  We investigate decidability of equivalence of register transducers, also
called copyful Streaming String Transducers in case of string input, extended
with an operation of substituting a register for all occurrences of a given
letter in another register. We reduce to zeroness of polynomial grammars (over
ring of polynomials) extended with analogous substitution operation by encoding
strings into polynomials; a similar method was used successfully by Seidl et
al. in 2018. We give two restrictions under which register transducers with
substitution have decidable equivalence. They seem to be very restrictive but
on the other hand, they seem to be on the edge of the scope of this
"polynomial" method, as in the third result we give a rather restricted model
of polynomial grammars with substitution that has undecidable equivalence.
