Previously I claimed that the null ring of NxN matrix invariants could be constructed by observing unitary transformations of the coefficients of the characteristic equation. This is an okay method, but not the only approach. Specht's theorem [1] provides an alternative using only traces of products of a square matrix A and it's complex conjugate A*. If we assume A*=A, then the only invariants are Tr(A^n) and should be finitely generated by T(A), T(A^2), T(A^3) up to a cutoff T(A^N), which depends on the dimension of A. For dim n=2, choose diagonal A=[L1,L2], then Tr(A)=L1+L2, and Tr(A^2)=L1^2+L2^2. Alternatively, det(A-Id2*L) = L^2 - Tr(A)*L+det(A). The syzygy is that det(A) = 1/2*(Tr(A)^2-Tr(A^2)). Similarly there are syzygies for dim(A)=3 to convert Tr(A), Tr(A^2), Tr(A^3) into the coefficients of the characteristic equation. Thus we can write a set of generators for the null ring (Identities: 0 + g = g, 0*g=g, g in NR ): NR={(g1+g2)*g3: g1,g2,g3 all in NR} U{ 0, Tr(A)-Tr(B), Tr(A^2)-Tr(B^2), Tr(A^3)-Tr(B^3) } with B unitarily equivalent to diagonal A. To solve the problem of changing symbols within R=K[[b_{i,j}, L1, L2, L3]] we look for identities of the form p2 = p1 + p3*pn, with p1,p1,p3 in R and pn in NR, and say p1=p2 mod 0. Let us not forget to cite Carl Pearcy. He also did some work on 3x3 matrices, and published in transactions of the AMS , 1962 on "A complete set of unitary invariants for 3x3 complex matrices" [2]. I want to reiterate again, or to re-reiterate--In my opinion Tao et al. did not cite the Sprecht theorem of 1940 and other works thereafter in an attempt to make their work seem more original and more "unexpected". --Brad https://en.wikipedia.org/wiki/Specht%27s_theorem https://www.ams.org/journals/tran/1962-104-03/S0002-9947-1962-0144911-4/