Lower bounds for decomposable univariate wild polynomials

Authors:
Joachim Von Zur Gathen
Affiliations:
B-IT, Universität Bonn, D-53113 Bonn, Germany
Venue:
Journal of Symbolic Computation
Year:
2013

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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. The tame case, where the field characteristic p does not divide the degree n of f, is reasonably well understood. The wild case, where p divides n, is more challenging. We present an efficient algorithm for this case that computes a decomposition, if one exists. It works for most but not all inputs, and provides a reasonable lower bound on the number of decomposable polynomials over a finite field. This is a central ingredient in finding a good approximation to this number.