p进整数环上的整二次型
Integral quadratic forms over a ring of $p$-adic integers
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中文总结 AI 辅助
本文研究p进整数环上整二次型的泛性，推广已有关于整数环上二次型泛性的结论，给出M₂(ℤₚ)上二次型泛性的充要条件，并确定n≥3时相关最小m值f(n)的界。
中文摘要 AI 辅助
2018年，Jungin Lee证明了整二次型∑(i=1到m)a_iX_i^2在M₂(ℤ)上是泛型的充要条件。对于正整数n≥2，Lee定义f(n)为最小的正整数m，使得对每一对互素的a₁,a₂,…,a_m∈ℤ，∑(i=1到m)a_iX_i^2在Mₙ(ℤ)上是泛型的，他还给出了f(n)的界。Koo和Lee在2025年进一步改进了这些界。本文结构如下：设ℤₚ为p进整数环，第一节给出二次型∑(i=1到m)a_iX_i^2在p=2及奇素数p时在M₂(ℤₚ)上是泛型的充要条件，由此将ℤₚ上的矩阵表示为平方和；对于正整数n≥3，定义f(n)为最小的正整数m，使得对每一个至少有三个为单位的p进整数a₁,a₂,…,a_m，对角二次型∑(i=1到m)a_iX_i^2在Mₙ(ℤₚ)上是泛型的，第二节给出n≥3时f(n)的界。
英文摘要
Jungin Lee in 2018 proved a necessary and sufficient condition that an integral quadratic form $\sum_{i=1}^{m} a_iX_i^2$ is universal over $M_2(\mathbb{Z})$. For a positive integer $n \geq 2$, Lee defined $f(n)$ to be the smallest positive integer $m$ such that for every pairwise coprime $a_1, a_2, \ldots a_m \in \mathbb{Z}$, $\sum_{i=1}^{m}a_iX_i^2$ is universal over $M_n(\mathbb{Z})$. He gave bounds on $f(n)$ too. Koo and Lee further improved the bounds in 2025. This paper is organized as follows. Let $\mathbb{Z}_p$ be the ring of $p$-adic integers. In the first section we give necessary and sufficient condition for a quadratic form $\sum_{i=1}^{m}a_iX_i^2$ to be universal over $M_2(\mathbb{Z}_p)$ for $p=2$ and for an odd prime $p$. Consequently we express matrices of over $\mathbb{Z}_p$ as sum of squares. For a positive integer $n \geq 3$, we define $f(n)$ to be the smallest positive integer $m$ such that for every $p$-adic integers $a_1,a_2, \ldots a_m$ with at least three of them units, the diagonal quadratic form $\sum_{i=1}^{m}a_iX_i^2$ is universal over $M_n(\mathbb{Z}_p)$. In the next section we find bounds on $f(n)$ for $n \geq 3$.
