  We consider natural polynomial truncations of hypergeometric power series
defined over finite fields. For these truncations, we establish asymptotic
upper bounds of order $O(p^{11/12})$ on the number of roots in the prime field
$\mathbb{F}_p$. We discuss the correspondence to families of elliptic curves
and K3 surfaces of certain such hypergeometric polynomials, for which sharp
bounds are obtained in some cases. We include some computations to illustrate
and supplement our results.
