Irreducibility of random polynomials: general measures

• 成果类型: Article
• 署名作者: Bary-Soroker, Lior; Koukoulopoulos, Dimitris; Kozma, Gady
• 署名单位: Tel Aviv University; Universite de Montreal; Weizmann Institute of Science
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-023-01193-6
• 发表日期: 2023
• 页码: 1041-1120
• 关键词: integers; divisor; primes; number

摘要

Let mu be a probability measure on Z that is not a Dirac mass and that has finite support. We prove that if the coefficients of a monic polynomial f (x) is an element of Z[x] of degree n are chosen independently at random according to mu while ensuring that f (0) not equal 0, then there is a positive constant theta = theta (mu) such that f (x) has no divisors of degree <= theta n with probability that tends to 1 as n -> infinity. Furthermore, in certain cases, we show that a random polynomial f (x) with f (0) not equal 0 is irreducible with probability tending to 1 as n -> infinity. In particular, this is the case if mu is the uniform measure on a set of at least 35 consecutive integers, or on a subset of [- H, H] boolean AND Z of cardinality >= H-4/5(log H)(2) with H sufficiently large. In addition, in all of these settings, we show that the Galois group of f (x) is either A(n) or S-n with high probability. Finally, when mu is the uniform measure on a finite arithmetic progression of at least two elements, we prove a random polynomial f (x) as above is irreducible with probability >= delta for some constant delta = delta (mu)> 0. In fact, if the arithmetic progression has step 1, we prove the stronger result that the Galois group of f (x) is A(n) or S-n with probability >= delta.