PreprintCC BY 4.0arXiv
Galois groups of random polynomials of large degree
Abstract
We study random polynomials of the form R(x)=xn+ωn−1xn−1+⋯+ω0, where ω0,…,ωn−1 are independent, uniformly bounded integer-valued random variables, and ω1,…,ωn−1 have a fixed common law μ. We prove (unconditionally) that, if the Rényi entropy of order $2$ satisfies H2(μ)=−log|μ|22>12, then ℙ(disc(R) is a square)=Oμ(1/log n). Combined with previous results, this shows that, for such measures μ and under additional hypotheses, R has full Galois group Sym(n) with high probability, when conditioned on ω0≠ 0.
Related
- Galois Groups of Shifted Exponential Polynomials
- Galois groups of twisted reciprocal polynomials: a uniform asymptotic
- Galois groups of integer polynomials with prescribed coefficients
- Galois groups of polynomials over a fixed number field: determinant methods in general boxes, resolvents, and quantitative counting
- Eigenvalue and Eigenvector Approximation for Random Matrices Using Low-Degree Polynomials
BibTeX
@misc{blachar2026galois,
title = {{Galois groups of random polynomials of large degree}},
author = {Guy Blachar and Emmanuel Breuillard and Gady Kozma},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.40123},
eprint = {2609.40123},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.40123},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Galois groups of random polynomials of large degree
AU - Guy Blachar
AU - Emmanuel Breuillard
AU - Gady Kozma
PY - 2026
DA - 2026/09/30
PB - arXiv
DO - 10.48550/arxiv.2609.40123
UR - https://arxiv.org/abs/2609.40123
AB - We study random polynomials of the form R(x)=x^n+ω_n−1x^n−1+⋯+ω_0, where ω_0,…,ω_n−1 are independent, uniformly bounded integer-valued random variables, and ω_1,…,ω_n−1 have a fixed common law μ. We prove (unconditionally) that, if the Rényi entropy of order $2$ satisfies H_2(μ)=−log|μ|_2^2>12, then ℙ(disc(R) is a square)=O_μ(1/log n). Combined with previous results, this shows that, for such measures μ and under additional hypotheses, R has full Galois group Sym(n) with high probability, when conditioned on ω_0≠ 0.
ER -