Extensions of the Bloch-Pólya theorem on the number of real zeros of polynomials (II)
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866909350925172736 |
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| author | Erdélyi, Tamás |
| author_facet | Erdélyi, Tamás |
| contents | We prove that there is an absolute constant $c > 0$ such that for every $$a_0,a_1, \ldots,a_n \in [1,M]\,, \qquad 1 \leq M \leq \frac 14 \exp \left( \frac n9 \right)\,,$$ there are $$b_0,b_1,\ldots,b_n \in \{-1,0,1\}$$ such that the polynomial $P$ of the form $\displaystyle{P(z) = \sum_{j=0}^n{b_ja_jz^j}}$ has at least $\displaystyle{c \left( \frac{n}{\log(4M)} \right)^{1/2}-1}$ distinct sign changes in $I_a := (1-2a,1-a)$, where $\displaystyle{a := \left( \frac{\log(4M)}{n} \right)^{1/2} \leq 1/3}$. This improves and extends earlier results of Bloch and Pólya and Erdélyi and, as a special case, recaptures a special case of a more general recent result of Jacob and Nazarov. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2407_16120 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Extensions of the Bloch-Pólya theorem on the number of real zeros of polynomials (II) Erdélyi, Tamás Number Theory Classical Analysis and ODEs 26C10 12D10 We prove that there is an absolute constant $c > 0$ such that for every $$a_0,a_1, \ldots,a_n \in [1,M]\,, \qquad 1 \leq M \leq \frac 14 \exp \left( \frac n9 \right)\,,$$ there are $$b_0,b_1,\ldots,b_n \in \{-1,0,1\}$$ such that the polynomial $P$ of the form $\displaystyle{P(z) = \sum_{j=0}^n{b_ja_jz^j}}$ has at least $\displaystyle{c \left( \frac{n}{\log(4M)} \right)^{1/2}-1}$ distinct sign changes in $I_a := (1-2a,1-a)$, where $\displaystyle{a := \left( \frac{\log(4M)}{n} \right)^{1/2} \leq 1/3}$. This improves and extends earlier results of Bloch and Pólya and Erdélyi and, as a special case, recaptures a special case of a more general recent result of Jacob and Nazarov. |
| title | Extensions of the Bloch-Pólya theorem on the number of real zeros of polynomials (II) |
| topic | Number Theory Classical Analysis and ODEs 26C10 12D10 |
| url | https://arxiv.org/abs/2407.16120 |