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

2006/02/21 by Steven N. Evans, Evans, Steven N.
Computer Science · Mathematics · #11S80 #30G06 #30G15 #60B99 #Commutative Algebra (math.AC) #FOS: Mathematics #Probability (math.PR) #Stochastic processes and statistical mechanics #Topological and Geometric Data Analysis #advanced mathematical theories #math.AC #math.PR #msc:11S80 #msc:30G06 #msc:30G15 #msc:60B99

paper · pdf · doi:10.48550/arxiv.math/0602478

13 pages, no figures, revised to incorporate referees' comments

openalex publication_date 2006/02/21 · arxiv created 2006/10/06 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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} (1 + p^-1 + p^-2 + ... + p^-d)^-1 for the simplest such model.

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