Sharp bounds for the valence of certain logharmonic polynomials
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866916898197733376 |
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| author | Lazebnik, Kirill Lundberg, Erik |
| author_facet | Lazebnik, Kirill Lundberg, Erik |
| contents | Consider a logharmonic polynomial; that is, a product of the form $p(z)\overline{q(z)}$, where $p$, $q$ are holomorphic polynomials. Assume $q$ is linear and denote by $n$ the degree of $p$. It was recently shown in arXiv:2302.04339 [math.CV] that the valence of such a logharmonic polynomial is at most $3n-1$; in this paper we show that their $3n-1$ upper bound is sharp. Together with the work of arXiv:2302.04339 [math.CV], this resolves a conjecture of Bshouty and Hengartner. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_10151 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Sharp bounds for the valence of certain logharmonic polynomials Lazebnik, Kirill Lundberg, Erik Complex Variables Dynamical Systems 30C55, 37F10 Consider a logharmonic polynomial; that is, a product of the form $p(z)\overline{q(z)}$, where $p$, $q$ are holomorphic polynomials. Assume $q$ is linear and denote by $n$ the degree of $p$. It was recently shown in arXiv:2302.04339 [math.CV] that the valence of such a logharmonic polynomial is at most $3n-1$; in this paper we show that their $3n-1$ upper bound is sharp. Together with the work of arXiv:2302.04339 [math.CV], this resolves a conjecture of Bshouty and Hengartner. |
| title | Sharp bounds for the valence of certain logharmonic polynomials |
| topic | Complex Variables Dynamical Systems 30C55, 37F10 |
| url | https://arxiv.org/abs/2508.10151 |