Equations in the Hadamard ring of rational functions

Let k be a number field. It is well known that the set of sequences composed by Taylor coefficients of rational functions over k is closed under component-wise operations, and so it can be equipped with a ring structure. A conjecture due to Pisot asks if (after enlarging the field) one can take d-th roots in this ring, provided d-th roots of coefficients can be taken in k. This was proved true in a preceding paper of the second author; in this article we generalize this result to more general equations, monic in Y, where the former case can be recovered for g(X,Y)=X^d-Y=0. Combining this with the Hadamard quotient theorem by Pourchet and Van der Poorten, we are able to get rid of the monic restriction, and have a theorem that generalizes both results.