On Functional Decomposition of Multivariate Polynomials with Differentiation and Homogenization

TL;DR

This paper provides a theoretical analysis of algorithms for functional decomposition of multivariate polynomials, demonstrating high-probability success for quartic homogenous polynomials and confirming a longstanding conjecture.

Contribution

It proves that the algorithms can compute degree proper decompositions for randomly decomposable quartic polynomials with high probability, confirming a conjecture and extending understanding of polynomial decomposition.

Findings

High-probability success for quartic homogenous polynomials

Confirmation of a conjecture on decomposition from homogenization

Decomposition algorithms work with probability one over characteristic zero fields

Abstract

In this paper, we give a theoretical analysis for the algorithms to compute functional decomposition for multivariate polynomials based on differentiation and homogenization which are proposed by Ye, Dai, Lam (1999) and Faug$\mu$ere, Perret (2006, 2008, 2009). We show that a degree proper functional decomposition for a set of randomly decomposable quartic homogenous polynomials can be computed using the algorithm with high probability. This solves a conjecture proposed by Ye, Dai, and Lam (1999). We also propose a conjecture such that the decomposition for a set of polynomials can be computed from that of its homogenization with high probability. Finally, we prove that the right decomposition factors for a set of polynomials can be computed from its right decomposition factor space. Combining these results together, we prove that the algorithm can compute a degree proper decomposition for a set of randomly decomposable quartic polynomials with probability one when the base field is of characteristic zero, and with probability close to one when the base field is a finite field with sufficiently large number under the assumption that the conjeture is correct.