Every real-rooted exponential polynomial is the restriction of a Lee-Yang polynomial

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.