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

James McKee, Pavlo Yatsyna

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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.
Original languageEnglish
Pages (from-to)409–417
Number of pages9
JournalJournal of Number Theory
Early online date2 Nov 2015
Publication statusPublished - Mar 2016


  • Salem numbers, minimal polynomials

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