Counting degenerate polynomials of fixed degree and bounded height

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• 作者：Artūras Dubickas ; Min Sha
• 关键词：Degenerate polynomial ; Linear recurrence sequence ; Mahler measure ; Resultant ; 11C08 ; 11B37 ; 11R06
• 刊名：Monatshefte f篓鹿r Mathematik
• 出版年：2015
• 出版时间：August 2015
• 年：2015
• 卷：177
• 期：4
• 页码：517-537
• 全文大小：502 KB
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• 作者单位：Artūras Dubickas (1)
  Min Sha (2)
  
  1. Department of Mathematics and Informatics, Vilnius University, Naugarduko 24, 03225, Vilnius, Lithuania
  2. School of Mathematics and Statistics, University of New South Wales, Sydney, NSW, 2052, Australia
  
• 刊物主题：Mathematics, general;
• 出版者：Springer Vienna
• ISSN：1436-5081

文摘

In this paper, we give sharp upper and lower bounds for the number of degenerate monic (and arbitrary, not necessarily monic) polynomials with integer coefficients of fixed degree \(n \ge 2\) and height bounded by \(H \ge 2\). The polynomial is called degenerate if it has two distinct roots whose quotient is a root of unity. In particular, our bounds imply that non-degenerate linear recurrence sequences can be generated randomly.