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

A Lee-Yang polynomial $p(z_1,\dots ,z_n)$ is a polynomial that has no zeros in the polydisc ${\mathbb{D}}^n$ and its inverse $(\mathbb{C}\setminus \bar{\mathbb{D}}{)}^n$. 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.