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Decomposition of Generic Multivariate Polynomials


We consider the composition $f = g \circ h$ of two systems $g = (g_0, \dots, g_t)$ and $h = (h_0, \dots, h_s)$ of homogeneous multivariate polynomials over a field $K$, where each $g_j \in K[y_0, \dots, y_s]$ has degree $\ell$ each $h_k \in K[x_0, \dots, x_r]$ has degree $m$, and $f_i = g_i(h_0, \dots, h_s) \in K[x_0, \dots, x_r]$ has degree $n = \ell . m$, for $0 \leq i \leq t$. The motivation of this paper is to investigate the behavior of the decomposition algorithm Multi-ComPoly proposed at ISSAC'09 [18]. We prove that the algorithm works correctly for generic decomposable instances -- in the special cases where $\ell$ is $2$ or $3$, and $m$ is $2$ -- and investigate the issue of uniqueness of a generic decomposable instance. The uniqueness is defined w.r.t. the "normal form" of a multivariate decomposition, a new notion introduced in this paper, which is of independent interest.
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hal-01292620 , version 1 (23-03-2016)



Jean-Charles Faugère, Joachim Gathen, Ludovic Perret. Decomposition of Generic Multivariate Polynomials. ISSAC 2010 - 35th International Symposium on Symbolic and Algebraic Computation, Jul 2010, Munich, Germany. pp.131-137, ⟨10.1145/1837934.1837963⟩. ⟨hal-01292620⟩
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