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 $e^{-i(m-n)θ}Q_{m,n}(e^{iθ})$ 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.