Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
4593413 | Journal of Number Theory | 2016 | 9 Pages |
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≥6d≥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≥12d≥12.
Keywords
Related Topics
Physical Sciences and Engineering
Mathematics
Algebra and Number Theory
Authors
James McKee, Pavlo Yatsyna,