Going through my Commutative Algebra notes I found out that I don't know the answer to this question.
Let $A$ be a commutative ring with unit and let $f(T):=\sum_{k=0}^\infty a_k T^k \in A[[T]]$ be a power series. It is a standard fact from Commutative Algebra that $1/f$ exists in $A[[T]]$ iff $a_0\in U(A)$, where $U(A)$ denotes the group of units of $A$.
Is there some characterization of all $f\in U(A[[T]])$ with the property that $1/f\in A[T]$?
Even though I have posed the problem for general $A$, I'd be happy to know the answer even for $A=\mathbb{C}$ only.
Some remarks:
- Such $f$ form a commutative submonoid of $U(A[[T]])$.
- If $f=\sum_{k=0}^n a_k T^k$ is in fact polynomial, then it is again a standard fact from Commutative Algebra that $1/f\in A[T]$ iff $a_0\in A^\times$ and $a_1,\dots,a_n\in\operatorname{nil}A$, so this special case is settled. Furthermore one might be inclined to conjecture for general $f=\sum_{k=0}^\infty a_k T^k$ that one needs $a_1, a_2, \ldots \in \operatorname{nil}A$, but this is clearly false as the simplest example $$ \frac{1}{1-T} = \sum_{k=0}^\infty T^k $$ shows.
- Writing $f=1/p$ for $p\in A[T]$ and $$ f(1/T) = \sum_{k=0}^\infty a_k T^{-k} = \frac{1}{p(1/T)} = \frac{T^m}{q(T)} $$ for some $q\in A[T]$ and $m=\deg p=\deg q$, this becomes a very special case of the problem of determining when a power series is a rational function in disguise.
- If $A=\kappa$ is an algebraically closed field, then $p(T)$ always splits into linear factors $p(T)=(T-\alpha_1)\ldots(T-\alpha_m)$, where wlog. we have assumed $a_0=1$, so $f$ is a product of a bunch of geometric series $$ f(T) = \sum_{k=0}^\infty a_k T^k = (-1)^m \bigg( \prod_{j=1}^m \frac{1}{\alpha_j} \bigg) \bigg(\prod_{j=1}^m \sum_{k_j=0}^\infty \frac{1}{\alpha_j^{k_j}} T^{k_j} \bigg) $$ In particular, $$ (-1)^m \bigg( \prod_{j=1}^m \frac{1}{\alpha_j} \bigg) = 1 $$ and $$ a_n = \sum_{k_1+\dots+k_m=n} \frac{1}{\alpha_1^{k_1} \ldots \alpha_m^{k_m}}, $$ but I don't see how this helps (if at all).
Is there at least an a priori way to determine $m$ in the case $A=\kappa$?