# Solving linear inequalities over rings

The concrete trouble: for any kind of offered $N\ge 1$ I have a system of $2^N-1$ straight inequalities over $\mathbb{Z}_6^N$ which resembles this: for every single nonempty $S\subseteq[N]$ there is some $b_S\in\mathbb{Z}_6$ and also the inequality $\sum_{i\in S}x_i\ne b_S$. I intend to locate a remedy to all the inequalities simultaneously, certainly.

Exists a reliable means to locate remedy to such a system? To count the variety of remedies? To examine whether a remedy exists? Additionally, what around extra basic instances (any kind of variety of inequalities, any kind of sort of ring, no constraint on the coefficients of the variables in the inequalities)?

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2022-07-25 20:46:55
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