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On Functional Decomposition of Multivariate Polynomials with Differentiation and Homogenization
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 Faugere,
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