Abstract
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 language | English |
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Pages (from-to) | 409–417 |
Number of pages | 9 |
Journal | Journal of Number Theory |
Volume | 160 |
Early online date | 2 Nov 2015 |
DOIs | |
Publication status | Published - Mar 2016 |
Keywords
- Salem numbers, minimal polynomials