论文标题
关于$ \ mathbb {z}^n $ prime Power索引的子环数
On the number of subrings of $\mathbb{Z}^n$ of prime power index
论文作者
论文摘要
令$ n $和$ k $为正整数，$ f_n（k）$（resp。$ g_n（k）$）是$ \ mathbb {z}^n $ k $ k $ $ \ mathbb {z}^n $ $ \ mathbb {z}^n $的Unital子圈数量（resp。Unititalnorital不可约合的子环）。 $ f_n（k）$的数字是某些自然兴趣函数的系数。函数$ k \ mapsto f_n（k）$是乘法的，对数字$ f_n（k）$的研究减少了计算Prime Powers $ k = p^e $的值。给定$ e $ $ e $的成分$α=（α_1，\ dots，α_{n-1}）$ $ e $在$ n-1 $阳性整数中，令$g_α（p）$表示$ \ mathbb {z}^n $的不可减至的子圈的数量\ dots，p^{α_{n-1}}，1）$。通过组合分析，$ f_n（p^e）$的计算减少到$g_α（p）$的计算中，用于$ i $的所有组合成$ j $ parts，其中$ i \ i \ leq e $和$ j \ j \ leq n-1 $。我们扩展了Liu和Atanasov-Kaplan-Krakoff-Menzel的结果，他们明确计算$ f_n（p^e）$，以$ e \ e \ leq 8 $。事实证明，$ e = 9 $参与其中。我们根据n和p的多项式对$ f_n（e^9）$进行评估，直到单个术语，这是一个猜想的多项式。我们的结果提供了猜想的进一步证据，该猜想指出，对于任何固定对$（n，e）$，功能$ p \ mapsto f_n（p^e）$是$ p $中的多项式。 Bhargava在$ f_n（k）$的渐近学上的猜想是$ k $的函数，激发了某些无限构图$ a $g_α$ $g_α$ $g_α$的渐近性研究，为此，我们可以使用数量的几何来从数量的几何学中获得一般的估计。
Let $n$ and $k$ be positive integers, and $f_n(k)$ (resp. $g_n(k)$) be the number of unital subrings (resp. unital irreducible subrings) of $\mathbb{Z}^n$ of index $k$. The numbers $f_n(k)$ are coefficients of certain zeta functions of natural interest. The function $k\mapsto f_n(k)$ is multiplicative, and the study of the numbers $f_n(k)$ reduces to computing the values at prime powers $k=p^e$. Given a composition $α=(α_1, \dots, α_{n-1})$ of $e$ into $n-1$ positive integers, let $g_α(p)$ denote the number of irreducible subrings of $\mathbb{Z}^n$ for which the associated upper triangular matrix in Hermite normal form has diagonal $(p^{α_1}, \dots, p^{α_{n-1}},1)$. Via combinatorial analysis, the computation of $f_n(p^e)$ reduces to the computation of $g_α(p)$ for all compositions of $i$ into $j$ parts, where $i\leq e$ and $j\leq n-1$. We extend results of Liu and Atanasov-Kaplan-Krakoff-Menzel, who explicitly compute $f_n(p^e)$ for $e\leq 8$. The case $e=9$ proves to be significantly more involved. We evaluate $f_n(e^9)$ explicitly in terms of a polynomial in n and p up to a single term which is conjecturally a polynomial. Our results provide further evidence for a conjecture, which states that for any fixed pair $(n,e)$, the function $p\mapsto f_n(p^e)$ is a polynomial in $p$. A conjecture of Bhargava on the asymptotics for $f_n(k)$ as a function of $k$ motivates the study of the asymptotics for $g_α(p)$ for certain infinite families of compositions $α$, for which we are able to obtain general estimates using techniques from the geometry of numbers.
