  We study the simultaneous zeros of a random family of $d$ polynomials in $d$
variables over the $p$-adic numbers. For a family of natural models, we obtain
an explicit constant for the expected number of zeros that lie in the $d$-fold
Cartesian product of the $p$-adic integers. Considering models in which the
maximum degree that each variable appears is $N$, this expected value is \[
p^{d \lfloor \log_p N \rfloor} (1 + p^{-1} + p^{-2} + ... + p^{-d})^{-1} \] for
the simplest such model.
