# A series that is rational?

Let $$k=\mathbb F_q(T)$$. Can one prove (or disprove) that the series $$\sum_{n\ge0}(1-TX^{q^n})Y^{q^n}\in k[[X,Y]]$$ belongs to $$k(X,Y)$$? At first, it looked like it was simple. But in fact, I have no clue to attack this question. I thought about Dwork-Polya-Bertrandias theorem, but I did not find a several variables version of this theorem.

If you set $$T=0$$ or $$X=0$$ then you get the series $$\sum_{n\geq 0} Y^{q^n}$$. This cannot be rational because a rational power series in one variable that is not a polynomial cannot have arbitrarily long sequences of 0 coefficients (since the coefficients satisfy a linear recurrence relation with constant coefficients).
• @joaopa letting $k$ be a field of characteristic $p$, the formal power series $f(Y) = \sum_{n \geq 0} Y^{q^n}$ in $k[[Y]]$, where $q$ is a power of $p$, is not in $k(Y)$ since $f^q - f - Y = 0$ and this relation is impossible for $f$ being in $k(Y)$: that $Z^q - Z - Y$ is monic in $Z$ and vanishes at $Z = f$ would force $f$ to be in $k[Y]$ with positive degree, but then $\deg(f^q - f) > 1 = \deg(Y)$, so $f^q - f - Y$ can't be $0$. Jul 11, 2020 at 1:39