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How random is the characteristic polynomial of a random matrix ?

  • Jennie C. Hansen (a1) and Eric Schmutz (a2)

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.’

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[7] J. P. S. Kung . The cycle structure of a linear transformation over a finite field. Linear Algebra and its Applications 36 (1981), 141155.

[13] R. Stong . Some asymptotic results on finite vector spaces. Advances in applied mathematics 9 (1988), 167199.

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Mathematical Proceedings of the Cambridge Philosophical Society
  • ISSN: 0305-0041
  • EISSN: 1469-8064
  • URL: /core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society
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