The expected number of zeros of a random system of $p$-adic polynomials

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.