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Let $G$ be a free abelian group (not necessarily finitely-generated) and $F$ be the fraction field of the group ring of $G$. Let $\Theta$ be the set of power series in $F[[t]]$ such that each nonzero coefficient is in $G$.

The following theorem of Larsen and Lunts ([1], Proposition 2.6) characterizes rational power series in $\Theta$.

Theorem. Let $f=\sum_{i\ge 0} g_i t^i\in \Theta$. Then $f$ is rational if and only if there exists a positive integer $n$ and sequence $h_0,h_1,\ldots,h_{n-1}\in G$ such that $g_{i+n}=h_{i \bmod n}g_i$ for $i$ large enough.

In their proof, they use the following lemma without proof (in second paragraph of the proof of Lemma 2.8).

Lemma. Let $f\in \Theta$ be rational. Let $q(t)$ be a polynomial in $F[t]$ such that $q(t) f(t)\in F[t]$. Let $m$ be the degree of $q$.

Let $f_1,\ldots,f_r\in \Theta$ be a sequence such that

(1) $f_1+\cdots+f_r=f$.

(2) Write $f_i=\sum_{j\ge 0} g_j^i t^j$. There exists a finite set $K\subset G$ such that for all $|j-k|\le m$ such that $g_j^i$ and $g_k^i$ are nonzero, we have $g_j^i(g_k^i)^{-1}\in K$.

(3) $r$ is minimal among all such sequences.

Then for any finite set $S\subset G$, there exists infinitely many intervals $I\subset \mathbb{N}$ of length $m+1$ such that for any $i,j\in I$, $k\ne l$ such that $g_i^k$ and $g_j^l$ are nonzero, we have $g_i^k (g_j^l)^{-1}\not \in S$.

I am trying to prove this lemma but fail. The obvious approach is to assume the converse, and produce a shorter sequence. So let us assume that there exists finitely such intervals. This means that for all $I$ with endpoints large enough, there exists $k\ne l$ and $i,j\in I$ such that $g_i^k$ and $g_j^l$ are nonzero, and $g_i^k (g_j^l)^{-1}\in S$. Then, for any such $I$ and $k\ne l$, for all $i,j\in I$, we have $g_i^k (g_j^l)^{-1} \in SK^2$.

However, this is not enough to combine $f_k$ and $f_l$. We need to know that there exists $k\ne l$ such that for all $I\subset \mathbb{N}$ with length $m+1$ and endpoints large enough, there exists $i,j\in I$ such that $g_k^i,g_l^j\ne 0$ and $g_i^k (g_j^l)^{-1}\in S$. This is true when $r=2$, but does not seem to be true in general.

Question: How to prove the lemma?

[1] Michael J. Larsen, Valery A. Lunts. Rationality criteria for motivic zeta-functions.

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  • $\begingroup$ Remark. In the published version of the paper, the authors removed this lemma and proved the theorem in a different way. $\endgroup$ – Yuzhou Gu Apr 25 '16 at 2:32

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