Where are the zeroes of a random p-adic polynomial?
Where are the zeroes of a random p-adic polynomial?
复制标题
随机 p 进多项式的零点在哪里？
DOI：
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发表时间：
2020
期刊：
影响因子：
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通讯作者：
X. Caruso
中科院分区：
文献类型：
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作者：
X. Caruso
本文研究了随机p-adic多项式的根在代数闭包中的分布。
${mathbb Q}_p$
.我们证明了生成一个固定的有限扩张K的根的平均数
${mathbb Q}_p$
主要取决于K的判别式，当K变得更加分歧时，K是包含更少根的扩展。进一步证明了对任意正整数r，一个足够大次数的随机p-adic多项式在至多r次的扩张中平均有r个根。除了平均值，我们还研究了高阶矩和两个给定子集的根的数量之间的相关性，
${mathbb Q}_p$
(or更一般地说，
${mathbb Q}_p$
).从这个角度来看，我们显着建立的结果强调，根往往相互排斥，并量化这一现象。
Abstract We study the distribution of the roots of a random p-adic polynomial in an algebraic closure of
${mathbb Q}_p$
. We prove that the mean number of roots generating a fixed finite extension K of
${mathbb Q}_p$
depends mostly on the discriminant of K, an extension containing fewer roots when it becomes more ramified. We prove further that for any positive integer r, a random p-adic polynomial of sufficiently large degree has about r roots on average in extensions of degree at most r. Beyond the mean, we also study higher moments and correlations between the number of roots in two given subsets of
${mathbb Q}_p$
(or, more generally, of a finite extension of
${mathbb Q}_p$
). In this perspective, we notably establish results highlighting that the roots tend to repel each other and quantify this phenomenon.
Zp${mathbb {Z}}_p$ 上的 n$n$ 次多项式的密度恰好有 r$r$ 根在 Qp${mathbb {Q}}_p$ 中
影响因子：
1.8
作者：
Manjul Bhargava;John Cremona;Tom Fisher;Stevan Gajović
通讯作者：
Stevan Gajović