  We define the generalized basic hypergeometric polynomial of degree $N \geq
1$ in terms of the generalized basic hypergeometric function, which depends on
(arbitrary, generic, possibly complex) parameters $q \neq 1$, the $r \geq 0$
parameters $\alpha _{j}$ and the $s \geq 0$ parameters $\beta _{k}$. In this
paper we obtain a set of $N$ nonlinear algebraic equations satisfied by the $N$
zeros $\zeta _{n}\equiv \zeta _{n}\left( \underline{\alpha },\underline{\beta
};q;N\right) $ of this polynomial. We moreover identify an $\left( N\times
N\right) $-matrix $\underline{M}\equiv \underline{M}\left( \underline{\alpha
},\underline{\beta };\underline{\zeta };q;N\right) $ featuring the $N$
eigenvalues $\mu _{n}=-q^{\left( s-r\right) \left( N-n\right)
}\left(q^{-n}-1\right) ~\prod\limits_{j=1}^{r}\left( \alpha
_{j}~q^{N-n}-1\right)$, where $n=1,2,...,N.$ These $N$ eigenvalues depend only
on the $r$ parameters $\alpha _{j}$ (besides $q$ and $N$), implying that the
$\left( N\times N\right) $-matrix $\underline{M}$ is isospectral for variations
of the $s$ parameters $\beta _{k}$; and they clearly are rational numbers if
$q$ and the $r$ parameters $\alpha _{j}$ are themselves rational numbers: a
nontrivial Diophantine property.
