Jean-Charles Faugère
ABSTRACT
We consider the composition f =g o h of two systems g= (g0, ..., gt) and h=(h0, ..., hs) of homogeneous multivariate polynomials over a field K, where each gj ∈ K[y0, ..., ys] has degree ℓ each hk ∈ K[x0, ..., xr] has degree m, and fi = gi(h0, ..., hs) ∈ K[x0, ..., xr] has degree n = ℓ · m, for 0 ≤ i ≤ 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 ℓ 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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Index Terms
- Decomposition of generic multivariate polynomials
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