Counting decomposable univariate polynomials
Commutative Algebra
2019-02-20 v2 Combinatorics
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 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.
Cite
@article{arxiv.0901.0054,
title = {Counting decomposable univariate polynomials},
author = {Joachim von zur Gathen},
journal= {arXiv preprint arXiv:0901.0054},
year = {2019}
}