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

   
@article{ECP1230,
	author = {Steven Evans},
	title = {The expected number of zeros of a random system of $p$-adic polynomials},
	journal = {Electron. Commun. Probab.},
	fjournal = {Electronic Communications in Probability},
	volume = {11},
	year = {2006},
	keywords = {co-area formula, Kac-Rice formula, local field, Gaussian,\$q\$-binomial formula, random matrix},
	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 $$ p^{d \lfloor \log_p N \rfloor} \left(1 + p^{-1} + p^{-2} + \cdots + p^{-d}\right)^{-1} $$ for the simplest such model.},
	pages = {no. 29, 278-290},
	issn = {1083-589X},
	doi = {10.1214/ECP.v11-1230},    
        url = {http://ecp.ejpecp.org/article/view/1230}}