多项式的整特殊化：希尔伯特不可约性、互素性与密度
Integral Specializations of Polynomials: Hilbert Irreducibility, Coprimality, and Density
AI总结：
本文重新证明并强化了整体域整数环$\boldsymbol{\textit{O}}_K$的整希尔伯特性质，证明其整希尔伯特子集具正密度，推广了Ekedahl–Poonen公式并得到算术级数有界高度点估计。
摘要：
域的希尔伯特特殊化性质指，在部分变量进行特殊化时多项式的不可约性得以保留，其存在一个整版本，即特殊化在环内进行且要求多项式在该环上不可约。已知任意整体域K的整数环$\boldsymbol{\textit{O}}_K$满足该整希尔伯特性质。本文采用不同方法重新证明该结果并加以强化，证明$\boldsymbol{\textit{O}}_K^k$对应的整希尔伯特子集不仅满足整希尔伯特性要求的扎里斯基稠密性，还具有正密度。核心要素是将最初为$\boldsymbol{\textit{Z}}$和$\boldsymbol{\textit{F}}_q[t]$证明的Ekedahl–Poonen公式推广到环$\boldsymbol{\textit{O}}_K$，该公式具有独立研究价值，用于计算$\boldsymbol{\textit{O}}_K^k$中满足系数在$\boldsymbol{\textit{O}}_K$的k元互素多项式取互素值的元组密度。该推广需要算术级数中有界高度点的数量估计，本文将其作为第三个主要结果得到此类估计，从而为高度理论作出贡献。
英文摘要：
The Hilbert specialization property of a field, whereby irreducibility of polynomials is preserved under specialization of some of the variables, admits an integral counterpart where specialization takes place inside a ring and irreducibility is required over the ring. It is known that the ring $\mathcal{O}_K$ of integers of any global field $K$ satisfies this integrally Hilbertian property. Using a different approach, we reprove this result and strengthen it by showing that the corresponding integral Hilbert subsets of $\mathcal{O}_K^k$ are not only Zariski-dense, as required by integral hilbertianity, but also of positive density. A central ingredient is a generalization to the ring $\mathcal{O}_K$ of the Ekedahl--Poonen formula originally proved for $\mathbb{Z}$ and $\mathbb{F}_q[t]$. This formula, of independent interest, gives the density of tuples in $\mathcal{O}_K^k$ for which coprime polynomials in $k$ variables and with coefficients in ${\mathcal O}_K$ take coprime values. Our generalization requires estimates for the number of points of bounded height in an arithmetic progression. We obtain such estimates as our third main result, thereby contributing to the theory of heights.
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