2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/109061We 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.13 pages, no figures, revised to incorporate referees' commentsProbabilityCommutative Algebra60B99; 30G15; 11S80; 30G06text