Universality theorems for zeros of random real polynomials with fixed coefficients
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866909311209308160 |
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| author | King, Matthew C. Swaminathan, Ashvin |
| author_facet | King, Matthew C. Swaminathan, Ashvin |
| contents | Consider a monic polynomial of degree $n$ whose subleading coefficients are independent, identically distributed, nondegenerate random variables having zero mean, unit variance, and finite moments of all orders, and let $m \geq 0$ be a fixed integer. We prove that such a random monic polynomial has exactly $m$ real zeros with probability $n^{-3/4+o(1)}$ as $n\to \infty$ through integers of the same parity as $m$. More generally, we determine conditions under which a similar asymptotic formula describes the corresponding probability for families of random real polynomials with multiple fixed coefficients. Our work extends well-known universality results of Dembo, Poonen, Shao, and Zeitouni, who considered the family of real polynomials with all coefficients random.
As a number-theoretic consequence of these results, we deduce that an algebraic integer $α$ of degree $n$ has exactly $m$ real Galois conjugates with probability $n^{-3/4+o(1)}$, when such $α$ are ordered by the heights of their minimal polynomials. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_02717 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Universality theorems for zeros of random real polynomials with fixed coefficients King, Matthew C. Swaminathan, Ashvin Probability Number Theory 60G99, 12D10 (primary), 11R04, 11R09 (secondary) Consider a monic polynomial of degree $n$ whose subleading coefficients are independent, identically distributed, nondegenerate random variables having zero mean, unit variance, and finite moments of all orders, and let $m \geq 0$ be a fixed integer. We prove that such a random monic polynomial has exactly $m$ real zeros with probability $n^{-3/4+o(1)}$ as $n\to \infty$ through integers of the same parity as $m$. More generally, we determine conditions under which a similar asymptotic formula describes the corresponding probability for families of random real polynomials with multiple fixed coefficients. Our work extends well-known universality results of Dembo, Poonen, Shao, and Zeitouni, who considered the family of real polynomials with all coefficients random. As a number-theoretic consequence of these results, we deduce that an algebraic integer $α$ of degree $n$ has exactly $m$ real Galois conjugates with probability $n^{-3/4+o(1)}$, when such $α$ are ordered by the heights of their minimal polynomials. |
| title | Universality theorems for zeros of random real polynomials with fixed coefficients |
| topic | Probability Number Theory 60G99, 12D10 (primary), 11R04, 11R09 (secondary) |
| url | https://arxiv.org/abs/2409.02717 |