Let $f(x), g(x), h(x)$ be randomly chosen irreducible polynomials over the finite field $GF(2^n)$. 


What would the probability be for $\sum\_{(i,j,k:i,j,k\in\mathbb{N},i+j+k=C)} f^i(x)g^j(x)h^k(x)=0$, where $C$ is much smaller than $n$?


I want to prove that this probability tends to zero as $n$ increases.



Probability of summing products of irreducible polynomials in a finite field to zero
Taking a view from old-fashioned algebra: your sum can be interpreted from a generating function of a product of three geometric series. That generating function has a partial fraction decomposition, with numerators say P, Q, and R. If these were rational functions in x alone the issue would be like 


$Pf^m(x) + Qg^m(x) +Rh^m(x) = 0$


for certain values of m. There is no reason to expect this can happen more than three times, I think. 


Is there a proof in there? This is too long for a comment, really.


