Decomposition of polynomials and approximate roots

TL;DR

This paper introduces a polynomial decomposition method based on a Euclidean division theorem, providing algorithms to compute approximate roots and determine polynomial decomposability.

Contribution

It presents a novel Euclidean division theorem for polynomials, defines approximate roots, and offers an algorithm for polynomial decomposition and decomposability testing.

Findings

Existence of a Euclidean division theorem for polynomials

Algorithm for computing polynomial decompositions

Application to polynomial decomposability decision

Abstract

We state a kind of Euclidian division theorem: given a polynomial P(x) and a divisor d of the degree of P, there exist polynomials h(x),Q(x),R(x) such that P(x) = h(Q(x)) +R(x), with deg h=d. Under some conditions h,Q,R are unique, and Q is the approximate d-root of P. Moreover we give an algorithm to compute such a decomposition. We apply these results to decide whether a polynomial in one or several variables is decomposable or not.