Irreducibility of random polynomials of ℤ[X]

2024/01/30 by Pierre-Alexandre Bazin, Bazin, Pierre-Alexandre · 1 citation
Mathematics · #Analytic Number Theory Research #Mathematical Dynamics and Fractals #Advanced Algebra and Geometry

paper · pdf · doi:10.48550/arxiv.2402.06643

Abstract

In a recent paper, Bary-Soroker, Koukoulopoulos and Kozma proved that when A is a random monic polynomial of ℤ[X] of deterministic degree n with coefficients a_j drawn independently according to measures μ_j, then A is irreducible with probability tending to 1 as n→∞ under a condition of near-uniformity of the μ_j modulo four primes (which notably happens when the μ_j are uniform over a segment of \mathbb Z of length N≥ 35.) We improve here the speed of convergence when we have the near-uniformity condition modulo more primes. Notably, we obtain the optimal bound ℙ(A irreducible) = 1 -O(1/√ n) when we have near-uniformity modulo twelve primes, which notably happens when the μ_j are uniform over a segment of length N≥ 10⁸.

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