  A well-known generalisation of positional numeration systems is the case
where the base is the residue class of $x$ modulo a given polynomial $f(x)$
with coefficients in (for example) the integers, and where we try to construct
finite expansions for all residue classes modulo $f(x)$, using a suitably
chosen digit set. We give precise conditions under which direct or fibred
products of two such polynomial number systems are again of the same form. The
main tool is a general form of the Chinese Remainder Theorem. We give
applications to simultaneous number systems in the integers.
