I want to approach linear equations of the following form over the integers $\mathbb{Z}$:
$$x_1 + \cdots + x_n = 0.$$
I stepped over the sum set, which is defined as follows:
$$S + T = \{ x + y \mid x \in S, y \in T \}.$$
Now assume we know that $x_i \in S_i^0$, where each $S_i^0$ is a finite integer set. One can define the following mapping which sends $(S_1^k,\dots,S_n^k)$ to $(S_1^{k+1},\dots,S_n^{k+1})$:
$$S_i^{k+1} = (- \sum_{j \neq i} S_j^k) \cap S_i^k.$$
How many iterations $l$ does it take until the mapping reaches a fixed point, i.e. what is the smallest $l$ such that $(S_1^l,\dots,S_n^l) = (S_1^{l+1},\dots,S_n^{l+1})$? Is there a more direct way to compute the fixed point?
Bye