Counting decomposable univariate polynomials

Details Counting decomposable univariate polynomials Counting decomposable univariate polynomials

2008-12-31

arxiv.org/abs/0901.0054v2

Mathematics

Commutative Algebra

Scientific paper

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 the dimension (over an algebraically closed field) of the set of decomposables, and an approximation to their number 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 error bounds. The wild case, where p divides n, is more challenging and our error bounds are weaker.

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