Permanent URL to this publication: http://dx.doi.org/10.5167/uzh22660
Arratia, R; Barbour, A D; Tavaré, S (1993). On random polynomials over finite fields. Mathematical Proceedings of the Cambridge Philosophical Society, 114(2):347368.

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Abstract
We consider random monic polynomials of degree n over a finite field of q elements, chosen with all qn possibilities equally likely, factored into monic irreducible factors. More generally, relaxing the restriction that q be a prime power, we consider that multiset construction in which the total number of possibilities of weight n is qn. We establish various approximations for the joint distribution of factors, by giving upper bounds on the total variation distance to simpler discrete distributions. For example, the counts for particular factors are approximately independent and geometrically distributed, and the counts for all factors of sizes 1, 2, …, b, where b = O(n/log n), are approximated by independent negative binomial random variables. As another example, the joint distribution of the large factors is close to the joint distribution of the large cycles in a random permutation. We show how these discrete approximations imply a Brownian motion functional central limit theorem and a PoissonDirichiet limit theorem, together with appropriate error estimates. We also give Poisson approximations, with error bounds, for the distribution of the total number of factors.
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Additional indexing
Item Type:  Journal Article, refereed, original work 

Communities & Collections:  07 Faculty of Science > Institute of Mathematics 
DDC:  510 Mathematics 
Language:  English 
Date:  1993 
Deposited On:  12 Feb 2010 15:32 
Last Modified:  27 Nov 2013 23:32 
Publisher:  Cambridge University Press 
ISSN:  03050041 
Additional Information:  Copyright: Cambridge University Press 
Publisher DOI:  10.1017/S0305004100071620 
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