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Abstract:
A classical theorem of Gerstenhaber and Motskin-Taussky states that
the dimension of the algebra generated by a pair of commuting n x n
matrices is at most n. We study the geometry underlying the proof and
discuss the status of the following open problem: is the dimension of
the algebra generated by a triple of commuting n x n matrices also
bounded by n? Along the way, we will discuss algebraic varieties, the
Zariski topology, irreducible components, and related wondrous beasts.
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