Universality of Persistence of Random Polynomials
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| author | Ghosal, Promit Mukherjee, Sumit |
| author_facet | Ghosal, Promit Mukherjee, Sumit |
| contents | We investigate the probability that a random polynomial with independent, mean-zero and finite variance coefficients has no real zeros. Specifically, we consider a random polynomial of degree $2n$ with coefficients given by an i.i.d. sequence of mean-zero, variance-1 random variables, multiplied by an $\fracα{2}$-regularly varying sequence for $α>-1$. We show that the probability of no real zeros is asymptotically $n^{-2(b_α+b_0)}$, where $b_α$ is the persistence exponents of a mean-zero, one-dimensional stationary Gaussian processes with covariance function as $\mathrm{sech}((t-s)/2)^{α+1}$. Our work generalizes the previous results of Dembo et al. [DPSZ02] and Dembo \& Mukherjee [DM15] by removing the requirement of finite moments of all order or Gaussianity. In particular, in the special case $α= 0$, our findings confirm a conjecture by Poonen and Stoll [PS99, Section 9.1] concerning random polynomials with i.i.d. coefficients. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_20714 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Universality of Persistence of Random Polynomials Ghosal, Promit Mukherjee, Sumit Probability Number Theory 62F12, 60F10 We investigate the probability that a random polynomial with independent, mean-zero and finite variance coefficients has no real zeros. Specifically, we consider a random polynomial of degree $2n$ with coefficients given by an i.i.d. sequence of mean-zero, variance-1 random variables, multiplied by an $\fracα{2}$-regularly varying sequence for $α>-1$. We show that the probability of no real zeros is asymptotically $n^{-2(b_α+b_0)}$, where $b_α$ is the persistence exponents of a mean-zero, one-dimensional stationary Gaussian processes with covariance function as $\mathrm{sech}((t-s)/2)^{α+1}$. Our work generalizes the previous results of Dembo et al. [DPSZ02] and Dembo \& Mukherjee [DM15] by removing the requirement of finite moments of all order or Gaussianity. In particular, in the special case $α= 0$, our findings confirm a conjecture by Poonen and Stoll [PS99, Section 9.1] concerning random polynomials with i.i.d. coefficients. |
| title | Universality of Persistence of Random Polynomials |
| topic | Probability Number Theory 62F12, 60F10 |
| url | https://arxiv.org/abs/2410.20714 |