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Factorization in Fq[x] and Brownian Motion

Published online by Cambridge University Press:  12 September 2008

Jennie C. Hansen
Jennie C. Hansen
Affiliation:
Actuarial Mathematics and Statistics Department, Heriot-Watt University, Edinburgh, Scotland
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Abstract

We consider the set of polynomials of degree n over a finite field and put the uniform probability measure on this set. Any such polynomial factors uniquely into a product of its irreducible factors. To each polynomial we associate a step function on the interval [0,1] such that the size of each jump corresponds to the number of factors of a certain degree in the factorization of the random polynomial. We normalize these random functions and show that the resulting random process converges weakly to Brownian motion as n → ∞. This result complements earlier work by the author on the order statistics of the degree sequence of the factors of a random polynomial.

Type
Research Article
Information
Combinatorics, Probability and Computing , Volume 2 , Issue 3 , September 1993 , pp. 285 - 299
Copyright
Copyright © Cambridge University Press 1993

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