Decomposition of Polynomials

TL;DR

This paper presents a polynomial-time algorithm for decomposing polynomials, establishes bounds on minimal collisions, and proves a conjecture classifying a special class of decomposable polynomials.

Contribution

It introduces an efficient decomposition algorithm, provides bounds on collisions, and proves a conjecture on polynomial classification.

Findings

Algorithm computes polynomial decompositions in polynomial time

Derived bounds for the number of minimal collisions

Proved a classification conjecture for a special class of decomposable polynomials

Abstract

This diploma thesis is concerned with functional decomposition $f = g \circ h$ of polynomials. First an algorithm is described which computes decompositions in polynomial time. This algorithm was originally proposed by Zippel (1991). A bound for the number of minimal collisions is derived. Finally a proof of a conjecture in von zur Gathen, Giesbrecht & Ziegler (2010) is given, which states a classification for a special class of decomposable polynomials.