Every realrooted exponential polynomial is the restriction of a LeeYang polynomial
Abstract
A LeeYang 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 realrooted 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 LeeYang polynomial to a positive line in the torus. Together with previous work by Olevskii and Ulanovskii, this implies that the KurasovSarnak construction of $ \mathbb{N} $valued Fourier quasicrystals from stable polynomials comprises every possible $ \mathbb{N} $valued Fourier quasicrystal.
 Publication:

arXiv eprints
 Pub Date:
 March 2023
 DOI:
 10.48550/arXiv.2303.03201
 arXiv:
 arXiv:2303.03201
 Bibcode:
 2023arXiv230303201A
 Keywords:

 Mathematics  Complex Variables;
 Mathematical Physics;
 45B10;
 52C23