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Density of Special Classes of Polynomials with Squarefree Discriminant

Gian Cordana Sanjaya

TL;DR

This work develops a Fourier-analytic framework to compute local densities of monic polynomials over $\mathbb{Z}_p$ with squarefree discriminant and with the corresponding maximal orders, under fixed-coefficient restrictions. By encoding coefficient constraints via $\Sigma$-admissible triples $(G,\psi,w)$ and leveraging $p$-adic Haar measure alongside generating series, the authors derive explicit formulas for $P_0^{\mathrm{sqf}}(\Sigma)$, $P_1^{\mathrm{sqf}}(\Sigma)$, and $P^{\max}(\Sigma)$ for broad families of subsets $\Sigma$ defined by congruence conditions; they further adapt these formulas to subsets with unit or fixed nonleading coefficients. The paper provides detailed local-density results for several natural subcases: fixing a single coefficient $a_1=b_1$, fixing two coefficients $a_1=b_1,a_2=b_2$, and variants where the constant term is a unit or fixed, including coprime-to-$p$ cases. The approach yields not only asymptotic densities as $n\to\infty$ but also exact densities, expressed through finite-group character sums and Gauss sums, enriching the understanding of local contributions to global densities in discriminant problems. The results unify and extend previous global and local density computations (e.g., Yamamura, Lenstra, ABZ) and provide a robust toolkit for analyzing densities under partial coefficient restrictions, with potential applications to arithmetic statistics and related counting problems.

Abstract

In this paper, we consider the problem of determining the density of monic polynomials over $\mathbb{Z}_p$ with squarefree discriminant over various subsets of the set of monic polynomials over $\mathbb{Z}_p$ of fixed degree. We compute the density of polynomials in each subset whose discriminant is squarefree, and we compute the density of polynomials $f$ in each subset such that $\mathbb{Z}_p[x]/(f(x))$ is the maximal order of $\mathbb{Q}_p[x]/(f(x))$.

Density of Special Classes of Polynomials with Squarefree Discriminant

TL;DR

This work develops a Fourier-analytic framework to compute local densities of monic polynomials over with squarefree discriminant and with the corresponding maximal orders, under fixed-coefficient restrictions. By encoding coefficient constraints via -admissible triples and leveraging -adic Haar measure alongside generating series, the authors derive explicit formulas for , , and for broad families of subsets defined by congruence conditions; they further adapt these formulas to subsets with unit or fixed nonleading coefficients. The paper provides detailed local-density results for several natural subcases: fixing a single coefficient , fixing two coefficients , and variants where the constant term is a unit or fixed, including coprime-to- cases. The approach yields not only asymptotic densities as but also exact densities, expressed through finite-group character sums and Gauss sums, enriching the understanding of local contributions to global densities in discriminant problems. The results unify and extend previous global and local density computations (e.g., Yamamura, Lenstra, ABZ) and provide a robust toolkit for analyzing densities under partial coefficient restrictions, with potential applications to arithmetic statistics and related counting problems.

Abstract

In this paper, we consider the problem of determining the density of monic polynomials over with squarefree discriminant over various subsets of the set of monic polynomials over of fixed degree. We compute the density of polynomials in each subset whose discriminant is squarefree, and we compute the density of polynomials in each subset such that is the maximal order of .
Paper Structure (13 sections, 43 theorems, 168 equations)

This paper contains 13 sections, 43 theorems, 168 equations.

Table of Contents

  1. Introduction
  2. Common notations
  3. General density formula
  4. Proof of Theorem \ref{['thm-density:sqf-0']} and Theorem \ref{['thm-density:sqf-1']}
  5. Proof of Theorem \ref{['thm-density:max']}
  6. Proof of Theorem \ref{['thm-local:sub1']}
  7. Proof of Theorem \ref{['thm-local:sub2']}
  8. Proof of Theorem \ref{['thm-sub2:step1']}
  9. Proof of Theorem \ref{['thm-sub2:step2']}
  10. Density formula for polynomials with unit constant coefficient
  11. Proof of Theorem \ref{['thm-local:copconst']}
  12. Proof of Theorem \ref{['thm-local:fixconst']}
  13. Proof of Theorem \ref{['thm-local:sub1-copconst']}

Key Result

Theorem 1.1

Let $n \geq 2$ and $p$ be a prime number. Fix $b_1 \in \mathbb{Z}_p$, and denote If $p = 2$, then If $p$ is odd, then where the implied constants are absolute. If $p$ is odd, then $P^{\mathop{\mathrm{sqf}}\nolimits}(V_n(\mathbb{Z}_p)_{a_1 = b_1})$ is positive, independent of $b_1$ if $p \nmid n$, and depends strictly on whether $b_1$ is a unit or not a unit if $p \mid n$.

Theorems & Definitions (64)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Theorem 1.9
  • Definition 2.1
  • ...and 54 more