Integer Symmetric Matrices of Small Spectral Radius and Small Mahler Measure

Christopher Smyth, James McKee

Research output: Contribution to journalArticlepeer-review

Abstract / Description of output

In a previous paper, we completely described cyclotomic matrices — integer symmetric matrices of spectral radius at most 2. In this paper, we find all minimal noncyclotomic matrices. As a consequence, we are able to determine all integer symmetric matrices of spectral radius at most 2.019, and to determine all integer symmetric matrices whose Mahler measure is at most 1.3. In particular, we solve the strong version of Lehmer’s problem for integer symmetric matrices: all noncyclotomic matrices have Mahler measure at least ‘Lehmer’s number’ λ_0 = 1.17628….
Original languageEnglish
Pages (from-to)102-136
Number of pages34
JournalInternational Mathematics Research Notices
Volume2012
Issue number1
Early online date20 Feb 2011
Publication statusPublished - 2012

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