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


 
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1. Title Title of document The expected number of zeros of a random system of $p$-adic polynomials
 
2. Creator Author's name, affiliation, country Steven N. Evans; University of California at Berkeley
 
3. Subject Discipline(s)
 
3. Subject Keyword(s) co-area formula, Kac-Rice formula, local field, Gaussian,$q$-binomial formula, random matrix
 
3. Subject Subject classification Primary: 60B99, 30G15; Secondary: 11S80, 30G06
 
4. Description 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.
 
5. Publisher Organizing agency, location
 
6. Contributor Sponsor(s) Supported in part by NSF grant DMS-0405778. Part of the research was conducted during a visit to the American Institute of Mathematics for a Workshop on Random Analytic Functions.
 
7. Date (YYYY-MM-DD) 2006-11-30
 
8. Type Status & genre Peer-reviewed Article
 
8. Type Type
 
9. Format File format PDF
 
10. Identifier Uniform Resource Identifier http://ecp.ejpecp.org/article/view/1230
 
10. Identifier Digital Object Identifier 10.1214/ECP.v11-1230
 
11. Source Journal/conference title; vol., no. (year) Electronic Communications in Probability; Vol 11
 
12. Language English=en
 
14. Coverage Geo-spatial location, chronological period, research sample (gender, age, etc.)
 
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