  We define the generalized hypergeometric polynomial of degree N in terms of
the generalized hypergeometric function that depends on p parameters a_1, ...,
a_p and q parameters b_1, ..., b_q. The parameters are "generic", possibly
complex, numbers. In this paper we obtain a set of N nonlinear algebraic
equations satisfied by the N zeros z_n of this polynomial. We moreover
manufacture an NxN matrix L in terms of the 1+p+q parameters N, a_j, b_k
characterizing this polynomial, and of its N zeros z_n. We show that the matrix
L features N eigenvalues that depend only on the q parameters b_k, implying
that this matrix is isospectral for the variations of the p parameters a_j.
These eigenvalues are integer (or rational) numbers if the q parameters b_k are
themselves integer (or rational) numbers: a nontrivial diophantine property.
