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

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...

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Bibliographic Details
Main Author Evans, Steven N
Format Journal Article
LanguageEnglish
Published 21.02.2006
Subjects
Mathematics - Commutative Algebra
Mathematics - Probability
Online AccessGet full text
DOI10.48550/arxiv.math/0602478

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Summary: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.
Bibliography:American Institute of Mathematics preprint number AIM 2006 - 9
DOI:10.48550/arxiv.math/0602478