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Given two matrices $A,B \in M_n(\mathbb{C})$, is there any way of telling if there exists a single $P \in M_n(\mathbb{C})$ such that $PAP^{-1}$ and $PBP^{-1}$ are integer matrices?

This post describes the necessary and sufficient conditions for any single matrix to be conjugate to an integer matrix, so I would imagine $A$ and $B$ must satisfy at least these conditions. However, I am aware that this problem of simultaneous conjugacy at least when it comes to diagonalizability is very difficult without further restrictions, so it's possible this problem may also be very hard. I am actually looking to apply this when $n=3$ in particular, so maybe answering just the $3\times 3$ case may be more feasible.

Dalop
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    @MishaLavrov I fixed my question. I meant to ask when A and B are conjugate to some integer matrices (not necessarily the same) via the same conjugating matrix P. – Dalop Mar 22 '24 at 01:56
  • @Misha, it is possible for $A$ to be conjugate to two different integer matrices, no? – Gerry Myerson Mar 22 '24 at 01:56
  • One observation: if $PAP^{-1}$ and $PBP^{-1}$ are integer matrices, so are $P(A+B)P^{-1}$ and $PABP^{-1}$ and $P(A^3 + 3A^2B + B^4)P^{-1}$, and so we can try to check the condition in the linked post for any integer polynomial in $A$ and $B$. – Misha Lavrov Mar 22 '24 at 02:08

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