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：1

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

共 23 条

[1]

[Anonymous], 1943, Rec. Math. Mat. Sbornik N.S.

[2] What is the Probability that a Random Integral Quadratic Form in n Variables has an Integral Zero? [J].

Bhargava, Manjul ;

Cremona, John E. ;

Fisher, Tom ;

Jones, Nick G. ;

Keating, Jonathan P. .

INTERNATIONAL MATHEMATICS RESEARCH NOTICES, 2016, 2016 (12) :3828-3848

[3]

Bloch A, 1932, P LOND MATH SOC, V33, P102

[4] The probability that a random monic p-adic polynomial splits [J].

Buhler, Joe ;

Goldstein, Daniel ;

Moews, David ;

Rosenberg, Joel .

EXPERIMENTAL MATHEMATICS, 2006, 15 (01) :21-32

[5]

Caruso X., 2021, PREPRINT

[6] Representation stability in cohomology and asymptotics for families of varieties over finite fields [J].

Church, Thomas ;

Ellenberg, Jordan S. ;

Farb, Benson .

ALGEBRAIC TOPOLOGY: APPLICATIONS AND NEW DIRECTIONS, 2014, 620 :1-54

[7]

Cohen S.D., 1970, Acta Arith., V17, P255, DOI [10.4064/aa-17-3-255-271, DOI 10.4064/AA-17-3-255-271]

[8] Random polynomials having few or no real zeros [J].

Dembo, A ;

Poonen, B ;

Shao, QM ;

Zeitouni, O .

JOURNAL OF THE AMERICAN MATHEMATICAL SOCIETY, 2002, 15 (04) :857-892

[9] Definable sets, motives and p-adic integrals [J].

Denef, J ;

Loeser, F .

JOURNAL OF THE AMERICAN MATHEMATICAL SOCIETY, 2001, 14 (02) :429-469

[10] A FUNCTIONAL-EQUATION OF IGUSA LOCAL ZETA FUNCTION [J].

DENEF, J ;

MEUSER, D .

AMERICAN JOURNAL OF MATHEMATICS, 1991, 113 (06) :1135-1152

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