The number of real zeros of polynomials with constrained coefficients
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866909347324362752 |
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| author | Erdélyi, Tamás |
| author_facet | Erdélyi, Tamás |
| contents | We prove that there is an absolute constant $c > 0$ such that every polynomial $P$ of the form $$P(z) = \sum_{j=0}^{n}{a_jz^j}\,, \quad |a_0| = 1\,, \quad |a_j| \leq M\,, \quad a_j \in \Bbb{C}\,, \quad M \geq 1\,,$$ has at most $cn^{1/2}(1+\log M)^{1/2}$ zeros in the interval $[-1,1]$. This result is sharp up to the multiplicative constant $c > 0$ and extends an earlier result of Borwein, Erdélyi, and Kós from the case $M=1$ to the case $M \geq $1. This has also been proved recently with the factor $(1+\log M)$ rather than $(1+\log M)^{1/2}$ in the Appendix of a recent paper by Jacob and Nazarov by using a different method. We also prove that there is an absolute constant $c > 0$ such that every polynomial $P$ of the above form has at most $(c/a)(1+\log M)$ zeros in the interval $[-1+a,1-a]$ with $a \in (0,1)$. Finally we correct a somewhat incorrect proof of an earlier result of Borwein and Erdélyi by proving that there is a constant $η> 0$ such that every polynomial $P$ of the above form with $M = 1$ has at most $ηn^{1/2}$ zeros inside any polygon with vertices on the unit circle, where the multiplicative constant $η> 0$ depends only on the polygon. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_09553 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The number of real zeros of polynomials with constrained coefficients Erdélyi, Tamás Classical Analysis and ODEs 26C10, 11C08 We prove that there is an absolute constant $c > 0$ such that every polynomial $P$ of the form $$P(z) = \sum_{j=0}^{n}{a_jz^j}\,, \quad |a_0| = 1\,, \quad |a_j| \leq M\,, \quad a_j \in \Bbb{C}\,, \quad M \geq 1\,,$$ has at most $cn^{1/2}(1+\log M)^{1/2}$ zeros in the interval $[-1,1]$. This result is sharp up to the multiplicative constant $c > 0$ and extends an earlier result of Borwein, Erdélyi, and Kós from the case $M=1$ to the case $M \geq $1. This has also been proved recently with the factor $(1+\log M)$ rather than $(1+\log M)^{1/2}$ in the Appendix of a recent paper by Jacob and Nazarov by using a different method. We also prove that there is an absolute constant $c > 0$ such that every polynomial $P$ of the above form has at most $(c/a)(1+\log M)$ zeros in the interval $[-1+a,1-a]$ with $a \in (0,1)$. Finally we correct a somewhat incorrect proof of an earlier result of Borwein and Erdélyi by proving that there is a constant $η> 0$ such that every polynomial $P$ of the above form with $M = 1$ has at most $ηn^{1/2}$ zeros inside any polygon with vertices on the unit circle, where the multiplicative constant $η> 0$ depends only on the polygon. |
| title | The number of real zeros of polynomials with constrained coefficients |
| topic | Classical Analysis and ODEs 26C10, 11C08 |
| url | https://arxiv.org/abs/2409.09553 |