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Bergman Kernel Zeros on Rational Hartogs Triangles
Abstract
We prove the Edholm--Mathew nonvanishing conjecture for the palindromic polynomials associated with rational Hartogs triangles. For relatively prime positive integers m>n, we show that e−i(m−n)θQm,n(eiθ) is strictly positive for every real θ and derive an explicit lower bound. The proof expresses this trigonometric polynomial as a finite average of products of unnormalized Fejér kernels. It follows that Qm,n has exactly m−n zeros in the punctured unit disc, counted with multiplicity. As a consequence, a rational Hartogs triangle has a zero-free Bergman kernel if and only if its exponent is the reciprocal of a positive integer.
Related
- Vanishing theorems for L2 Dolbeault cohomology groups of projective Hartogs triangles
- Bergman Kernel Asymptotics for Semipositive Line Bundles near Curvature-Degenerate Points
- Explicit Estimates for the Bergman Kernel Form
- Cesàro operator induced by a Bergman kernel
- Beyond the unique pair: a two-parameter family of rational triangles with equal perimeter and area
BibTeX
@misc{deng2026bergman,
title = {{Bergman Kernel Zeros on Rational Hartogs Triangles}},
author = {Guantie Deng and Qian Fu},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.07480},
eprint = {2611.07480},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.07480},
}
RIS
TY - UNPB
TI - Bergman Kernel Zeros on Rational Hartogs Triangles
AU - Guantie Deng
AU - Qian Fu
PY - 2026
DA - 2026/11/05
PB - arXiv
DO - 10.48550/arxiv.2611.07480
UR - https://arxiv.org/abs/2611.07480
AB - We prove the Edholm--Mathew nonvanishing conjecture for the palindromic polynomials associated with rational Hartogs triangles. For relatively prime positive integers m>n, we show that e^−i(m−n)θQ_m,n(e^iθ) is strictly positive for every real θ and derive an explicit lower bound. The proof expresses this trigonometric polynomial as a finite average of products of unnormalized Fejér kernels. It follows that Q_m,n has exactly m−n zeros in the punctured unit disc, counted with multiplicity. As a consequence, a rational Hartogs triangle has a zero-free Bergman kernel if and only if its exponent is the reciprocal of a positive integer.
ER -