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Weakly Divisible Rings

Gaurav Digambar Patil

Abstract

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.

Weakly Divisible Rings

Abstract

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 . 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 . Our results improve an estimate of Bhargava-Shankar-Wang in \cite{bhargava2022squarefree}. In particular we show the following: When the number of rings of rank over with discriminant less than or equal to is When the number of number fields of degree with discriminant less than is where and where is if is odd and is when is even.

Paper Structure

This paper contains 9 sections, 16 theorems, 110 equations.

Table of Contents

  1. Introduction
  2. Main theorems
  3. Defining Weakly Divisible Rings
  4. Effective injectivity of the map: $(f,m,l)\longrightarrow R'_{(f_l,m)}$
  5. Fundamentals of Minkowski Reduction
  6. Reduced-m-Polynomials
  7. Rings and Orders when ordered by discriminant
  8. A lower bound for the number of number fields with bounded discriminant
  9. Matrix of transformation of canonical basis:$R_f\longrightarrow R_{f_l}$

Key Result

Theorem 1

When $n\ge 4,$ the number of isomorphism classes of integral domains of rank $n$ over $\mathbb{Z},$ with discriminant less than or equal to $X$ is

Theorems & Definitions (58)

  • Definition 1
  • Theorem 1: Ring Count
  • Theorem 2: Number-field count
  • Definition 2
  • Remark 1
  • Theorem 3
  • Definition 3
  • Definition 4
  • Proposition 1
  • Remark 2
  • ...and 48 more