小谱半径的双对称矩阵与小Mahler测度
Integer symmetric matrices of small spectral radius and small Mahler measure
摘要：
在前一篇文章中，我们完全描述了分圆矩阵--谱半径至多为2的整数对称矩阵。本文给出了所有极小非分圆矩阵。因此，我们能够确定所有的整数对称矩阵的谱半径最多为2.019，并确定所有的整数对称矩阵的Mahler测度最多为1.3。特别是我们解决了强版本的莱默问题的整数对称矩阵：所有noncyclotomic矩阵的马勒措施至少“莱默数”1.17628. .
英文摘要：
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" 1.17628... .
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