Irreducibility of the characteristic polynomials of random tridiagonal matrices
Fuente:
arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866909906986074112 |
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| author | Bary-Soroker, Lior Garzoni, Daniele Sodin, Sasha |
| author_facet | Bary-Soroker, Lior Garzoni, Daniele Sodin, Sasha |
| contents | 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--Varjú (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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_17218 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Irreducibility of the characteristic polynomials of random tridiagonal matrices Bary-Soroker, Lior Garzoni, Daniele Sodin, Sasha Number Theory Probability 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--Varjú (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. |
| title | Irreducibility of the characteristic polynomials of random tridiagonal matrices |
| topic | Number Theory Probability |
| url | https://arxiv.org/abs/2502.17218 |