The number of solutions to a set of polynomial equations defined over a
finite field of $q=p^a$ elements is encoded by its zeta function, which is a
rational function in one variable. A question of fundamental interest is
how to compute this function efficiently. We describe a method to solve
this problem (due to Wan) using a $p$-adic trace formula of Dwork. We
examine how well this method works in practice on some explicit examples
coming from a certain class of varieties known as $\Delta$-regular
hypersurfaces. Our experience suggests several potential avenues of further
research.