Salem numbers of trace -2, and a conjecture of Estes and Guralnick. / McKee, James; Yatsyna, Pavlo.

In: Journal of Number Theory, Vol. 160, 03.2016, p. 409–417.

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Salem numbers of trace -2, and a conjecture of Estes and Guralnick. / McKee, James; Yatsyna, Pavlo.

In: Journal of Number Theory, Vol. 160, 03.2016, p. 409–417.

Research output: Contribution to journalArticle

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McKee, James ; Yatsyna, Pavlo. / Salem numbers of trace -2, and a conjecture of Estes and Guralnick. In: Journal of Number Theory. 2016 ; Vol. 160. pp. 409–417.

BibTeX

@article{6363e7d2a27943359283e7b19e62ffcc,
title = "Salem numbers of trace -2, and a conjecture of Estes and Guralnick",
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.",
keywords = "Salem numbers, minimal polynomials",
author = "James McKee and Pavlo Yatsyna",
note = "Email of 29/10/15 Ms. Ref. No.: JNT-D-15-00377R1 Title: Salem numbers of trace -2, and a conjecture of Estes and Guralnick Journal of Number Theory Dear Dr. James McKee, A final disposition of {"}Accept{"} has been registered for the above-mentioned manuscript. Your manuscript is now entering production. With kind regards, Elsevier Editorial System Journal of Number Theory E-mail: jnt@elsevier.com ",
year = "2016",
month = mar
doi = "10.1016/j.jnt.2015.09.019",
language = "English",
volume = "160",
pages = "409–417",
journal = "Journal of Number Theory",
issn = "0022-314X",
publisher = "Academic Press Inc.",

}

RIS

TY - JOUR

T1 - Salem numbers of trace -2, and a conjecture of Estes and Guralnick

AU - McKee, James

AU - Yatsyna, Pavlo

N1 - Email of 29/10/15 Ms. Ref. No.: JNT-D-15-00377R1 Title: Salem numbers of trace -2, and a conjecture of Estes and Guralnick Journal of Number Theory Dear Dr. James McKee, A final disposition of "Accept" has been registered for the above-mentioned manuscript. Your manuscript is now entering production. With kind regards, Elsevier Editorial System Journal of Number Theory E-mail: jnt@elsevier.com

PY - 2016/3

Y1 - 2016/3

N2 - 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.

AB - 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.

KW - Salem numbers, minimal polynomials

U2 - 10.1016/j.jnt.2015.09.019

DO - 10.1016/j.jnt.2015.09.019

M3 - Article

VL - 160

SP - 409

EP - 417

JO - Journal of Number Theory

JF - Journal of Number Theory

SN - 0022-314X

ER -