I'm working on a ring $R=\mathbb{F}_7[x,y,z]/(x^2+y^3+z^5)$ and want to prove $x\notin(y,z)^*$, the tight closure.
First I want to find a test element, which can be obtained from the Jacobian ideal. So $x,y^2,z^4$ are all test elements, and I pick $x$.
So I need to prove that, for some $q=7^e$ arbitrarily large, $x^{q+1}\notin (y^q,z^q)$.
The method I use is to observe $q=2k+1$ and what we want in equivalent to
$$ x^{q+1}= (y^3+z^5)^{k+1}\notin (y^{2k+1},z^{2k+1}) $$
Then I want to show that there are some monomial $(y^3)^a(z^5)^{k+1-a}$ such that the powers of $y,z$ are both less than $2k+1$. This can be achieved with $$a= \text{integer part of } \frac{2k+1}{3}-1$$
Here's the trouble: I don't know if this monomial is indeed NONZERO? The coefficient is too big for huge $q$ so as of right now, it's impossible for me to check. For $q=7^2,7^3$, I have verified that such a nonzero monomial exists, but I cannot generalize it.