Arithmetic properties and zeros of the Bergman kernel on a class of quotient domains
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866913130134634496 |
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| author | Edholm, Luke D. Mathew, Vikram T. |
| author_facet | Edholm, Luke D. Mathew, Vikram T. |
| contents | An effective formula for the Bergman kernel on $\mathbb{H}_γ = \{|z_1|^γ< |z_2| < 1 \}$ is obtained for rational $γ= \frac{m}{n} >1$. The formula depends on arithmetic properties of $γ$, which uncovers new symmetries and clarifies previous results. The formulas are then used to study the Lu Qi-Keng problem. We produce sequences of rationals $γ_j \searrow 1$, where each $\mathbb{H}_{γ_j}$ has a Bergman kernel with zeros (while $\mathbb{H}_1$ is known to have a zero-free kernel), resolving an open question on this domain class. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2505_20489 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Arithmetic properties and zeros of the Bergman kernel on a class of quotient domains Edholm, Luke D. Mathew, Vikram T. Complex Variables 32A25 (Primary), 32A60, 30C15 (Secondary) An effective formula for the Bergman kernel on $\mathbb{H}_γ = \{|z_1|^γ< |z_2| < 1 \}$ is obtained for rational $γ= \frac{m}{n} >1$. The formula depends on arithmetic properties of $γ$, which uncovers new symmetries and clarifies previous results. The formulas are then used to study the Lu Qi-Keng problem. We produce sequences of rationals $γ_j \searrow 1$, where each $\mathbb{H}_{γ_j}$ has a Bergman kernel with zeros (while $\mathbb{H}_1$ is known to have a zero-free kernel), resolving an open question on this domain class. |
| title | Arithmetic properties and zeros of the Bergman kernel on a class of quotient domains |
| topic | Complex Variables 32A25 (Primary), 32A60, 30C15 (Secondary) |
| url | https://arxiv.org/abs/2505.20489 |