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

@article{Evans2006TheEN,
title={The expected number of zeros of a random system of \$p\$-adic polynomials},
author={Steven N. Evans},
journal={Electronic Communications in Probability},
year={2006},
volume={11},
pages={278-290}
}
• S. Evans
• Published 21 February 2006
• Mathematics
• Electronic Communications in Probability
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 $$p^{d \lfloor \log_p N \rfloor} \left(1 + p^{-1} + p^{-2} + \cdots + p^{-d}\right)^{-1}$$ for the simplest such…
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