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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