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

@article{Evans2006TheEN,
title={The expected number of zeros of a random system of \$p\$-adic polynomials},
author={Steven N. Evans},
journal={Electronic Communications in Probability},
year={2006},
volume={11},
pages={278-290}
}
• S. Evans
• Published 21 February 2006
• Mathematics
• Electronic Communications in Probability
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…
Zeros of random tropical polynomials, random polytopes and stick-breaking
• Mathematics
• 2014
For $i = 0, 1, \ldots, n$, let $C_i$ be independent and identically distributed random variables with distribution $F$ with support $(0,\infty)$. The number of zeros of the random tropical
Ultrametric Root Counting
• Mathematics
• 2009
Let $K$ be a complete non-archimedean field with a discrete valuation, $f\in K[X]$ a polynomial with non-vanishing discriminant, $A$ the valuation ring of $K$, and $\M$ the maximal ideal of $A$. The
The density of polynomials of degree $n$ over $\mathbb{Z}_p$ having exactly $r$ roots in $\mathbb{Q}_p$
• Mathematics
• 2021
We determine the probability that a random polynomial of degree n over Z p has exactly r roots in Q p , and show that it is given by a rational function of p that is invariant under replacing p by 1
The Number of Roots of a Random Polynomial over The Field of $p$-adic Numbers
. We study the roots of a random polynomial over the ﬁeld of p-adic numbers. For a random monic polynomial with coeﬃcients in Z p , we obtain an asymptotic formula for the factorial moments of the
Probabilistic enumerative geometry over $p$-adic numbers: linear spaces on complete intersections
• Mathematics
• 2020
We compute the expectation of the number of linear spaces on a random complete intersection in $p$-adic projective space. Here "random" means that the coefficients of the polynomials defining the
Multivariate ultrametric root counting
• Mathematics
• 2011
Let $K$ be a field, complete with respect to a discrete non-archimedian valuation and let $k$ be the residue field. Consider a system $F$ of $n$ polynomial equations in $K\vars$. Our first result is
Where are the zeroes of a random p-adic polynomial?
We study the repartition of the roots of a random p-adic polynomial in an algebraic closure of Qp. We prove that the mean number of roots generating a fixed finite extension K of Qp depends mostly on
The density of polynomials of degree n$n$ over Zp${\mathbb {Z}}_p$ having exactly r$r$ roots in Qp${\mathbb {Q}}_p$
• Mathematics
Proceedings of the London Mathematical Society
• 2022
We determine the probability that a random polynomial of degree n$n$ over Zp${\mathbb {Z}}_p$ has exactly r$r$ roots in Qp${\mathbb {Q}}_p$ , and show that it is given by a rational function of p$p$
The Expected Number of Roots over The Field of p-adic Numbers
We study the roots of a random polynomial over the field of p-adic numbers. For a random monic polynomial with i.i.d. coefficients in Zp, we obtain an estimate for the expected number of roots of