  This paper addresses two questions regarding N-polyregular functions, that
forms a proper subset of N-rational series. We show that given a Z-rational
series, it is decidable whether it is computable via a commutative
N-polyregular function, and provide a counter-example to the theorem of
Karhum\"aki that studied the same question in the case of polynomials. We also
prove that it is decidable whether a commutative N-polyregular function is
star-free, by proving the stronger statement that star-free Z-polyregular
functions that are N-polyregular are in fact computable using a star-free
N-polyregular function. Building towards answering the same questions in the
non-commutative case, we present a canonical model of computation of
N-polyregular functions by generalizing the notion of residual transducers
previously introduced in Z-polyregular functions.
