The density of polynomials of degree n$n$ over Zp${\mathbb {Z}}_p$ having exactly r$r$ roots in Qp${\mathbb {Q}}_p$

被引：2

作者：

Bhargava, Manjul ^{[1
]}

Cremona, John ^{[2
]}

Fisher, Tom ^{[3
]}

Gajovic, Stevan ^{[4
]}

机构：

[1] Princeton Univ, Dept Math, Princeton, NJ 08544 USA

[2] Univ Warwick, Math Inst, Coventry, W Midlands, England

[3] Univ Cambridge, Ctr Math Sci, DPMMS, Wilberforce Rd, Cambridge CB3 0WB, England

[4] Univ Groningen, Fac Sci & Engn, Groningen, Netherlands

来源：

PROCEEDINGS OF THE LONDON MATHEMATICAL SOCIETY | 2022年 / 124卷 / 05期

基金：

美国国家科学基金会;

关键词：

EQUATION; NUMBER;

D O I：

10.1112/plms.12438

中图分类号：

O1 [数学];

学科分类号：

0701 ; 070101 ;

摘要：

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$p$ that is invariant under replacing p$p$ by 1/p$1/p$.

引用

页码：713 / 736

页数：24

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