[Paper Review] Squarefree values of polynomial discriminants II

Manjul Bhargava, Arul Shankar+1 more|arXiv (Cornell University)|Jul 12, 2022

Analytic Number Theory Research4 citations

TL;DR

This paper establishes the first positive lower density for integral binary $n$-ic forms with squarefree discriminant, proving the density exists and computing it explicitly for all $n \geq 2$. It further determines the density of forms whose associated rings are maximal orders, proving an arithmetic Bertini theorem for $\mathbb{P}^1_{\mathbb{Z}}$, and improves lower bounds on the number of $S_n$-number fields of bounded discriminant.

ABSTRACT

We determine the density of integral binary forms of given degree that have squarefree discriminant, proving for the first time that the lower density is positive. Furthermore, we determine the density of integral binary forms that cut out maximal orders in number fields. The latter proves, in particular, an ``arithmetic Bertini theorem'' conjectured by Poonen for $\mathbb{P}^1_\mathbb{Z}$. Our methods also allow us to prove that there are $\gg X^{1/2+1/(n-1)}$ number fields of degree~$n$ having associated Galois group~$S_n$ and absolute discriminant less than $X$, improving the best previously known lower bound of $\gg X^{1/2+1/n}$. Finally, our methods correct an error in and thus resurrect earlier (retracted) results of Nakagawa on lower bounds for the number of totally unramified $A_n$-extensions of quadratic number fields of bounded discriminant.

Motivation & Objective

• To determine the natural density of integral binary $n$-ic forms with squarefree discriminant when ordered by height.
• To compute the density of irreducible binary $n$-ic forms for which the associated ring is the maximal order in its field of fractions.
• To establish an unconditional arithmetic Bertini theorem for $\mathbb{P}^1_{\mathbb{Z}}$, confirming a conjecture of Poonen.
• To improve the lower bound on the number of degree-$n$ number fields with Galois group $S_n$ and bounded absolute discriminant.
• To correct and resurrect earlier results of Nakagawa on $A_n$-extensions of quadratic fields by fixing an error in his analysis.

Proposed method

• Uses $p$-adic and local density methods to compute the proportion of binary $n$-ic forms that are maximal at each prime $p$, leveraging Hensel’s lemma and lifting of factorizations modulo $p$.
• Applies sieve-theoretic techniques to control the global density of forms with squarefree discriminant by combining local densities at all primes.
• Introduces a refined counting method for forms where the associated ring $R_f$ is maximal, based on analyzing the reduction type modulo $p$ and the behavior of the discriminant under $p$-adic lifting.
• Computes the local density $\beta_n(p)$ of forms maximal at $p$ by decomposing the space of forms according to the $p$-adic valuation of coefficients and the factorization type modulo $p$, using results from $p$-adic analysis.
• Employs the product formula over primes to derive the global density of squarefree discriminants and maximal orders, using Euler products and zeta function identities.
• Corrects an error in Nakagawa’s earlier work by refining the local density computation for $A_n$-extensions, restoring validity to his results on unramified $A_n$-extensions of quadratic fields.

Experimental results

Research questions

• RQ1 What is the natural density of integral binary $n$-ic forms with squarefree discriminant when ordered by height, for $n \geq 2$?
• RQ2 What is the density of irreducible integral binary $n$-ic forms for which the associated ring $R_f$ is the maximal order in its field of fractions?
• RQ3 Does an arithmetic Bertini theorem hold for $\mathbb{P}^1_{\mathbb{Z}}$, i.e., is the density of hyperplane sections that are regular equal to $\zeta(2)^{-1}\zeta(3)^{-1}$?
• RQ4 What is the best known lower bound for the number of degree-$n$ number fields with Galois group $S_n$ and absolute discriminant less than $X$?
• RQ5 Can the erroneous results of Nakagawa on unramified $A_n$-extensions of quadratic fields be corrected and resurrected?

Key findings

• The density of integral binary $n$-ic forms with squarefree discriminant exists and is explicitly computed as an Euler product over primes, with values approximately 38.97% for $n=2$, 24.64% for $n=3$, 21.18% for $n=4$, and 20.83% for $n \geq 5$.
• The density of irreducible binary $n$-ic forms for which the associated ring $R_f$ is the maximal order in its field of fractions is $\zeta(2)^{-1}\zeta(3)^{-1} \approx 50.57\%$ for $n \geq 3$, and $\approx 53.59\%$ for $n=2$, proving an arithmetic Bertini theorem for $\mathbb{P}^1_{\mathbb{Z}}$.
• The number of $S_n$-number fields of degree $n$ and absolute discriminant less than $X$ is $\gg X^{1/2 + 1/(n-1)}$, improving the prior lower bound of $\gg X^{1/2 + 1/n}$.
• The paper corrects an error in Nakagawa’s retracted work on $A_n$-extensions of quadratic fields, restoring the validity of Theorems 3–4 in [19] and Theorem 2 in [21] for unramified $A_n$-extensions.
• The local density $\beta_n(p)$ of forms maximal at $p$ is computed as $\left(1 - \frac{1}{p^2}\right)\left(1 - \frac{1}{p^3}\right)$ for $n \geq 3$, and $\left(1 - \frac{1}{p}\right)\left(1 + \frac{1}{p} - \frac{1}{p^3}\right)$ for $n=2$, which underlies the global density computations.
• All number fields constructed in the main lower bound result can be taken to have squarefree discriminant, linking the results on squarefree discriminants and maximal orders.