The characteristic polynomial of a random matrix

Abstract

Form an $n\times n$ matrix by drawing entries independently from $\{\pm 1\}$ (or another fixed nontrivial finitely supported distribution in $\mathbf{Z}$) and let $\varphi$ be the characteristic polynomial. Conditionally on the extended Riemann hypothesis, with high probability $\varphi$ is irreducible and $\mathrm{Gal}(\varphi )\ge A_n$.

27 pages. Final version following referee's comments

References in corpus (3)

• Irreducibility of random polynomials: general measures
• Four Random Permutations Conjugated by an Adversary Generate $S_n$ with High Probability
• The singularity probability of a random symmetric matrix is exponentially small

Cited by in corpus (6)

• Winding Number Statistics for Chiral Random Matrices: Averaging Ratios of Determinants with Parametric Dependence
• Winding Number Statistics for Chiral Random Matrices: Averaging Ratios of Parametric Determinants in the Orthogonal Case
• Galois groups of random additive polynomials
• Universality for low degree factors of random polynomials over finite fields
• Irreducibility of the characteristic polynomials of random tridiagonal matrices
• Counting matrices over finite rank multiplicative groups