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 ∑ m = 0 ∞ 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.