I am looking for a general theorem which would tell me when a formal series solution exists for an equation over a semiring. One may assume that the semiring is equipped with a (formal) derivative. One may further restrict to the case $$A(X) = H(X, A(X))$$ where $A$ is a formal series in $X$ and $H(X,Y)$ is a polynomial.
Motivation: Trying to find a more precise set of conditions for which Joyal's implicit species theorem holds. [For example, one can easily lift the condition $H(0,0)=0$ to $H(0,0)$ is constant].