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The trace algebra C_{nd} is generated by all traces of products of d genericn x n matrices. Minimal generating sets of C_{nd} and their defining relationsare known for n < 3 and n = 3, d=2. This paper states a minimal generating setand their defining relations for n=d=3. Furthermore the computations yield adescription of C_{33} as a free module over the ring generated by a homogeneoussystem of parameters.

A presentation of the trace algebra of three 3x3 matrices