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

February, 2006
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Steven N. Evans

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.

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