I would like to know what are the formal power series $$f(t) = \sum\_a \omega\_a t^{-a}$$ over an algebraicially closed field of characteristic 2, with two properties: (1) The series represents a rational function, i.e. the coefficients satisfy a linear recursion, and (2) $\omega\_{2a} = \omega\_a^2$ for $a \ge 0$.


One family of solutions is $\omega\_a = p\_a(u\_1, \dots, u\_r)$ where $p\_a$ is the $a$-th power sum symmetric function in some finite subset of $F$, $p\_a = \sum\_{i = 1}^r u\_i^a$.


Are these (more or less) all the solutions?



some rational functions over a field of characteristic 2
Kevin Buzzard gave the solution. Here it is with a little more detail:


Our assumptions include $\omega\_0 = \omega\_0^2$. Thus $\omega\_0 \in \{0, 1\}$.


The linear homogeneous recursion only kicks in eventually; say the $\omega\_a$ for $a \ge N$ satisfy such a recursion.


Let $v\_1, \dots, v\_m$ be the distinct roots of the characteristic polynomial of the linear recursion. Then there exist polynomials $h\_1, \dots, h\_m$ such that
$\omega\_a = \sum\_{i = 1} ^m h\_i(a) v\_i^a $ for $a \ge N$. Let $\alpha\_i$ be the constant term of 
$h\_i$ for each $i$. Since the characteristic is $2$, we have $h\_i(2a) = \alpha\_i$ for all $a$.  

For $a \ge N$,  

\begin{equation} 
 \sum\_i \alpha\_i v\_i^{4a} = \omega\_{4 a} = \omega\_{2a}^2 = \sum\_i \alpha\_i^2 v\_i^{4a}.
\end{equation}
Because the characteristic of $F$ is $2$, each element has a unique $2^k$--th root for all $k \ge 1$; in particular all the $v\_i^4$ are distinct, so the displayed equation implies that $\alpha\_i^2 = \alpha\_i$ for all $i$, i.e. $\alpha\_i \in \{0, 1\}$. Let $u\_1, \dots, u\_d$ be the list of those $v\_j$ such that $\alpha\_j = 1$. Then we have $\omega\_{2a} = \sum\_i u\_i^{2a}$ for
$a \ge N$. For an arbitrary $a \ge 1$, chose $k$ such that $2^{k-1} a \ge N$. Then
$\omega\_a$ is the unique $2^k$--th root of $\omega\_{2^k a} = \sum\_i u\_i^{2^k a}$, namely
$\omega\_a = \sum\_i u\_i^a$. 


Thus we have $\omega\_0 \in \{0, 1\}$ and $\omega\_a = p\_a(u\_1, \dots, u\_d)$ for $a \ge 1$.


THANKS, KEVIN ! 


