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The Frobenius--Cecioni theorem states that if $A$ and $B$ are square matrices with entries in a field $k$ then the dimension of the $k$ vector space of solutions of $$ AX=XB $$ is given by the sum $$ \sum_{i,j} \deg( \gcd(d_i(A),d_j(B))) $$ Here $d_i(\cdot)$ denotes the $i$th invariant factor of its argument. My question: How well known is this theorem in any particular context? (E.g., "I use it all the time and I work in number theory ...", "I've never heard of it and I taught abstract algebra for years ...") Are there any well known applications of this result?

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Great, so now I know how it is called. I came across a corollary some time ago (C. Koc, AMM 2002, solution of problem #10813) and posted it on the MathLinks forum. From this corollary either "loup blanc" (another poster in that topic) or me reconstructed the theorem, at least in the case of an algebraically closed field.

I still do not know an answer to the natural question: For a given $n$, what are the possible values of $m$ such that there exist two $n\times n$ matrices $A$ and $B$ with $\dim\left\lbrace X\in k^{n\times n}\mid AX=XB\right\rbrace = m$ ? This is equivalent to asking for the possible values of $\sum\limits_{i} a_ib_i$ where $\left(a_i\right)$ and $\left(b_i\right)$ are two finite sequences of nonnegative integers satisfying $\sum\limits_i a_i\leq n$ and $\sum\limits_i b_i\leq n$. This is a purely combinatorial question. Maybe you happen to know an answer to it, since I am hardly the first person to pose it?...

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