On random polynomials with an intermediate number of real roots
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
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2023
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| _version_ | 1866909161569124352 |
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| author | Michelen, Marcus O'Rourke, Sean |
| author_facet | Michelen, Marcus O'Rourke, Sean |
| contents | For each $α\in (0, 1)$, we construct a bounded monotone deterministic sequence $(c_k)_{k \geq 0}$ of real numbers so that the number of real roots of the random polynomial $f_n(z) = \sum_{k=0}^n c_k \varepsilon_k z^k$ is $n^{α+ o(1)}$ with probability tending to one as the degree $n$ tends to infinity, where $(\varepsilon_k)$ is a sequence of i.i.d. (real) random variables of finite mean satisfying a mild anti-concentration assumption. In particular, this includes the case when $(\varepsilon_k)$ is a sequence of i.i.d. standard Gaussian or Rademacher random variables. This result confirms a conjecture of O. Nguyen from 2019. More generally, our main results also describe several statistical properties for the number of real roots of $f_n$, including the asymptotic behavior of the variance and a central limit theorem. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2310_16966 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On random polynomials with an intermediate number of real roots Michelen, Marcus O'Rourke, Sean Probability For each $α\in (0, 1)$, we construct a bounded monotone deterministic sequence $(c_k)_{k \geq 0}$ of real numbers so that the number of real roots of the random polynomial $f_n(z) = \sum_{k=0}^n c_k \varepsilon_k z^k$ is $n^{α+ o(1)}$ with probability tending to one as the degree $n$ tends to infinity, where $(\varepsilon_k)$ is a sequence of i.i.d. (real) random variables of finite mean satisfying a mild anti-concentration assumption. In particular, this includes the case when $(\varepsilon_k)$ is a sequence of i.i.d. standard Gaussian or Rademacher random variables. This result confirms a conjecture of O. Nguyen from 2019. More generally, our main results also describe several statistical properties for the number of real roots of $f_n$, including the asymptotic behavior of the variance and a central limit theorem. |
| title | On random polynomials with an intermediate number of real roots |
| topic | Probability |
| url | https://arxiv.org/abs/2310.16966 |