(This question arose during group work on classifying modular tensor categories.)
Let $p$ and $q$ be two distinct primes. We seek integral solutions $\lbrace x_{i,j} \rbrace$ to the equation
\begin{align} 0 = \sum_{i=0}^{i=m} \sum_{j=m-i}^{j=m} p^i q^j x_{i,j} \end{align} subject to the requirements
\begin{align} x_{m,m}=-1, \\ p \nmid x_{i,j}, \\ q \nmid x_{i,j}. \end{align}
For example, when $p=5$, $q=7$, and $m=3$, we want to find integral solutions to the equation
\begin{align} 5^3 7^3 = 5^1 7^2 x_{1,2} + 5^2 7^1 x_{2,1} + 5^3 7^0 x_{3,0} + 5^0 7^3 x_{0,3} + 5^2 7^2 x_{2,2} + 5^3 7^1 x_{3,1} + 5^1 7^3 x_{1,3} + 5^3 7^2 x_{3,2} + 5^2 7^3 x_{2,3} \end{align}
such that none of the non-zero $x_{i,j}$ are divisible by 5 or 7. (We do allow $x_{i,j}$ to equal $0$.)
Question 1. What algebraic geometry would be necessary to attack this problem? It seems related to "finding integral points," so it could be much more difficult than I appreciate.
Question 2. If determining integral solutions is too hard: Is there a way we could at least give a bound for the sum of all the $x_{i,j}$ in terms of $m$? That is, can we say
\begin{align} \sum_{i=0}^{i=m} \sum_{j=m-i}^{j=m} x_{i,j} \, < \, f(m) \end{align}
for some function $f(m)$ that doesn't grow too fast?