TY - JOUR

T1 - Irreducibility of random polynomials

T2 - general measures

AU - Bary-Soroker, Lior

AU - Koukoulopoulos, Dimitris

AU - Kozma, Gady

N1 - Publisher Copyright:
© 2023, The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature.

PY - 2023/9

Y1 - 2023/9

N2 - Let μ 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) ∈ Z[x] of degree n are chosen independently at random according to μ while ensuring that f(0) ≠ 0 , then there is a positive constant θ= θ(μ) such that f(x) has no divisors of degree ≤ θn with probability that tends to 1 as n→ ∞ . Furthermore, in certain cases, we show that a random polynomial f(x) with f(0) ≠ 0 is irreducible with probability tending to 1 as n→ ∞ . In particular, this is the case if μ is the uniform measure on a set of at least 35 consecutive integers, or on a subset of [− H, H] ∩ Z of cardinality ≥ H4 / 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 An or Sn with high probability. Finally, when μ 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 ≥ δ for some constant δ= δ(μ) > 0 . In fact, if the arithmetic progression has step 1, we prove the stronger result that the Galois group of f(x) is An or Sn with probability ≥ δ .

AB - Let μ 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) ∈ Z[x] of degree n are chosen independently at random according to μ while ensuring that f(0) ≠ 0 , then there is a positive constant θ= θ(μ) such that f(x) has no divisors of degree ≤ θn with probability that tends to 1 as n→ ∞ . Furthermore, in certain cases, we show that a random polynomial f(x) with f(0) ≠ 0 is irreducible with probability tending to 1 as n→ ∞ . In particular, this is the case if μ is the uniform measure on a set of at least 35 consecutive integers, or on a subset of [− H, H] ∩ Z of cardinality ≥ H4 / 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 An or Sn with high probability. Finally, when μ 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 ≥ δ for some constant δ= δ(μ) > 0 . In fact, if the arithmetic progression has step 1, we prove the stronger result that the Galois group of f(x) is An or Sn with probability ≥ δ .

UR - http://www.scopus.com/inward/record.url?scp=85160815163&partnerID=8YFLogxK

U2 - 10.1007/s00222-023-01193-6

DO - 10.1007/s00222-023-01193-6

M3 - ???researchoutput.researchoutputtypes.contributiontojournal.article???

AN - SCOPUS:85160815163

SN - 0020-9910

VL - 233

SP - 1041

EP - 1120

JO - Inventiones Mathematicae

JF - Inventiones Mathematicae

IS - 3

ER -