auxiliary polynomials to problems involving the Weil height. The height appears as a solution to a certain extremal problem involving polynomials. We further generalize the above techniques to acquire both the projective height and the height on subspaces in the same way. We further obtain lower bounds on the heights of points on some subvarieties of $\mathbb P^{N-1}(\overline{\mathbb Q})$…
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.
On the height of polynomials that split completely over a fixed number field
本文证明了在固定数域上完全分裂的 $n$ 次多项式的最小高度随 $n$ 呈指数级增长，并精确确定了全实数域及特定复二次域情况下的隐含常数。
Canonical heights and division polynomials
讨论了数域上超椭圆雅可比矩阵上代数点的标准高的一种新的计算方法。该方法不需要任何几何模型，既不需要$p$ -adic也不需要复杂的解析模型。在亏格2的情况下，我们也提出了一个版本，根本不需要因式分解。该方法是基于递归关系的“司多项式”相关的超椭圆雅可比，和一个丢番图逼近的结果，由于Faltings。
An explicit height bound for the classical modular polynomial
对于素数m，设Phi_m为经典模多项式，h（Phi_m）表示其对数高.通过推广Cohen的一个定理，证明了h（Phi_m）≤ 6m log m +16m +14m log m.作为推论，我们发现h（Phi_m）<= 6mlogm +18m也成立.提供了m <= 3607时的h（Phi_m）值表。
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