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

Let $H_m(z)$ be a sequence of polynomials whose generating function ${\sum }_{m=0}^{\mathrm{\infty }}{H}_m(z)t^m$ is the reciprocal of a bivariate polynomial $D(t,z)$. We show that in the three cases $D(t,z)=1+B(z)t+A(z)t^2$, $D(t,z)=1+B(z)t+A(z)t^3$ and $D(t,z)=1+B(z)t+A(z)t^4$, where $A(z)$ and $B(z)$ are any polynomials in $z$ with complex coefficients, the roots of $H_m(z)$ 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.