Linear equations over multiplicative groups recurrences, and mixing III

Given an algebraic $Z^d$-action corresponding to a prime ideal of a Laurent ring of polynomials in several variables, we show how to find the smallest order n+1 of non-mixing. It is known that this is determined by the non-mixing sets of size n+1, and we show how to find these in an effective way. When the underlying characteristic is positive and $n\ge 2$, we prove that there are at most finitely many classes under a natural equivalence relation. We work out two examples, the first with 5 classes and the second with 134 classes