Around a Conjecture of K. Tran

We study the root distribution of a sequence of polynomials $\{P_n(z){\}}_{n=0}^{\mathrm{\infty }}$ with the rational generating function $$\sum _{n=0}^{\mathrm{\infty }}{P}_n(z)t^n=\frac{1}{1+B(z)t^{\mathrm{\ell }}+A(z)t^k}$$ for $(k,\mathrm{\ell })=(3,2)$ and $(4,3)$ where $A(z)$ and $B(z)$ are arbitrary polynomials in $z$ with complex coefficients. We show that the zeros of $P_n(z)$ which satisfy $A(z)B(z)\mathrm{\ne }0$ lie on a real algebraic curve which we describe explicitly.