Chomsky Schützenberger enumeration theorem In many textbooks the Chomsky-Schützenberger enumeration theorem is stated as that the characteristic formal power series of a language is $\mathbb N$-algebraic, if the grammar is unambigious. In some other books the formulation is given, that the structure describing function $$f_L(x)=\sum_{n \in \mathbb N}l_nx^n$$ is algebraic over $\mathbb Q$ iff the grammar is unambigious, where $l_n = \vert \Sigma^n \cap L \vert$.
I understand that by replacing every terminal by the same terminal in the system of equations associated with a grammar $G$ it is possible, to obtain the structure describing function, but I couldn't find a proof anywhere,(and don't know how I could construct one)  that then the (analytical) power series $f_L(x)$ is not transcendental.
There is a proof in the book of Kuich &amp; Salomaa, Semirings, Automata, Languages and another one in the paper of Panholzer, "Gröbner Bases and the Defining Polynomial of a Context-free Grammar Generating Function", J. of Automata, Languages and Combinatorics 10 (2005), 79–97.   I wish there were a simple and clear proof of the result.