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p-adic root separation and the discriminant of integer polynomials

Victor Beresnevich, Bethany Dixon

Abstract

In this paper we investigate the following related problems: (A) the separation of $p$-adic roots of integer polynomials of a fixed degree and bounded height; and (B) counting integer polynomials of a fixed degree and bounded height with discriminant divisible by a (large) power of a fixed prime. One of the consequences of our findings is the existence, for all large $Q>1$, of $Q^{2/n}$ integer irreducible polynomials $P$ of degree $n$ and height $\asymp Q$ with an almost prime power discriminant of maximal size, that is $|D(P)|\asymp Q^{2n-2}$ and $D(P)=p^kC_P$ with $C_P\in\mathbb{Z}$ satisfying $|C_P|\ll1$. The method we use generalises the techniques used in the study of the real case [Beresnevich, Bernik and Götze, 2010 and 2016] and relies on a quantitative non-divergence estimate developed by Kleinbock and Tomanov.

p-adic root separation and the discriminant of integer polynomials

Abstract

In this paper we investigate the following related problems: (A) the separation of -adic roots of integer polynomials of a fixed degree and bounded height; and (B) counting integer polynomials of a fixed degree and bounded height with discriminant divisible by a (large) power of a fixed prime. One of the consequences of our findings is the existence, for all large , of integer irreducible polynomials of degree and height with an almost prime power discriminant of maximal size, that is and with satisfying . The method we use generalises the techniques used in the study of the real case [Beresnevich, Bernik and Götze, 2010 and 2016] and relies on a quantitative non-divergence estimate developed by Kleinbock and Tomanov.

Paper Structure

This paper contains 16 sections, 25 theorems, 164 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Root separation: past and new results
  3. The quantitative theory
  4. Counting discriminants: past and new results
  5. Previous results
  6. New results
  7. Further remarks
  8. Key Lemma on polynomials
  9. Outlining the approach
  10. A quantitative non-divergence estimate
  11. A result of Kleinbock and Tomanov
  12. Verifying conditions (1) and (2)
  13. Proof of the Key Lemma
  14. Finding close roots
  15. Root separation: proof of Theorem \ref{['Thm:Main2']}
  16. ...and 1 more sections

Key Result

Theorem 2.1

For any $n\ge 2$ and any prime $p$, we have that

Figures (1)

  • Figure 1: (a) $j_0$ does not exist, (b) $j_0=0$, (c) $j_0>0$, $\Pi_0 \ge \Pi_{n-1}$, (d) $j_0>0$, $\Pi_0 \le \Pi_{n-1}$

Theorems & Definitions (38)

  • Theorem 2.1
  • Theorem 2.2
  • Corollary 2.3
  • proof
  • Corollary 2.4
  • proof
  • proof : Proof of Theorem \ref{['thm:short']}
  • Theorem 3.1
  • Corollary 3.2: Almost prime power discriminants
  • Corollary 3.3: The cubic case
  • ...and 28 more