Let $q$ be a prime power and $\mathbb{F}_q$ be the field of $q$ elements. Let $\mathbb{F}_q[t]$ be the polynomial ring over $\mathbb{F}_q$. I am working on a research project, where I bounded a certain expectation $E$ by $E \ll S + S'$, where $$ S = \sum_{ \substack{ x_i \in \mathbb{F}_q[t] \backslash \{ 0 \} \ (1 \leq i \leq 6) \\ x_1 + x_2 + x_3 =n \\ x_1 + x_4 = x_5 + x_6 } } q^{ -\gamma (\deg x_1 + \deg x_2 + \deg x_3 + \deg x_4 + \deg x_5 + \deg x_6 ) }, $$ $$ S' = \sum_{ \substack{ x_i \in \mathbb{F}_q[t] \backslash \{ 0 \} \ (i = 1,2,3,5,6 ) \\ x_1 + x_2 + x_3 =n \\ x_1 + x_2 = x_5 + x_6 } } q^{ -\gamma (\deg x_1 + \deg x_2 + \deg x_3 + \deg x_5 + \deg x_6 ) }. $$ I managed to find a bound for $S$ and $S'$ by "brute force" and found a bound for $E$. What I Was wondering was is there possibly an easy or obvious way to see that $S' \ll S$? The reason why I ask is I had many cases to consider and I am looking for a way to cut down on the amount of computation I wrote up (and many of the cases look like what I have here). I would greatly appreciate any advice. Thanks!

ps Let $\gamma > 0$ be a positive real number sufficiently large so that both sums converge.


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