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arXiv · 2502.17218

Irreducibility of the characteristic polynomials of random tridiagonal matrices

Lior Bary-Soroker·Daniele Garzoni·Sasha Sodin
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Abstract

Conditionally on the Riemann hypothesis for certain Dedekind zeta functions, we show that the characteristic polynomial of a class of random tridiagonal matrices of large dimension is irreducible, with probability exponentially close to one; moreover, its Galois group over the rational numbers is either the symmetric or the alternating group. This is the counterpart of the results of Breuillard--Varj\'u (for polynomials with independent coefficients), and with those of Eberhard and Ferber--Jain--Sah--Sawhney (for full random matrices). We also analyse a related class of random tridiagonal matrices for which the Galois group is much smaller.

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Lior Bary-Soroker, Daniele Garzoni, Sasha Sodin. 2025-02-24. Irreducibility of the characteristic polynomials of random tridiagonal matrices. https://doi.org/10.1016/j.jnt.2025.09.008

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