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High order derivatives and decomposition of multivariate polynomials

In this paper, we present an improved method for decomposing multivariate polynomials. This problem, also known as the Functional Decomposition Problem (FDP) [17, 9, 27], is classical in computer algebra (e.g. [17, 18, 19, 23, 24, 7, 25]). Here, we propose to use high order partial derivatives to improve the algorithm described in [14]. Our new approach is more simple, and in some sense more natural. From a practical point of view, this new approach will lead to more efficient algorithms. The complexity of our algorithms will depend of the degree of the input polynomials, and the ratio n/u between the number of variables/polynomials.