General polynomial decomposition and the s-1-decomposition are NP-Hard

Abstract

In the past few years, much work has been done on the functional decomposition of polynomials. Beginning with the first polynomial time algorithm of Kozen and Landau1 for the decomposition of a univariate polynomial in the “tame” case, significant progress has been made toward polynomial time algoithms for the more general cases: decomposition of multivariate polynomials, and decomposition in the “wild” case.2−8 However it has remained an open problem whether general multivariate decomposition is in P. In this paper, we present a basic form for the general polynomial decomposition problem which encompasses most forms of previously examined decomposition problems, and then prove that the problem is NP-Hard by proving that a sub-problem called the S-1-Decomposition problem is NP-Hard.