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A Note on Integer Symmetric Matrices and Mahler's Measure

Published online by Cambridge University Press:  20 November 2018

Edward Dobrowolski
Edward Dobrowolski*
Affiliation:
College of New Caledonia, Prince George, BC, V2N 1P8 e-mail: dobrowolski@cnc.bc.ca
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Abstract

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We find a lower bound on the absolute value of the discriminant of the minimal polynomial of an integral symmetric matrix and apply this result to find a lower bound on Mahler's measure of related polynomials and to disprove a conjecture of D. Estes and R. Guralnick.

Keywords

11C2011R06Integer matricesLehmer's problemMahler's measure
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 51 , Issue 1 , 01 March 2008 , pp. 57 - 59
Copyright
Copyright © Canadian Mathematical Society 2008

References

[1] Dobrowolski, E., On a question of Lehmer and the number of irreducible factors of a polynomial. Acta Arith. 34(1979), no. 4, 391–401.Google Scholar
[2] Estes, D. and Guralnick, R., Minimal polynomials of integral symmetric matrices. Computational linear algebra in algebraic and related problems. Linear Algebra Appl. 192(1993), 83–89.Google Scholar
[3] McKee, J. and Smyth, C. J. Salem numbers, Pisot numbers, Mahler measure and graphs. Experiment. Math. 14(2005), no. 2, 211–229.Google Scholar
[4] Simon, D., Équations dans les corps de nombres et discriminants minimaux. Ph.D. thesis, Université Bordeaux-I, 1998.Google Scholar