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Let $f$ be a homogeneous irreducible element in a graded commutative Noetherian ring. What is the possible set of elements $f_1$ and $f_2$ such that $f=f_1g_1+ f_2g_2$?

Even in very particular cases this is very interesting e.g. $f=x^4+y^4+z^4+t^4$ in $C[x,y,z,t]$. One may wish to study the Picard group or Neron-Severi, but a kind of algorithms or Grobner bases would be nice.

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  • $\begingroup$ I am not sure what kind of answer you are looking for. I will explain this with your example of $C[x,y,z,t]$. If $f$ is a very general homogeneous polynomial of degree at least 4, by Noether-Lefschetz, the associated projective surface has Picard group $Z$. You can see that this implies $f$ can not be written in any non-trivial (that is all the four terms above have positive degree) way as above. $\endgroup$
    – Mohan
    Apr 23, 2015 at 22:51
  • $\begingroup$ Thanks Mohan, the question concerns a given polynomial. The Fermat polynomial is of particular interest. For example if \alpha is a fourth root of unity then f_1=(x-\alpha y) and f_2=(z-\alpha t) are solutions. I wonder if there exists any other solution of different type. Of course in higher dimension, f=x_1^n+...x_r^n=\sum_1^m f_ig_i implies that m\geq n/2, by looking on the gradient. $\endgroup$ Apr 24, 2015 at 23:38

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