Hostname: page-component-7dd5485656-wxk4p Total loading time: 0 Render date: 2025-10-30T18:43:02.452Z Has data issue: false hasContentIssue false
  • English
  • Français

How random is the characteristic polynomial of a random matrix ?

Published online by Cambridge University Press:  24 October 2008

Jennie C. Hansen  and
Eric Schmutz
Jennie C. Hansen
Affiliation:
Actuarial Mathematics and Statistics Department, Heriot-Watt University, Edinburgh, Scotland
Eric Schmutz
Affiliation:
Mathematics and Computer Science Department, Drexel University, Philadelphia, PA 19104, USA
Get access Rights & Permissions [Opens in a new window]

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.’

Information

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 114 , Issue 3 , November 1993 , pp. 507 - 515
Copyright
Copyright © Cambridge Philosophical Society 1993

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

REFERENCES

[1]Arratia, R., Barbour, A. D. and Tavaré, S.. On Random Polynomials over finite fields (preprint).Google Scholar
[2]Berlekamp, E.. Algebraic Coding Theory (2nd ed.). Aegean Park Press (1984).Google Scholar
[3]Gerstenhaber, M.. On the number of nilpotent matrices with entries in a finite field. Illinois J. Math. 5 (2) (1961), 330–333.CrossRefGoogle Scholar
[4]Goh, W. and Schmutz, E.. Random Matrices and Brownian Motion (to appear in Combinatorics, Probability, and Computing).Google Scholar
[5]Hansen, J.. Factorization in GF(q m)[x] and Brownian motion (to appear in Combinatorics, Probability and Computing).Google Scholar
[6]Hansen, J.. Order statistics for decomposable combinatorial structures (to appear in Random Structures and Algorithms).Google Scholar
[7]Kung, J. P. S.. The cycle structure of a linear transformation over a finite field. Linear Algebra and its Applications 36 (1981), 141–155.CrossRefGoogle Scholar
[8]Lehmer, D. H.. On reciprocally weighted partitions. Acta Arithmetica XXI (1972), 379–388.CrossRefGoogle Scholar
[9]Mignotte, M. and Nicolas, J. L.. Statistiques sur Fq[x]. Ann. Inst. Henri Poincaré XIX no. 2 (1983), 113–121.Google Scholar
[10]Nicolas, J. L.. A Gaussian law on Fq[x]. Colloq. Math. Soc. Janos Bolyai (Topics in classical number theory) 34 (1984), 1127–1162.Google Scholar
[11]Odlyzko, A. M., Discrete logarithms in finite fields and their cryptographic significance, Lecture Notes in Computer Science 209 (Springer Verlag, 1984).Google Scholar
[12]Schmutz, E.. The order of a typical matrix with entries in a finite field (submitted).Google Scholar
[13]Stong, R.. Some asymptotic results on finite vector spaces. Advances in applied mathematics 9 (1988), 167–199.CrossRefGoogle Scholar
[14]Wilf, H. S.. Generatingfunctionology (Academic Press, 1990).Google Scholar