Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely:  consider the composition  f =g o h  of two systems  g= (g 0 , ..., g t )  and  h=(h 0 , ..., h s )  of homogeneous multivariate polynomials over a field K, where each  g   j   ∈ K[ y  0 , ...,  y   s  ] has degree e each  h   k   ∈ K[ x  0 , ...,  x   r  ] has degree  m , and  f   i   =  g   i  ( h  0 , ...,  h   s  ) ∈ K[ x  0 , ...,  x   r  ] has degree  n  = e ·  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].  prove that the algorithm works correctly for generic decomposable instances -- in the special cases where e 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.
['Jean-Charles Faugère', 'Joachim von zur Gathen', 'Ludovic Perret']
I am writing a research paper on a idea Decomposition of generic multivariate polynomials
[{"from": "human", "value": "I am writing a research paper on a idea Decomposition of generic multivariate polynomials"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely:  consider the composition  f =g o h  of two systems  g= (g 0 , ..., g t )  and  h=(h 0 , ..., h s )  of homogeneous multivariate polynomials over a field K, where each  g   j   \u2208 K[ y  0 , ...,  y   s  ] has degree e each  h   k   \u2208 K[ x  0 , ...,  x   r  ] has degree  m , and  f   i   =  g   i  ( h  0 , ...,  h   s  ) \u2208 K[ x  0 , ...,  x   r  ] has degree  n  = e \u00b7  m , for 0 \u2264  i \u2264 t . The motivation of this paper is to investigate the behavior of the decomposition algorithm  Multi-ComPoly  proposed at ISSAC'09 [18].  prove that the algorithm works correctly for generic decomposable instances -- in the special cases where e 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."}]