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The Gerstenhaber problem for commuting triples of matrices is “decidable”

Abstract The Gerstenhaber Problem (GP) asks whether Gerstenhaber’s surprising 1961 theorem for two commuting matrices extends to three commuting n × n matrices A, B, C over a field F: must the (unital) subalgebra of generated by A, B, C have dimension at most n? We show the GP reduces to the prime fields and . Moreover, the GP is “decidable” in terms of Turing computability (a huge reduction in complexity). We also provide some more evidence pointing to a negative answer to the problem.