Salem numbers of trace-2, and a conjecture of Estes and Guralnick

JOURNAL OF NUMBER THEORY

卷:160

Article

McKee, James¹  Yatsyna, Pavlo¹ 

[1] Univ London, Dept Math, Egham TW20 0EX, Surrey, England

关键词: Salem numbers;    Minimal polynomials;   

DOI  :  10.1016/j.jnt.2015.09.019

来源: Elsevier

【 摘 要 】

In 1993 Estes and Guralnick conjectured that any totally real separable monic polynomial with rational integer coefficients will occur as the minimal polynomial of some symmetric matrix with rational integer entries. They proved this to be true for all such polynomials that have degree at most 4. In this paper, we show that for every d >= 6 there is a polynomial of degree d that is a counterexample to this conjecture. The only case still in doubt is degree 5. One of the ingredients in the proof is to show that there are Salem numbers of degree 2d and trace -2 for every d >= 12. (C) 2015 Elsevier Inc. All rights reserved.

【 授权许可】

Free