Mathematical Proceedings of the Cambridge Philosophical Society

Research Article

How random is the characteristic polynomial of a random matrix ?

Jennie C. Hansena1 and Eric Schmutza2

a1 Actuarial Mathematics and Statistics Department, Heriot-Watt University, Edinburgh, Scotland

a2 Mathematics and Computer Science Department, Drexel University, Philadelphia, PA 19104, USA

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.’

(Received October 26 1992)

(Revised January 06 1993)

Footnotes

† Supported by N.S.F. grants DMS 90-099074 and 9101753.