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

On Functional Decomposition of Multivariate Polynomials with Differentiation and Homogenization