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I've got a finite set of vectors in $Z^3$, which I know to be of the form $A^{i_1}B^{i_2}A^{i_3}B^{i_4}\dots v$ for some unknown vector $v\in Z^3$ and two unknown matrices $A$ and $B$ with integer entries, and where $i_j$ are small non-negative integers. How do I compute $v$ and $A$ and $B$?

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  • $\begingroup$ What does "small" mean? $\endgroup$
    – Igor Rivin
    Commented Mar 20, 2015 at 20:08
  • $\begingroup$ Maybe 50 or 100. $\endgroup$
    – KeithB
    Commented Mar 26, 2015 at 16:40

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I don't have an answer, but with very high probability this question is undecidable, since it is clearly closely connected with the vector reachability problem for integral matrix semigroups, which is in general undecidable. For a very nice survey, see this paper by Halava, Harju, Hirvensalo (2006, but doubt that any great advances have occured since then).

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