具有组合和拓扑意义的多元算术函数

多元算术函数与向量微积分

Multivariate arithmetical functions and vector calculus

本文的目的是建立多元算术函数的向量微积分理论。我们证明类似的积分定理和庞加莱引理。

算术多元笛卡尔法则

Arithmetic Multivariate Descartes' Rule

设L是任意数域或$\mathfrak{p}$ -adic域，并考虑F：=（f_1，.，f_k），其中f_i在L[x_1，...，x_n]\{0}，并且恰好有m个不同的指数向量出现在f_1，...，f_k。证明了F在L^n中有不超过1+（cmn（m-1）^2 log m）^n个几何孤立根，其中c是一个仅依赖于L的显式可计算常数.这给出了Khovanski关于Fewnomials的定理的一个明显更清晰的算术模拟，并给出了Hendrik W.小伦斯特拉对于单个一元多项式的情况。我们还提出了一些进一步的改进，我们的新的界限，并简要讨论了寻找孤立的有理根的复杂性。

关于保守矩阵场：连续渐近性与算术

On Conservative Matrix Fields: Continuous Asymptotics and Arithmetic

该研究提出保守矩阵场用于分析和计算D - 有限函数，借助离散Levinson型框架建立周期线性形式族渐近性质，还对相关线性形式族做实验分析并形成关于连续渐近性和算术性质的猜想。

Asymptotics of arithmetic functions of GCD and LCM of random integers in hyperbolic regions

We prove limit theorems for the greatest common divisor and the least common multiple of random integers. While the case of integers uniformly distributed on a hypercube with growing size is classical, we look at the uniform distribution on sublevel sets of multivariate symmetric polynomials, which we call hyperbolic regions. Along the way of deriving our main results, we obtain some asymptotic estimates for the number of integer points in these hyperbolic domains, when their size goes to infinity.

Positive definite functions on semilattices

We introduce a notion of positive definiteness for functions $f\!:P\to\mathbb{R}$ defined on meet semilattices $(P,\preceq,\wedge)$ and prove several properties for these functions. In addition, we utilize the $LDL^{\rm T}$ decomposition of meet matrices in order to explore the properties of multivariate positive definite arithmetic functions $f\!:\mathbb{Z}_+^d\to\mathbb{R}$. Finally, we give a series of examples and counterexamples of positive definite functions.

广义Châtelet簇的局部可解性

Local solubility in generalised Châtelet varieties

该研究利用Rydin Myerson等人的工作得到多元算术函数平均值的渐近公式，将其应用于解析数论与算术几何，改进了多项式系统Hasse原理的变量要求，并计数高维Châtelet簇族中带有理点的纤维数。