  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 is
explicitly given. The proofs involve the $q$-analogue of the discriminant, a
concept introduced by Mourad Ismail.
