Suppose that we have the following expression $$p^{-2}|E|^6+16p^{-1}|E|^4+p^{2}|E|^2-8p^{-\frac{3}{2}}|E|^5-2|E|^4+8p^{\frac{1}{2}}|E|^3,$$ where $p$ is a prime number.


1. I was wondering is it possible to find some $C>0$ such that if $|E|\geq Cp$ then the above inequality is positive?
2. If the above is not possible then can we find some $C>0$ and $d\in (1,2)$ such that if $|E|\geq Cp^d$ then the above inequality is positive?


I would very thankful for your help because I have tried the first case but I failed. Maybe I am doing something it in a wrong way.


And a more general question: what is method to find this kind of $C$ and $d$?



Finding the optimal size of $|E|$ for specific inequality
Let's handle the lowest-order terms, treating $p$ and $|E|$ as degree $1$, using $p\geq 2$


$$p^{-2}|E|^6+16p^{-1}|E|^4+p^{2}|E|^2-8p^{-\frac{3}{2}}|E|^5-2|E|^4+8p^{\frac{1}{2}}|E|^3 $$ $$\geq p^{-2}|E|^6+0+p^{2}|E|^2-4 \sqrt{2} p^{-1}|E|^5-2|E|^4+0 $$
$$ = p^4 ( (|E|/p)^6 - 4\sqrt{2} (|E|/p)^5 - 2 (|E|/p)^4 + (|E|/p)^2 )$$ and we observe that the inner polynomial in $|E|/p$ has leading term positive, so clearly there is some $C$ (the largest real roo) such that this polynomial has positive value of $(|E|/p)>C$.


If wolfram alpha calculated the roots correctly, we can take $C=6$.


The only way to improve this value of $C$ is by shrinking the term $ -4 \sqrt{2} p^{-1}|E|^5$, as the other inequalities we used are sharp for $p$ large.


One general principle for bounds of this type is is "find the right notion of "leading term", then examine the leading term". In this case, we're looking for the term that is largest when $|E|/p$ grows as $p$, potentially grows very large, so we look for terms that have the highest joint degree in $|E|$ and $p$, and, among those, the highest degree in $|E|$.


