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Squarefree values of polynomial discriminants II

Manjul Bhargava, Arul Shankar, Xiaoheng Wang

TL;DR

The paper proves that the density of integral binary $n$-ic forms with squarefree discriminant exists when ordered by height and provides explicit Euler-product values across all $n$ (e.g., for $n=2$, $3$, $4$, and $n\u2265 5$). It also determines the density of irreducible forms for which the attached rank-$n$ ring $R_f$ is maximal, delivering a form of arithmetic Bertini for ${f P}^1_{f Z}$. Central to the method are invariant-theoretic parametrizations via representations like $W=2\otimes\text{Sym}_2(n)$, uniform tail bounds (the key geometric-sieve ingredient), and squarefree sieving to pass from discriminant-divisibility tails to squarefree-discriminant densities. The authors develop a novel approach for even degrees, including new invariants ($q$) and lifting techniques to higher-dimensional representations, enabling power-saving estimates and leading to improved lower bounds for the number of $S_n$-number fields of given discriminant, as well as revived, previously retracted results on unramified $A_n$-extensions. These results connect discriminant statistics to maximal orders and to broader questions in arithmetic geometry, including level-of-distribution phenomena and arithmetic Bertini over $f Z$.

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.

Squarefree values of polynomial discriminants II

TL;DR

The paper proves that the density of integral binary -ic forms with squarefree discriminant exists when ordered by height and provides explicit Euler-product values across all (e.g., for , , , and ). It also determines the density of irreducible forms for which the attached rank- ring is maximal, delivering a form of arithmetic Bertini for . Central to the method are invariant-theoretic parametrizations via representations like , uniform tail bounds (the key geometric-sieve ingredient), and squarefree sieving to pass from discriminant-divisibility tails to squarefree-discriminant densities. The authors develop a novel approach for even degrees, including new invariants () and lifting techniques to higher-dimensional representations, enabling power-saving estimates and leading to improved lower bounds for the number of -number fields of given discriminant, as well as revived, previously retracted results on unramified -extensions. These results connect discriminant statistics to maximal orders and to broader questions in arithmetic geometry, including level-of-distribution phenomena and arithmetic Bertini over .

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 . Our methods also allow us to prove that there are number fields of degree~ having associated Galois group~ and absolute discriminant less than , improving the best previously known lower bound of . Finally, our methods correct an error in and thus resurrect earlier (retracted) results of Nakagawa on lower bounds for the number of totally unramified -extensions of quadratic number fields of bounded discriminant.
Paper Structure (23 sections, 59 theorems, 254 equations)

This paper contains 23 sections, 59 theorems, 254 equations.

Table of Contents

  1. Introduction
  2. Outline of proof
  3. Invariant theory on spaces associated to binary $n$-ic forms
  4. Arithmetic invariant theory for the representation $2\otimes{\rm Sym}_2(n)$ of ${\rm SL}_n$
  5. The representation $2\otimes g\otimes (g+1)$ of ${\rm SL}_2\times{\rm SL}_g\times{\rm SL}_{g+1}$ and the $Q$-invariant
  6. Divisibility properties of $\Delta$ when restricted to $W_{0}$
  7. Embedding ${\mathcal{W}}_{m,n}^{\rm {(2)}}$ into $W_n({\mathbb Z})$, for $n$ odd
  8. Embedding ${\mathcal{W}}_{m,n}^{(2),\,{\rm gen}}$ into $W_{n+1}({\mathbb Z})$, for $n$ even
  9. A uniformity estimate for odd degree polynomials
  10. Reduction theory and averaging over fundamental domains
  11. The number of orbits of distinguished elements with large $Q$-invariant
  12. A bound on the number of singular symmetric matrices in skewed boxes
  13. A uniformity estimate for even degree polynomials
  14. Setup and preliminary bounds
  15. Bounding the number of distinguished elements in the main body
  16. ...and 8 more sections

Key Result

Theorem 1

Let $n\geq 2$ be an integer. When integral binary $n$-ic forms $f(x,y)=a_0x^n+a_1x^{n-1}y+\cdots+a_n y^n$ are ordered by $H(f):={\rm max}\{|a_0|,\ldots,|a_n|\}.$, the density of forms having squarefree discriminant exists and is equal to

Theorems & Definitions (60)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Theorem 6
  • Proposition 3.1
  • Proposition 3.2
  • Lemma 3.3
  • Proposition 3.4
  • ...and 50 more