Integral points on singular toric varieties and cyclic normal polynomials

We establish a formula for the height zeta function for integral points on a class of projective toric varieties. Our method builds on the harmonic analysis approach of Batyrev--Tschinkel for rational points and is applicable even when the toric variety has cyclic quotient singularities. As an application, we determine the leading term in the asymptotic number of monic integral polynomials of bounded height with linearly independent roots and a given cyclic Galois group.