关于多项式无平方密度的说明

IF 0.8 3区数学 Q2 MATHEMATICS

Mathematika Pub Date : 2024-08-26 DOI:10.1112/mtk.12275

R. C. Vaughan, Yu. G. Zarhin

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引用次数: 0

摘要

猜想中的变量积分多项式的无平方密度是一个欧拉积，可以看作是局部密度的积。我们证明，当是一个整数上的变量多项式时，0 的必要条件和充分条件是：要么有一个质数使得所有整数点的值都能被除以，要么这个多项式不是无平方多项式。我们还证明，一般情况下，无平方上限密度满足 .

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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A note on the squarefree density of polynomials

The conjectured squarefree density of an integral polynomial in variables is an Euler product which can be considered as a product of local densities. We show that a necessary and sufficient condition for to be 0 when is a polynomial in variables over the integers, is that either there is a prime such that the values of at all integer points are divisible by or the polynomial is not squarefree as a polynomial. We also show that generally the upper squarefree density satisfies .

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来源期刊

Mathematika MATHEMATICS, APPLIED-MATHEMATICS

CiteScore

1.40

自引率

0.00%

发文量

审稿时长

>12 weeks

期刊介绍： Mathematika publishes both pure and applied mathematical articles and has done so continuously since its founding by Harold Davenport in the 1950s. The traditional emphasis has been towards the purer side of mathematics but applied mathematics and articles addressing both aspects are equally welcome. The journal is published by the London Mathematical Society, on behalf of its owner University College London, and will continue to publish research papers of the highest mathematical quality.