I created the following system of equations to solve a Sudoku puzzle.
It's easy to show that, if $a, b, c, \dots, i$ are distinct nonzero numbers, we have the following solution
$$S=\{(a_1,\dots,a_9) \mid a_i \in \{1,...,9\},a_i\neq a_j,\forall i\neq j\}$$
but how can I show this solution is unique?
I don't know the answer to your question. But I will note that you can achieve the same thing using just one Diophantine eqution. If you require that $$ x_1^{100}+x_2^{100} + \ldots x_{9}^{100} = 1^{100} +2^{100} + \ldots 9^{100}$$ then in fact $\{x_1, x_2, \ldots , x_9\} = \{1,2,3,4,5,6,7,8,9\}$ assuming you are looking for non-negative integer solutions. One can see first that it is impossible for any $x_i$ to exceed $9$ , since $10^{100}$ is in fact larger than the RHS of the equation. But if all the $x_i$ are at most $8$ , then the LHS is too small. So one of the variable equals $9$, say $x_9=9$. Subtract $9^{100}$ from both sides and proceed similarily to show to show that one variable equals $8$ and so on.