# Questions tagged [formal-power-series]

This tag is for questions relating to "formal power series" which can be considered either as an extension of the polynomial to a possibly infinite number of terms or as a power series in which the variable is not assigned any value.

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### Show that the ring of formal power series in a commutative ring $R$, $R[[x]]$ is noetherian.

Yes, I am aware that this has been answered (If $R$ is a Noetherian ring then $R[[x]]$ is also Noetherian), but the answers given did not answer my specific question regarding this: In my notes from a ...
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### Transcendence of meromorphic function vs formal power series

Consider the meromorphic function $f$ on $\mathscr D=\{z\in\mathbb C\mid|z|<1\}$ definied by $\displaystyle f(z)=\sum_{n\ge1}\frac{z^{2^n}}{z^{2^n}-\frac12}$. Obviously $f$ admits infinitely many ...
1 vote
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### Find a formula for summation of a increasing term (formal power series)

I am reading the exercises in the Combinatorics book written by Miklos Bona and trying to solve this exercise of 2.10. Find an formula for $a_n = \sum_{i=0}^{n-1} (i+1)a_i$ , where $a_0 = 1$ I've ...
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### Seeking more alternate proofs of a combinatorial generating function identity $G(x)=\overline{G}(-x)^{-1}$ related to counting strings.

Let $\mathcal{S}=[m]^*$ be the set of all strings on the alphabet $[m]=\{1, 2,\cdots, m\}$. Let $\Sigma\subset[m]^2$ be a set of strings of length $2$, and let $\overline{\Sigma}=[m]^2\backslash\Sigma$...
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### Definition of formal path in the group of diffeomorphisms

In M.Kontsevich's paper about deformation quantization https://arxiv.org/pdf/q-alg/9709040.pdf. Page3. He defines formal Poisson structure as the set of equivalence classes of Poisson structures ...
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1 vote
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### Lang's proof of Euclidean algorithm for power series

I have a question about the use of projections in Lang's proof of the Euclidean division algorithm for power series (Algebra - Serge Lang, Chapter IV, section 9, Theorem 9.1). Specifically, there is a ...
64 views

### Why do we care if a power series has roots

I am reading up on Christol's theoreom and an important part is that k-uniform transducers (where k is somehow related to prime numbers) preserve the algebricity of a formal power series (taking the ...
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### There is $f$ nonnilpotent in $A[[x]]$ with nilpotent coefficients -- solution verification [duplicate]

This is an exercise in Atiyah-MacDonald (Exercise 5.2). I have shown that if $f \in A[[x]]$ ($A$ commutative with unity) is nilpotent, then $f$'s coefficients are all nilpotent. The next question in ...
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### Why isn't the tensor series an algebra, while the tensor algebra is?

I am looking to understand why the free tensor series, denoted $T((V))$, is not an algebra, while the free tensor algebra, denoted $T(V)$, is. To clarify definitions: Let $V$ be a vector space over a ...
I am studying for an exam and I am stuck on solving this problem: Check the associativity for the formal series $F(G(H(z))) = F(G(z))(H(z))$ whenever it is defined. In one hand, it seems ...