Title: The expected number of zeros of a random system of $p$-adic polynomials
Author: Steven N. Evans
pub: PDF
Url: http://www.stat.berkeley.edu/users/evans/699.pdf
Date: February 2006
Abstract: 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.  This expected value, which is
\[
\left(1 + p^{-1} + p^{-2} + \cdots + p^{-d}\right)^{-1}
\]
for the simplest model, is independent of the degree of the polynomials.