Connections between discriminants and the root distribution of polynomials with rational generating function

Let Hm(z) be a sequence of polynomials whose generating function ∑∞ m=0 Hm(z)t m is the reciprocal of a bivariate polynomial D(t, z). We show that in the three cases D(t, z) = 1 + B(z)t + A(z)t, D(t, z) = 1 + B(z)t + A(z)t and D(t, z) = 1 + B(z)t + A(z)t, where A(z) and B(z) are any polynomials in z with complex coefficients, the roots of Hm(z) lie on a portion of a real algebraic curve whose equation is explicitly given. The proofs involve the q-analogue of the discriminant, a concept introduced by Mourad Ismail.