Distribution of the zeros of polynomials near the unit circle
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866916331588157440 |
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| author | Das, Mithun Kumar |
| author_facet | Das, Mithun Kumar |
| contents | We estimate the number of zeros of a polynomial in $\mathbb{C}[z]$ within any small circular disc centered on the unit circle, which improves and comprehensively extends a result established by Borwein, Erd{é}lyi, and Littmann~\cite{BE1} in 2008. Furthermore, by combining this result with Euclidean geometry, we derive an upper bound on the number of zeros of such a polynomial within a region resembling a gear wheel. Additionally, we obtain a sharp upper bound on the annular discrepancy of such zeros near the unit circle. Our approach builds upon a modified version of the method described in \cite{BE1}, combined with the refined version of the best-known upper bound for angular discrepancy of zeros of polynomials. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_15306 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Distribution of the zeros of polynomials near the unit circle Das, Mithun Kumar Complex Variables Number Theory 30C15, 11K38 We estimate the number of zeros of a polynomial in $\mathbb{C}[z]$ within any small circular disc centered on the unit circle, which improves and comprehensively extends a result established by Borwein, Erd{é}lyi, and Littmann~\cite{BE1} in 2008. Furthermore, by combining this result with Euclidean geometry, we derive an upper bound on the number of zeros of such a polynomial within a region resembling a gear wheel. Additionally, we obtain a sharp upper bound on the annular discrepancy of such zeros near the unit circle. Our approach builds upon a modified version of the method described in \cite{BE1}, combined with the refined version of the best-known upper bound for angular discrepancy of zeros of polynomials. |
| title | Distribution of the zeros of polynomials near the unit circle |
| topic | Complex Variables Number Theory 30C15, 11K38 |
| url | https://arxiv.org/abs/2407.15306 |