How random is the characteristic polynomial of a random matrix ?
How random is the characteristic polynomial of a random matrix ?
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随机矩阵的特征多项式有多随机？
DOI：
10.1017/s0305004100071796
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发表时间：
1993
影响因子：
0.8
通讯作者：
E. Schmutz
中科院分区：
文献类型：
--
作者：
J. Hansen;E. Schmutz
摘要 Fq[x;] 中的每一个 monic n 次多项式都是至少一个 n × n 矩阵（条目在有限域 Fq 中）的特征多项式，但它们出现的频率并不相同。没有先验的理由表明典型矩阵的特征多项式应该类似于典型的 monic 次数 n 多项式。尽管如此，我们证明了以下启发式陈述的精确版本：“除了它的小因素外，随机矩阵的特征多项式是随机的。”
Abstract Every monic, degree n polynomial in Fq[x;] is the characteristic polynomial of at least one n × n matrix (with entries in the finite field Fq), but they do not appear with equal frequency. There is no a priori reason that the characteristic polynomial of a typical matrix should resemble a typical monic degree n polynomial. Nevertheless, we prove a precise version of the following heuristic statement: ‘Excepting its small factors, the characteristic polynomial of a random matrix is random.’