Decomposition of generic multivariate polynomials

We consider the composition <i>f =g o h</i> of two systems <i>g= (g<sub>0</sub>, ..., g<sub>t</sub>)</i> and <i>h=(h<sub>0</sub>, ..., h<sub>s</sub>)</i> of homogeneous multivariate polynomials over a field K, where each <i>g</i><sub><i>j</i></sub> ∈ K[<i>y</i><sub>0</sub>, ..., <i>y</i><sub><i>s</i></sub>] has degree ℓ each <i>h</i><sub><i>k</i></sub> ∈ K[<i>x</i><sub>0</sub>, ..., <i>x</i><sub><i>r</i></sub>] has degree <i>m</i>, and <i>f</i><sub><i>i</i></sub> = <i>g</i><sub><i>i</i></sub>(<i>h</i><sub>0</sub>, ..., <i>h</i><sub><i>s</i></sub>) ∈ K[<i>x</i><sub>0</sub>, ..., <i>x</i><sub><i>r</i></sub>] has degree <i>n</i> = ℓ · <i>m</i>, for 0 ≤ <i>i ≤ t</i>. The motivation of this paper is to investigate the behavior of the decomposition algorithm <b>Multi-ComPoly</b> proposed at ISSAC'09 [18]. We prove that the algorithm works correctly for generic decomposable instances -- in the special cases where ℓ is 2 or 3, and <i>m</i> is 2 -- and investigate the issue of uniqueness of a <i>generic</i> 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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