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Squarefree discriminants of polynomials with prime coefficients

Valentio Iverson, Gian Cordana Sanjaya, Xiaoheng Wang

TL;DR

This work analyzes counts of polynomials with prime-coefficient inputs for which the discriminant is squarefree and for which the associated order is maximal. It combines a Browning-style inclusion-exclusion sieve with a detailed local-density analysis, yielding asymptotic formulas whose main terms factor into products of local densities $P_{n,p}^{\mathrm{sqf}}$ and $P_{n,p}^{\max}$, and providing explicit constants and positivity results. A central technical contribution is the quantitative equidistribution of nonzero coefficient residues modulo primes, established via a doubly stochastic matrix framework and the discrepancy function $\delta_{n,p}(d)$, which feeds into precise bounds and limits for the local densities. The paper also offers specialized results for the prime $p=2$ and proves a general polydisc density theorem that unifies the squarefree and maximality densities across families of polynomials with prime inputs, with implications for broader sieve problems in polynomial settings.

Abstract

In this paper, we consider the family of monic polynomials with prime coefficients and the family of all polynomials with prime coefficients. We determine the number of $f(x)$ in each of these families having: squarefree discriminant; $\mathbb{Z}[x]/(f(x))$ as the maximal order in $\mathbb{Q}[x]/(f(x))$.

Squarefree discriminants of polynomials with prime coefficients

TL;DR

This work analyzes counts of polynomials with prime-coefficient inputs for which the discriminant is squarefree and for which the associated order is maximal. It combines a Browning-style inclusion-exclusion sieve with a detailed local-density analysis, yielding asymptotic formulas whose main terms factor into products of local densities and , and providing explicit constants and positivity results. A central technical contribution is the quantitative equidistribution of nonzero coefficient residues modulo primes, established via a doubly stochastic matrix framework and the discrepancy function , which feeds into precise bounds and limits for the local densities. The paper also offers specialized results for the prime and proves a general polydisc density theorem that unifies the squarefree and maximality densities across families of polynomials with prime inputs, with implications for broader sieve problems in polynomial settings.

Abstract

In this paper, we consider the family of monic polynomials with prime coefficients and the family of all polynomials with prime coefficients. We determine the number of in each of these families having: squarefree discriminant; as the maximal order in .
Paper Structure (10 sections, 19 theorems, 135 equations)

This paper contains 10 sections, 19 theorems, 135 equations.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['thm:primesieve']}
  3. Density calculation
  4. Proof of Theorem \ref{['thm:Psqf']}
  5. Proof of Theorem \ref{['thm:Pmax']}
  6. Equidistribution of polynomials with nonzero coefficients
  7. Doubly stochastic matrices
  8. Proof of Theorem \ref{['thm3:matrix-p-large-bound']} and Theorem \ref{['thm3:matrix-p-small-bound']}
  9. Proof of Theorem \ref{['thm:polydisc']} for squarefree density
  10. Proof of Theorem \ref{['thm:polydisc']} for maximality density

Key Result

Theorem 1.1

For each $n \geq 2$ and real number $X>0$, let Then for any real number $A>0$, we have where for some local densities $P_{n,p}^{\mathop{\mathrm{sqf}}\nolimits},P_{n,p}^{\max}$ satisfying for $p>2$ and $n\geq 16$, with where $t$ is the odd part of $n + 1$ and where $o_d(2)$ for any odd positive integer $d$ denotes the order of $2$ in $(\mathbb{Z}/d\mathbb{Z})^\times$. Moreover, $C_{n, 2}^{\ma

Theorems & Definitions (31)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 3.1
  • Theorem 3.2
  • Lemma 3.3
  • Lemma 3.4
  • proof
  • Lemma 3.5: Dedekind's criterion
  • Lemma 3.6
  • ...and 21 more