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Every real-rooted exponential polynomial is the restriction of a Lee-Yang polynomial (2303.03201v3)
Published 6 Mar 2023 in math.CV, math-ph, and math.MP
Abstract: A Lee-Yang polynomial $ p(z_{1},\ldots,z_{n}) $ is a polynomial that has no zeros in the polydisc $ \mathbb{D}{n} $ and its inverse $ (\mathbb{C}\setminus\overline{\mathbb{D}}){n} $. We show that any real-rooted exponential polynomial of the form $f(x) = \sum_{j=0}s c_j e{\lambda_j x}$ can be written as the restriction of a Lee-Yang polynomial to a positive line in the torus. Together with previous work by Olevskii and Ulanovskii, this implies that the Kurasov-Sarnak construction of $ \mathbb{N} $-valued Fourier quasicrystals from stable polynomials comprises every possible $ \mathbb{N} $-valued Fourier quasicrystal.