Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2024
|
| Schlagworte: | |
| Online-Zugang: | https://arxiv.org/abs/2410.12970 |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866917806012891136 |
|---|---|
| author | Patil, Gaurav Digambar |
| author_facet | Patil, Gaurav Digambar |
| contents | We define a new class of rings parameterized by binary forms of a certain type, and give an effective lower bound for the number of such rings whose discriminant is less than a bound $X$. We also obtain a lower bound for the number of number fields whose ring of integers lies in the above class and whose discriminant is less than a bound $X$. Our results improve an estimate of Bhargava-Shankar-Wang in \cite{bhargava2022squarefree}. In particular we show the following:
$\bullet$ When $n\ge 4,$ the number of rings of rank $n$ over $\mathbb{Z}$ with discriminant less than or equal to $X$ is $$\gg_n X^{\frac{1}{2}+\frac{1}{n-\frac{4}{3}}}.$$
$\bullet$ When $n\ge 6,$ the number of number fields of degree $n$ with discriminant less than $X$ is $$\gg_{n,ε} X^{\frac{1}{2} +\frac{1}{n-1} + \frac{(n-3)r_n}{(n-2)(n-1)}-ε}$$ where $r_n=\frac{η_n}{n^2-4n+3-2η_n (n+\frac{2}{n-2})}$ and where $η_n$ is $\frac{1}{5n}$ if $n$ is odd and is $\frac{1}{88n^6}$ when $n$ is even. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_12970 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Weakly Divisible Rings Patil, Gaurav Digambar Number Theory Rings and Algebras We define a new class of rings parameterized by binary forms of a certain type, and give an effective lower bound for the number of such rings whose discriminant is less than a bound $X$. We also obtain a lower bound for the number of number fields whose ring of integers lies in the above class and whose discriminant is less than a bound $X$. Our results improve an estimate of Bhargava-Shankar-Wang in \cite{bhargava2022squarefree}. In particular we show the following: $\bullet$ When $n\ge 4,$ the number of rings of rank $n$ over $\mathbb{Z}$ with discriminant less than or equal to $X$ is $$\gg_n X^{\frac{1}{2}+\frac{1}{n-\frac{4}{3}}}.$$ $\bullet$ When $n\ge 6,$ the number of number fields of degree $n$ with discriminant less than $X$ is $$\gg_{n,ε} X^{\frac{1}{2} +\frac{1}{n-1} + \frac{(n-3)r_n}{(n-2)(n-1)}-ε}$$ where $r_n=\frac{η_n}{n^2-4n+3-2η_n (n+\frac{2}{n-2})}$ and where $η_n$ is $\frac{1}{5n}$ if $n$ is odd and is $\frac{1}{88n^6}$ when $n$ is even. |
| title | Weakly Divisible Rings |
| topic | Number Theory Rings and Algebras |
| url | https://arxiv.org/abs/2410.12970 |