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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?

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Yeah, I think so. One can solve a general linear recurrence over an alg closed field: the general solution is that $\omega_a$ is a sum of things of the form $h(a).x^a$ with $h$ a polynomial and $x\in F$. Now your second assumption implies $h(a)^2=h(2a)$ for all integers $a\geq0$, but you are in char 2 so $h(2a)=h(0)=c$, the constant term, and so $h(a)$ is the unique (again as you're in char 2) square root of $c$ for all $a$, so $h$ may as well be replaced by a constant function $c$ satisfying $c=c^2$, and $c=1$ is the only interesting solution, giving you the solution you already spotted. – Kevin Buzzard Oct 27 '10 at 22:02
up vote 0 down vote accepted

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 $a \ge 1$.


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No worries. Make your answer community-wiki and accept it, and then this stops the problem re-appearing on the front page in 6 months time. – Kevin Buzzard Oct 31 '10 at 8:43

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