Open Access
2006 The expected number of zeros of a random system of $p$-adic polynomials
Steven Evans
Author Affiliations +
Electron. Commun. Probab. 11: 278-290 (2006). DOI: 10.1214/ECP.v11-1230


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.


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Steven Evans. "The expected number of zeros of a random system of $p$-adic polynomials." Electron. Commun. Probab. 11 278 - 290, 2006.


Accepted: 30 November 2006; Published: 2006
First available in Project Euclid: 4 June 2016

zbMATH: 1130.60010
MathSciNet: MR2266718
Digital Object Identifier: 10.1214/ECP.v11-1230

Primary: 30G15 , 60B99
Secondary: 11S80 , 30G06

Keywords: $q$-binomial formula , co-area formula , Gaussian , Kac-Rice formula , Local Field , Random matrix

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