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Counting Decomposable Univariate Polynomials

Published online by Cambridge University Press:  05 September 2014

JOACHIM VON ZUR GATHEN
JOACHIM VON ZUR GATHEN*
Affiliation:
B-IT, Universität Bonn, D-53113 Bonn, Germany (e-mail: gathen@bit.uni-bonn.de) http://cosec.bit.uni-bonn.de/
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Abstract

A univariate polynomial f over a field is decomposable if it is the composition f = g ○ h of two polynomials g and h whose degree is at least 2. We determine an approximation to the number of decomposables over a finite field. The tame case, where the field characteristic p does not divide the degree n of f, is reasonably well understood, and we obtain exponentially decreasing relative error bounds. The wild case, where p divides n, is more challenging and our error bounds are weaker.

Keywords

Primary 68W30Secondary 11T0612E10

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Type
Paper
Information
Combinatorics, Probability and Computing , Volume 24 , Special Issue 1: Honouring the Memory of Philippe Flajolet - Part 3 , January 2015 , pp. 294 - 328
Copyright
Copyright © Cambridge University Press 2014 

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