# Probability of summing products of irreducible polynomials in a finite field to zero

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.

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## 1 Answer

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.

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