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Let $H_{m}(z)$ be a sequence of polynomials whose generating function$\sum_{m=0}^{\infty}H_{m}(z)t^{m}$ 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 isexplicitly given. The proofs involve the $q$-analogue of the discriminant, aconcept introduced by Mourad Ismail.

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