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.