  Equivalence testing for a polynomial family {g_m} over a field F is the
following problem: Given black-box access to an n-variate polynomial f(x),
where n is the number of variables in g_m, check if there exists an A in
GL(n,F) such that f(x) = g_m(Ax). If yes, then output such an A. The complexity
of equivalence testing has been studied for a number of important polynomial
families, including the determinant (Det) and the two popular variants of the
iterated matrix multiplication polynomial: IMM_{w,d} (the (1,1) entry of the
product of d many w $\times$ w symbolic matrices) and Tr-IMM_{w,d} (the trace
of the product of d many w $\times$ w symbolic matrices). The families Det, IMM
and Tr-IMM are VBP-complete, and so, in this sense, they have the same
complexity. But, do they have the same equivalence testing complexity? We show
that the answer is 'yes' for Det and Tr-IMM (modulo the use of randomness). The
result is obtained by connecting the two problems via another well-studied
problem called the full matrix algebra isomorphism problem (FMAI). In
particular, we prove the following:
  1. Testing equivalence of polynomials to Tr-IMM_{w,d}, for d$\geq$ 3 and
w$\geq$ 2, is randomized polynomial-time Turing reducible to testing
equivalence of polynomials to Det_w, the determinant of the w $\times$ w matrix
of formal variables. (Here, d need not be a constant.)
  2. FMAI is randomized polynomial-time Turing reducible to equivalence testing
(in fact, to tensor isomorphism testing) for the family of matrix
multiplication tensors {Tr-IMM_{w,3}}.
  These in conjunction with the randomized poly-time reduction from determinant
equivalence testing to FMAI [Garg,Gupta,Kayal,Saha19], imply that FMAI,
equivalence testing for Tr-IMM and for Det, and the $3$-tensor isomorphism
problem for the family of matrix multiplication tensors are randomized
poly-time equivalent under Turing reductions.
