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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2410.20714 |
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Table of 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.