The expected number of zeros of a random system of p-adic polynomials
Steven N. Evans, University of California at Berkeley
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.
Considering models in which the maximum degree that each variable appears is N,
this expected value is
pd ⌊
logp N ⌋ (1 +
p-1 + p-2 + ... + p-d)-1
for the simplest such model.
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