p-adic root separation and the discriminant of integer polynomials

2025/04/04 by Beresnevich, Victor, Dixon, Bethany · 1 citation
#Dynamical Systems (math.DS) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2504.03851

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∈ℤ satisfying |C_P|≪1. 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.

Cited by

• Polynomials assuming only local prime powers

Related

• Root separation for irreducible integer polynomials
• On the distribution of polynomial discriminants: totally real case
• Discriminators of quadratic polynomials
• Discriminants of Fields Generated by Polynomials of Given Height
• Integral polynomials with small discriminants and resultants