A presentation of the trace algebra of three 3×3 matrices

The trace algebra Cnd is generated by all traces of products of d generic n×n matrices. Minimal generating sets of Cnd and their defining relations are known for n<3 and n=3, d=2. This paper states a minimal generating set and their defining relations for n=d=3. Furthermore the computations yield a description of C33 as a free module over the ring generated by a homogeneous system of parameters.