Counting Ideals in $\mathbb{Z}[t]/(f)$

Index, Prime Ideal Factorization in simplest Quartic Fields and counting their discriminants

We consider the simplest quartic number fields $\mathbb{K}_m$ defined by the irreducible quartic polynomials $$x^4-mx^3-6x^2+mx+1,$$ where $m$ runs over the positive rational integers such that the odd part of $m^2+16$ is squarefree. In this paper, we study the common index divisor $I(\mathbb K_m)$ and determine explicitly the prime ideal decomposition for any prime number in any simplest quartic number fields $\mathbb{K}_m$. On the other hand, we establish an asymptotic formula for the number of simplest quartic fields with discriminant $\leq x$ and given index.

Variance of lattice point counting in some special shells in $\mathbb{R}^d$

We study the variance of the random variable that counts the number of lattice points in some shells generated by a special class of finite type domains in $\mathbb R^d$. The proof relies on estimates of the Fourier transform of indicator functions of convex domains.

整数环中除数计数公式：除数函数d₀(n)的推广

A formula of counting divisors in integers rings: a generalization of the divisor function $d_0(n)$

本文通过建立整数环中理想结构与加性组合学理论之间的联系，证明了在具有循环类群的整数环中，每个非零理想均可分解为一个最大主因子和一个总长度受达文波特常数限制的理想乘积。通过这种分解，找到了理想I=(α, β)生成元的除数。本文的核心成果是利用有限阿贝尔群上的特征理论，推导出一个闭合公式，用于精确计数给定序列的零和子序列数目。该公式在建立主理想与零和序列之间的对应关系下，提供了对任意给定理想主理想除数的精确计数，从而计数理想I=(α, β)生成元的公共除数。这一结果构成了从唯一分解环到具有有限类群的任何迪克因环的除数函数d₀(n)的自然推广。最后，本文基于这些零和子序列的计数，对O_K中的不可约元进行了特征化。

Counting Subrings of ${\mathbb Z}^n$ of finite index

In this article we investigate the number of subrings of $\Z^d$ using subring zeta functions and $p$-adic integration.

Counting Galois ${\mathbb U}_4({\mathbb F}_p)$-extensions using Massey products

We use Massey products and their relations to unipotent representations to parametrize and find an explicit formula for the number of Galois extensions of a given local field with the prescribed Galois group ${\mathbb U}_4({\mathbb F}_p)$ consisting of unipotent four by four matrices over ${\mathbb F}_p$. Further applications of this method involve the counting of certain Galois extensions with restricted ramifications, and counting the numbers of Galois ${\mathbb U}_4({\mathbb F}_p)$-extensions of some other fields. For each Demushkin pro-$p$-group, we find a very simple version of the condit…

正特征下互质w-的S-整数和S-整理想的计数

Counting w-coprime S-integers and S-integral ideals in positive characteristic

本文结合解析方法、Riemann-Roch定理和Weil定理，在正特征函数域中计数互质w-的S-整数和S-整理想。

数域中任意次数的理想显式计数

Explicit counting of ideals in number fields of arbitrary degree

本文利用数论几何方法，研究数域中积分理想数量的显式估计，通过最小化次数对误差项的影响，避免高阶项，并利用刘维尔空间中的基本域理论和施密特的分拆技巧改进误差估计。

理想类计数函数的矩

Moments of ideal class counting functions

本文针对d次数域中给定理想类的理想计数函数（对应GL(d)上非Hecke特征、非尖点的自守形式傅里叶系数），计算了其矩并研究了对应尖点投影的矩。

θ不变量与赋范$\boldsymbol{\text{Z}}$-模中的格点计数

Theta invariants and Lattice-Point Counting in Normed $\mathbb{Z}$-Modules

本文运用欧几里得格理论研究有限秩赋范$\text{Z}$-模，基于θ级数解析性质建立其格点计数函数的不等式及相关结果，拓展了数论与几何领域的相关理论。

Counting points on $\text{Hilb}^m\mathbb{P}^2$ over function fields

Let $K$ be a global field of positive characteristic. We give an asymptotic formula for the number of $K$-points of bounded height on the Hilbert scheme $\text{Hilb}^2\mathbb{P}^2$ and show that by eliminating an exceptional thin set, the constant in front of the main term agrees with the prediction of Peyre in the function field setting. Moreover, we extend the analogy between the integers and $0$-cycles on a variety $V$ over a finite field to $0$-cycles on a variety $V$ over $K$ and establish a version of the prime number theorem in the case when $V = \mathbb{P}^2$.

在堆叠（stacky）$\operatorname{Sym}^2\mathbb{P}^1$ 上计算有理点

Counting Rational Points on the Stacky $\operatorname{Sym}^2\mathbb{P}^1$

本文针对商堆叠 $\operatorname{Sym}^2\mathbb{P}^1$ 证明广义 Batyrev–Manin–Malle 猜想的弱形式，并确认其对角为累积子堆叠。

Counting subrings of $\mathbb Z^n$ of non-zero co-rank

In this paper we study subrings of $\mathbb Z^{n+k}$ of co-rank $k$.

通过阿廷导体计数Q上代数环面的一个上界

An upper bound for counting algebraic tori over $\mathbb{Q}$ by Artin conductor

该研究给出Q上n维代数环面同构类数的上界，其证明借助ChatGPT 5.6 Pro的迭代对话完成。

通过Artin导数计数有理数上的代数tori

Counting algebraic tori over $\mathbb{Q}$ by Artin conductor

本文通过Artin导数研究有理数上代数tori的计数问题，提出渐进行为猜想并证明其与Malle猜想的关系，同时给出二维tori的上界估计。

理想类群的 5-秩至少为 2 的虚二次域的计数

Counting Imaginary Quadratic Fields with an Ideal Class Group of 5-rank at least 2

本文利用 Howe、Leprévost 和 Poonen 的方法构造亏格 2 曲线，将理想类群 5-秩至少为 2 的虚二次域的计数下界从 $X^{1/4}$ 改进至 $X^{1/3}/(\log X)^2$。

Counting rational points near planar curves

We find an asymptotic formula for the number of rational points near planar curves. More precisely, if $f:\mathbb{R}\rightarrow\mathbb{R}$ is a sufficiently smooth function defined on the interval $[η,ξ]$, then the number of rational points with denominator no larger than $Q$ that lie within a $δ$-neighborhood of the graph of $f$ is shown to be asymptotically equivalent to $(ξ-η)δQ^2$.

关于有界域中格点计数的一个注记

A Note on Counting Lattice Points in Bounded Domains

本文利用$k$元组zeta函数的零点与极点，将正整数编码于成对互素的$k$格中，从而推导出计数素数幂$k$元组的显式公式，并应用于$\bb{R}^2$及$\bb{R}^n$中有界区域的格点计数，特别计算了圆内格点数。

计数有理数域上椭圆曲线的5-同源

Counting 5-isogenies of elliptic curves over $\mathbb{Q}$

本文通过建立模曲线栈与广义费马方程之间的显式同构，证明了有理数域上高度有界的椭圆曲线5-同源数量的渐近公式，从而解决了亏格零模曲线上有理点计数的渐近问题。

Counting multiplicative groups with prescribed subgroups

We examine two counting problems that seem very group-theoretic on the surface but, on closer examination, turn out to concern integers with restrictions on their prime factors. First, given an odd prime $q$ and a finite abelian $q$-group $H$, we consider the set of integers $n\le x$ such that the Sylow $q$-subgroup of the multiplicative group $(\mathbb Z/n\mathbb Z)^\times$ is isomorphic to $H$. We show that the counting function of this set of integers is asymptotic to $K x(\log\log x)^\ell/(\log x)^{1/(q-1)}$ for explicit constants $K$ and $\ell$ depending on $q$ and $H$. Second, we conside…

计数具有预设纠缠的椭圆曲线

Counting elliptic curves with prescribed entanglements

该研究通过数论几何和筛法技术，证明了具有特定分裂场纠缠的椭圆曲线数量随高度增长的渐近下界。

Counting periodic points over finite fields

Let $V$ be a quasiprojective variety defined over $\mathbb{F}_q$, and let $ϕ:V\rightarrow V$ be an endomorphism of $V$ that is also defined over $\mathbb{F}_q$. Let $G$ be a finite subgroup of $\operatorname{Aut}_{\mathbb{F}_q}(V)$ with the property that $ϕ$ commutes with every element of $G$. We show that idempotent relations in the group ring $\mathbb{Q}[G]$ give relations between the periodic point counts for the maps induced by $ϕ$ on the quotients of $V$ by the various subgroups of $G$. We also show that if $G$ is abelian, periodic point counts for the endomorphism on $V/G$ induced by $ϕ$…

计数具有平方自由行列式的整数矩阵

Counting integer matrices with square-free determinants

研究整数矩阵集合中行列式为平方自由数的矩阵数量，并推导出相关渐进行为，同时改进方法计算矩阵行列式与欧拉函数的和。

Poisson distribution of a prime counting function corresponding to elliptic curves

Let $E$ be an elliptic curve defined over rational field $\mathbb{Q}$ and $N$ be a positive integer. Now, $M_E(N)$ denotes the number of primes $p$, such that the group $E_p(\mathbb{F}_p)$ is of order $N$. We show that $M_E(N)$ follows Poisson distribution when an average is taken over a large class of curves.

数值半群中理想个数的计数

Counting Ideals in Numerical Semigroups

本文研究数值半群中按余维数和导子计数的理想个数，计算了嵌入维数为2及特定三类生成元情形的生成函数，并证明导子计数序列最终平稳且给出相应平稳项。

Counting elliptic curves with prescribed level structures over number fields

Harron and Snowden counted the number of elliptic curves over $\mathbb{Q}$ up to height $X$ with torsion group $G$ for each possible torsion group $G$ over $\mathbb{Q}$. In this paper we generalize their result to all number fields and all level structures $G$ such that the corresponding modular curve $X_G$ is a weighted projective line $\mathbb{P}(w_0,w_1)$ and the morphism $X_G\to X(1)$ satisfies a certain condition. In particular, this includes all modular curves $X_1(m,n)$ with coarse moduli space of genus $0$. We prove our results by defining a size function on $\mathbb{P}(w_0,w_1)$ follo…

Counting flags of primitive lattices

We count flags of primitive lattices, which are objects of the form ${0}=Λ^{(0)}<Λ^{(1)}< \cdots <Λ^{(\ell)}= \mathbb{Z}^n$, where every $Λ^{(i)}$ is a primitive lattice in $\mathbb{Z}^n$. The counting is with respect to two different natural height functions, allowing us to give a new proof of the Manin conjecture for flag varieties over rational numbers. We deduce the equidistribution of rational points in flag varieties, as well as the equidistribution of the shapes of the successive quotient lattices, $Λ^{(i)}/Λ^{(i-1)}$. In doing so, we generalize previous work of Schmidt, as well as our …

Counting solutions of special linear equations over finite fields

Let $q$ be a prime power, let $\mathbb F_q$ be the finite field with $q$ elements and let $d_1, \ldots, d_k$ be positive integers. In this note we explore the number of solutions $(z_1, \ldots, z_k)\in\overline{\mathbb F}_q^k$ of the equation \begin{equation*}L_1(x_1)+\cdots+L_k(x_k)=b,\end{equation*} with the restrictions $z_i\in \mathbb F_{q^{d_i}}$, where each $L_i(x)$ is a non zero polynomial of the form $\sum_{j=0}^{m_i}a_{ij}x^{q^j}\in \mathbb F_q[x]$ and $b\in \overline{\mathbb F}_q$. We characterize the elements $b$ for which the equation above has a solution and, in affirmative case, …

计算自由群到 $\mathrm{SL}_2(\mathbb{Z})$ 及其子群中的嵌入数

Counting embeddings of free groups into $\mathrm{SL}_2(\mathbb{Z})$ and its subgroups

本文计数自由群到 $\mathrm{SL}_2(\mathbb{Z})$ 及同余子群 $\Gamma_0(Q)$ 的嵌入，证明随机矩阵以高概率自由生成子群，并修正 Fuchs–Rivin 的相关结论。

Counting non-standard binary representations

Let $\mathcal{A}$ be a finite subset of $\mathbb{N}$ including $0$ and $f_\mathcal{A}(n)$ be the number of ways to write $n=\sum_{i=0}^{\infty}ε_i2^i$, where $ε_i\in\mathcal{A}$. We consider asymptotics of the summatory function $s_\mathcal{A}(r,m)$ of $f_\mathcal{A}(n)$ from $m2^r$ to $m2^{r+1}-1$ and show that $s_{\mathcal{A}}(r,m)\approx c(\mathcal{A},m)\left|\mathcal{A}\right|^r$ for some $c(\mathcal{A},m)\in\mathbb{Q}$.

Counting points on bilinear and trilinear hypersurfaces

Consider an irreducible bilinear form $f(x_1,x_2;y_1,y_2)$ with integer coefficients. We derive an upper bound for the number of integer points $(\mathbf{x},\mathbf{y})\in\mathbb{P}^1\times\mathbb{P}^1$ inside a box satisfying the equation $f=0$. Our bound seems to be the best possible bound and the main term decreases with a larger determinant of the form $f$. We further discuss the case when $f(x_1,x_2;y_1,y_2;z_1,z_2)$ is an irreducible non-singular trilinear form defined on $\mathbb{P}^1\times \mathbb{P}^1\times\mathbb{P}^1$, with integer coefficients. In this case, we examine the singular…

Counting the solutions of $λ_1 x_1^{k_1}+\cdots +λ_t x_t^{k_t}\equiv c\bmod{n}$

Given a polynomial $Q(x_1,\cdots, x_t)=λ_1 x_1^{k_1}+\cdots +λ_t x_t^{k_t}$, for every $c\in \mathbb{Z}$ and $n\geq 2$, we study the number of solutions $N_J(Q;c,n)$ of the congruence equation $Q(x_1,\cdots, x_t)\equiv c\bmod{n}$ in $(\mathbb{Z}/n\mathbb{Z})^t$ such that $x_i\in (\mathbb{Z}/n\mathbb{Z})^\times$ for $i\in J\subseteq I= \{1,\cdots, t\}$. We deduce formulas and an algorithm to study $N_J(Q; c,p^a)$ for $p$ any prime number and $a\geq 1$ any integer. As consequences of our main results, we completely solve: the counting problem of $Q(x_i)=\sum\limits_{i\in I}λ_i x_i$ for any prime…

Counting Finite Index Subrings of $\mathbb{Z}^n$

We count subrings of small index of $\mathbb{Z}^n$, where the addition and multiplication are defined componentwise. Let $f_n(k)$ denote the number of subrings of index $k$. For any $n$, we give a formula for this quantity for all integers $k$ that are not divisible by a 9th power of a prime, extending a result of Liu.

On a counting theorem for weakly admissible lattices

We give a precise estimate for the number of lattice points in certain bounded subsets of $\mathbb{R}^{n}$ that involve `hyperbolic spikes' and occur naturally in multiplicative Diophantine approximation. We use Wilkie's o-minimal structure $\mathbb{R}_{\exp}$ and expansions thereof to formulate our counting result in a general setting. We give two different applications of our counting result. The first one establishes nearly sharp upper bounds for sums of reciprocals of fractional parts, and thereby sheds light on a question raised by Lê and Vaaler, extending previous work of Widmer and of t…

Counting the number of solutions to certain infinite Diophantine equations

Let $r, v, n$ be positive integers. This paper investigate the number of solutions $s_{r,v}(n)$ of the following infinite Diophantine equations $$ n=1^{r}\cdot |k_{1}|^{v}+2^{r}\cdot |k_{2}|^{v}+3^{r}\cdot |k_{3}|^{v}+\ldots, $$ for ${\bf k}=(k_1,k_2,k_3,\dots)\in\mathbb{Z}^{\infty}$. For each $(r,v)\in\mathbb{N}\times\{1,2\}$, a generating function and some asymptotic formulas of $s_{r,v}(n)$ are established.

Counting the ideals of given codimension of the algebra of Laurent polynomials in two variables

We establish an explicit formula for the number $C_n(q)$ of ideals of codimension $n$ of the algebra ${\mathbb F}_q[x,y,x^{-1}, y^{-1}]$ of Laurent polynomials in two variables over a finite field of cardinality $q$. This number is a palindromic polynomial of degree $2n$ in $q$. Moreover, $C_n(q) = (q-1)^2 P_n(q)$, where $P_n(q)$ is another palindromic polynomial; the latter is a $q$-analogue of the sum of divisors of $n$, which happens to be the number of subgroups of ${\mathbb Z}^2$ of index $n$.

On rational singularities and counting points of schemes over finite rings

We study the connection between the singularities of a finite type $\mathbb{Z}$-scheme X and the asymptotic point count of X over various finite rings. In particular, if the generic fiber $X_{\mathbb{Q}}=X\times_{\mathrm{Spec}\mathbb{Z}}\mathrm{Spec}\mathbb{Q}$ is a local complete intersection, we show that the boundedness of $\frac{\left|X(\mathbb{Z}/p^{n}\mathbb{Z})\right|}{p^{n\mathrm{dim}X_{\mathbb{Q}}}}$ in p and n is in fact equivalent to the condition that $X_{\mathbb{Q}}$ is reduced and has rational singularities. This paper completes a result of Aizenbud and Avni.

有限域F_p上的Vieta图计数

Counting in Vieta graphs over $\mathbb{F}_p$

该研究引入有限域F_p上的Vieta图（Markoff图的推广），基于二次特征和系统方法研究其顶点数与度分布等计数问题，处理了三、四变量的低维显式计数情形。

具有指定韦尔奇和的子空间计数

Counting subspaces with prescribed Welch sums

本文针对有限域𝔽_q上的韦尔奇函数，确定了满足指定韦尔奇和的k维𝔽₂子空间数量，其中k=3的结果以二元Kloosterman和表示，进而证明了Hou与Zhao的相关猜想。

Counting Separable Polynomials in $\mathbb{Z}/n[x]$

For a commutative ring $R$, a polynomial $f\in R[x]$ is called separable if $R[x]/f$ is a separable $R$-algebra. We derive formulae for the number of separable polynomials when $R = \mathbb{Z}/n$, extending a result of L. Carlitz. For instance, we show that the number of separable polynomials in $\mathbb{Z}/n[x]$ that are separable is $ϕ(n)n^d\prod_i(1-p_i^{-d})$ where $n = \prod p_i^{k_i}$ is the prime factorisation of $n$ and $ϕ$ is Euler's totient function.

高等秩下的Lax-Phillips轨道计数

Lax-Phillips orbit counting in higher rank

本文提出一种抽象谱方法，用于研究离散晶格轨道中长度受限点的渐进行为，并与其他方法进行比较。

高阶双曲格点计数问题中误差项的逐点改进

Pointwise improvement of the error term in the higher rank hyperbolic lattice counting problem

该研究针对高阶双曲格点计数问题，在Blomer等人成果基础上，证明了SL(n,ℤ)对应误差项的改进上界，还得到符合Rudnick–Phillips风格的Ω-结果。

计数接近流形的有理点：一种改进的估计、一个猜想和一个变体

Counting rational points near manifolds: a refined estimate, a conjecture and a variant

本文改进了接近子流形的有理点计数界，并提出一个猜想，通过复结构重新表述曲率条件，进一步用高斯有理数验证该猜想。

按多变元计数 $D_4$-域扩张

Counting $D_4$-field extensions by multi-invariants

本文按 Gundlach 引入的多变元计数具有固定 Galois 群 $D_4$ 的 Galois 扩张，验证了 Malle 猜想的渐近行为，并将主项常数与 Loughran 和 Santens 的近期预测进行了比较。

Counting invertible sums of squares modulo $n$ and a new generalization of Euler totient function

In this paper we introduce and study a family $Φ_k$ of arithmetic functions generalizing Euler's totient function. These functions are given by the number of solutions to the equation $\gcd(x_1^2+\ldots +x_k^2, n)=1$ with $x_1,\ldots,x_k \in {\mathbb{Z}}/n{\mathbb{Z}}$ which, for $k=2,4$ and $8$ coincide, respectively, with the number of units in the rings of Gaussian integers, quaternions and octonions over ${\mathbb{Z}}/n{\mathbb{Z}}$. We prove that $Φ_k$ is multiplicative for every $k$, we obtain an explicit formula for $Φ_k(n)$ in terms of the prime-power decomposition of $n$ and derive an…

计数具有指定局部条件的$\mathbb{P}^1$上的有理映射

Counting rational maps on $\mathbb{P}^1$ with prescribed local conditions

本文研究了具有指定局部条件的有理映射计数问题，证明了具有最小余数的映射集合具有正密度，并对度数为2的映射进行了显式计算。

On counting cuspidal automorphic representations for $\mathrm{GSp}(4)$

We find the number $s_k(p,Ω)$ of cuspidal automorphic representations of $\mathrm{GSp}(4,\mathbb{A}_{\mathbb{Q}})$ with trivial central character such that the archimedean component is a holomorphic discrete series representation of weight $k\ge 3$, and the non-archimedean component at $p$ is an Iwahori-spherical representation of type $Ω$ and unramified otherwise. Using the automorphic Plancherel density theorem, we show how a limit version of our formula for $s_k(p,Ω)$ generalizes to the vector-valued case and a finite number of ramified places.

Counting Cubic Extensions with given Quadratic Resolvent

Given a number field $k$ and a quadratic extension $K_2$, we give an explicit asymptotic formula for the number of isomorphism classes of cubic extensions of $k$ whose Galois closure contains $K_2$ as quadratic subextension, ordered by the norm of their relative discriminant ideal. The main tool is Kummer theory. We also study in detail the error term of the asymptotics and show that it is $O(X^α)$, for an explicit $α<1$.

一种用于研究$\mathbb{Z}^{d}$上自回避行走数量的格点计数方法

A lattice point counting approach for the study of the number of self-avoiding walks on $\mathbb{Z}^{d}$

该研究通过将正方形格自回避行走计数简化为多维域整点计数，得到其长度为$n$的渐近估计，并为任意$d\geq 2$维$\mathbb{Z}^{d}$格自回避行走数量性质研究提供统一框架。

Counting rational points on quartic del Pezzo surfaces with a rational conic

Upper and lower bounds, of the expected order of magnitude, are obtained for the number of rational points of bounded height on any quartic del Pezzo surface over $\mathbb{Q}$ that contains a conic defined over $\mathbb{Q}$.

计数多项式映射φ_{d,c}的n-周期点数量及其在算术统计中的应用VI

Counting the number of $n$-periodic $\mathbb{Z}_{p}$-and $\mathbb{F}_{p}[t]$-points of a discrete dynamical system with applications from arithmetic statistics, VI

本文研究了多项式映射φ_{d,c}的n-周期点数量，揭示了其与系数c的关系，并在算术统计中应用了密度和等分布结果。

整数与 $p$-进整数的计数和 Hausdorff 测度

Counting and Hausdorff measures for integers and $p$-adic integers

本文利用 $p$-进整数的分形结构研究整数集合，建立计数测度界、计数维数与盒计数维数关系，并给出整数集为闭集投影的充要组合条件。

Counting elliptic curves of bounded Faltings height

We give an asymptotic formula for the number of elliptic curves over $\mathbb{Q}$ with bounded Faltings height. Silverman has shown that the Faltings height for elliptic curves over number fields can be expressed in terms of modular functions and the minimal discriminant of the elliptic curve. We use this to recast the problem as one of counting lattice points in a particular region in $\mathbb{R}^2$.

Divisibility on point counting over finite Witt rings

Let $\mathbb{F}_q$ denote the finite field of $q$ elements with characteristic $p$. Let $\mathbb{Z}_q$ denote the unramified extension of the $p$-adic integers $\mathbb{Z}_p$ with residue field $\mathbb{F}_q$. In this paper, we investigate the $q$-divisibility for the number of solutions of a polynomial system in $n$ variables over the finite Witt ring $\mathbb{Z}_q/p^m\mathbb{Z}_q$, where the $n$ variables of the polynomials are restricted to run through a combinatorial box lifting $\mathbb{F}_q^n$. The introduction of the combinatorial box makes the problem much more complicated. We prove a …

计数多项式动力系统中$\mathbb{Z}_{p}$-和$\mathbb{F}_{p}[t]$-固定点的数量及其在算术统计中的应用，III

Counting the number of $\mathbb{Z}_{p}$-and $\mathbb{F}_{p}[t]$-fixed points of a discrete dynamical system with applications from arithmetic statistics, III

本文研究多项式动力系统中$\mathbb{Z}_{p}$-和$\mathbb{F}_{p}[t]$-固定点的计数问题，并应用算术统计结果获得数域和函数场的统计结果。