A computing strategy and programs to resolve the Gerstenhaber Problem for commuting triples of matrices

2020/06/15 by Holbrook, John, O'Meara, Kevin C.
#13E10 #13E15 #15A21 #15A27 #68-04 #Combinatorics (math.CO) #Commutative Algebra (math.AC) #FOS: Mathematics

paper · doi:10.48550/arxiv.2006.08588

Abstract

We describe a MATLAB program that could produce a negative answer to the Gerstenhaber Problem by the construction of three commuting n × n matrices A,B,C over a field F such that the subalgebra F[A,B,C] they generate has dimension greater than n. This problem has remained open for nearly 60 years, following Gerstenhaber's surprising result (Annals Math.) that dim F[A,B] ≤ n for any two commuting matrices A,B. The property fails for four or more commuting matrices. We also make the MATLAB files freely available.

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