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 ...
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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
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Is $\mathbb{Z}_p \cong \mathbb{Z}[[X]]/(X-p)$?
Let $p$ be a prime number and denote by $\mathbb{Z}_p$ the ring of p-adic integers. Denote by $\mathbb{Z}[[X]]$ the ring of formal power series with integer coefficients and let $\frac{\mathbb{Z}[[X]]}...
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Metric Induced by order of formal power series
In the book I'm following the order of a formal real power series $a=\sum_{n=1}^{\infty} a_nx^n$ is defined as: $$o(a)=\begin{cases} \infty, \, \, a=0\\ \min\{n:a_n \neq 0\}, \, \, a \neq 0 \end{cases}...
- 143
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A certain formal power series equals the zeta function of a curve
Be $X$ a projective, geometrically connected and smooth curve over the finite field $\mathbb{F}_q$. Then one can define the following series, i.e. the zeta function of the curve: $$Z(t) = \exp\left(\...
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Isomorphism between quotients of two variable formal power series ring
$k$ is a field whose characteristic is not $2$, and $f(x,y)=x^2-y^2, g(x,y)=x^2+x^3-y^2$.
Exercise. Show that $k[[x,y]]/(f)\simeq k[[x,y]]/(g)$.
So far, I've shown the following.
$k[[x,y]]$ is an ...
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Freeness of the algebra of formal power series?
Let $\mathbb{k}$ be a field and $A$ be a commutative $\mathbb{k}$-algebra. Then the algebra of formal power series $A[[x]]$ can be viewed as a $\mathbb{k}[[x]]$-module in a natural way. My question is ...
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Is the ring of formal power series a localization of some non local ring at prime ideals?
Consider $F = K[[x]],$ the formal series over field $K.$ We know that $K$ is a local ring with maximal ideal $(x).$ Does there exist a non local ring $R$ and prime ideal $p$ such that $R_p = K[[x]]?$ ...
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Evaluation of a rational function vs formal power series
Let $K$ be a complete valued field and $X$ be an indeterminate $K[X]$ be the ring of polynomials over $K$ in the indeterminate $X$ and $K(X)$ be its quotient field.
For $a\in K$, consider the ...
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notation for creating formal power series and $R-module$
The following is taken from "Module Theory an approach to linear algebra" by T.S Blyth
$\color{Green}{Background:}$
$\textbf{(1) Example}$
Let $R$ be a unitary ring and let $R^N$ denote the ...
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Affineness of the algebra of formal power series?
Let $\mathbb{k}$ be a field and $A$ be a commutative $\mathbb{k}$-algebra. It is clear that when $A$ is an affine $\mathbb{k}$-algebra (that is, $A$ is finitely generated as a $\mathbb{k}$-algebra), ...
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Condition for convergence of series of formal power series
Proposition 1.1.8 in this answer, from "Enumerative Combinatorics" by Stanley, says
The infinite series $\sum_j F_j(x)$ converges if and only if $\lim_{j \to \infty} \deg(F_j(x)) = \infty$.
...
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Is a formal power series a fraction of polynomials?
We consider formal power series over a field. Power series such as $x+x^2+ x^3 +\cdots$ can be expressed as fraction of polynomials where the denominator is a unit in the power series ring. There are ...
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Convergence of formal powers seires implies convergence of evaluation
For every $n\in\mathbb N$, let $f_n(Z)=\sum_{k\ge0}f^{(n)}_kZ^k\in\mathbb R[[Z]]$ be a formal power series. One assumes that there exists a positive real number $C$ such that for all $(n,k)\in\mathbb ...
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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 ...
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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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Krull Dimension of $\mathbb{C}[[X,Y]]/(Y^2)$
I want to compute the Krull Dimension of the ring $\mathbb{C}[[X,Y]]/(Y^2)$. I have tried the following and would be glad if somebody could verify or point out mistakes in the following proof:
Let $y=...
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Power series expansion of $\frac{kx}{(x+1)^k-1}$ at $x=0$
The question is simple: for integer $k\geq 2$, how to calculate the power series expansion of $f(x)=\frac{kx}{(x+1)^k-1}$ at $x=0$?
As far as I am aware, there are two effective (operatable) ways to ...
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Do pro-objects in a monoidal category have a completed tensor product?
Given a monoidal category C it seems natural to define a completed tensor product in its category of pro objects Pro(C) by
$$("lim_{a\in A}" X_a) \hat\otimes ("lim_{b\in B}" X_b)="lim_{(a,b)\in A\...
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Motivation (intuition) about a formal group
A (one-dimensional) formal group over $\mathbb{C}$ is a formal power series $F(x,y)\in\mathbb{C}[[x,y]]$ such that
$$
F(x,y)=x+y + \text{terms of higher order}
$$
$$
F(x,F(y,z))=F(F(x,y),z))
$$
...
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Change of variable in Taylor Series Expansion
$$f(z) = \frac{e^z}{(z-2)^3}$$
I want to write Taylor series expansion for this function about $z=2$.
I know I can expand $e^z$ about $z=0$. So I proceed as follows:
Let $z-2=t$ so $z=2+t$
$$ f(t) = \...
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Idempotents in Laurent series
I think it’s true that if $R$ is any commutative, unital ring, then if the formal Laurent series ring $R((t))$ is isomorphic to a Cartesian product of two rings $R((t)) \cong S_1 \times S_2$, then ...
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Minimal field containing two field of formal Laurent series
Let $L/K$ be a field extension and $X,Y$ be two $K$-algebraically independent elements in $L$.
What is the minimal subfield of $L$ that contains both $K((X))$ and $K((Y))$?
The field $K((X,Y))=\...
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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 ...
- 1,009
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Check the associativity of formal series
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 ...
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Help solving a partial differential equation in queuing theory
I have been working on a problem in queuing theory, and in order to understand the steady state behaviour, I have obtained a PDE. The equation is
$(\mu + 2\lambda)f(x, y) + \displaystyle\frac{\partial ...
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Formal Power Series for Recursion where Successor is Linear Combination of all Predecessors
I am trying to solve a recursion of the form
\begin{equation*}a_n=\sum_{j=1}^{n-1} k_{j,n} \cdot a_j + d_n \end{equation*} where $k_{j,n}$ and $d_n$ are constants depending on $j,n$ and $n$, ...
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How to show that $\mathbb{C} ((z))((w))$ is a field
Notation: with $\mathbb{C} ((z))((w))$ we denote the space of formal bilateral power series which have bounded below powers of $w$ but not uniformly bounded below powers of $z$ (this is in the context ...
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Power Series with non integer exponent
I will use the following definitions:
(I apologize in advance, my Latex skills are subpar)
$$\binom{\lambda}{n} = \frac{(\lambda).(\lambda -1)...(\lambda-n+1)}{n!}$$
and $$\binom{\lambda}{0} = 1$$
...
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Automorphism of power series and Jacobian matrix [duplicate]
I have seen it claimed, for example here (in the case $n=2$) that if $k$ is a field and $f_1,\ldots,f_n \in k[[x_1,\ldots,x_n]]$ all of which have zero constant coefficient, the homomorphism
$$\phi: k[...
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Does it make sense to think about formal power series where the coefficients belong to a Ring?
Normally, one defines formal power series as below:
Let $F$ be a field. A formal power series is an expression of the form
$$ a_0 + a_1x + a_2x^2 + \dots = \sum_{n \geqslant 0} a_nx^n,$$
where $\{a_n\}...
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Proving convergence theorem on power series
When proving convergence of power series, one encounters the following term:
$$\left( 1 + \delta \right)\frac{\sqrt[k]{|a_k|}}{\lim\sup_{n\to\infty}\sqrt[n]{|a_n|}}$$
where $\delta>0$, and the ...
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Exponential of derivation of commutative associative algebra
Let $A$ be a commutative associative algebra with unit 1 and derivation $D$. For any $a\in A$, form the formal power series (vertex operator) $$Y(a,z)=\sum_{n\leq -1} \frac{D^{-n-1}a}{(-n-1)!}z^{-n-1}=...
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Does $\bigcap^{\infty}_{k=0} (I + (X^k)) = I$ hold in a ring of formal power series?
Let $A$ be a commutative ring, $A[[X]]$ be the ring of formal power series over $A$. Let $I$ be an ideal of $A[[X]]$, do we have
$$\bigcap^{\infty}_{k=0} (I + (X^k)) = I?$$
Thanks you in advance of ...
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In $V[[z]]$, why is it allowed to separate Taylor series to infinite sum of "smaller" sums?
I'll give you an example in $\mathbb{C}[[x, y]]$:
$$\sum_{n,m\in \mathbb{N}} \binom{n+m}{m} x^ny^m =\sum_{k\in \mathbb{N}}\sum_{n+m=k} \binom{n+m}{m}x^ny^m =\sum_{k\in\mathbb{N}} (x+y)^k=\frac{1}{1-x-...
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Tensor of ring of formal power series and a field
Let $L/K$ be (any) field extension. Is it true that we have an isomorphism $K[[t_1, \dots, t_n]] \otimes L \cong L[[t_1, \dots, t_n]]$?
We know that if we remove the field assumption, this is not true....
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Degree of an even/odd part of a formal power series over a polynomial ring
Consider a formal power series $f=f(x)\in\mathbb{R}[[x]]$ such that $[\,\mathbb{R}[x,f]:\mathbb{R}[x]\,]=d$. Suppose $f_e,f_o\in\mathbb{R}[[x]]$ are such that $f(x)=f_e(x^2)+xf_o(x^2)$. What can be ...
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The multiplication '$*$' of the ring $W(K)=1+tK[[t]]$ in C.Weibel's 'The K-book'
I am reading C. Weibel's 'The K-book', and in page 101, example 4.3, there is a construction of the ring $(W(K)=1+tK[[t]],\cdot,*)$. I could not understand how can we construct the multiplication '$*$'...
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Proof that ring of formal Dirichlet series is isomorphic to a ring of formal power series over countably many variables
I found this article of E.D. Cashwell and C.J. Everett "The ring of number-theoretic functions" and they said Dirichlet series ring is isomorphic to formal power series ring of countably ...
- 197
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Multiplication of a power series and a finite-order polynomial [duplicate]
I am trying to find a general expression for the coefficients of the power series that results from the multiplication of a polynomial and a power series. I have looked at this post Convolution and ...
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What is the neatest formula for the coefficients of the composition $f\circ g$ where $f,g$ are formal polynomials or generating functions?
There are well-known formulas for the coefficients resulting from multiplying two formal polynomials or generating functions (we define formal polynomial to be a generating function such that all ...
- 3,355
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Shift operator equality
Let $K/\Bbb Q_p$ be a finite extension with integers $\mathcal{O}$ and $\mathcal{O}[[T]]$ the ring of formal power series in one variable over $\mathcal{O}$. Define a "shift operator" by
$$ \...
2
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Find $f(x) \in \mathbb C[[x]]$, $g(x) \in\mathbb C[[x^{-1}]]$ infinitely many non-zero coeff. so that product $f(x)g(x)$ is defined (formally)?
the exercise is: Find $f(x)\in\mathbb C[[x]]$ and $g(x)\in\mathbb C[[x^{-1}]]$ with infinitely many non-zero coefficients such that product (in formal calculus, purely algebraically, i.e. summability ...
- 197
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Multiplication problem for formal power series given in a pseudobasis
My question is about the topological algebra $F$ of formal power series with coefficients in $\mathbb{C}$, which is the set of formal power series with multiplication given by the Cauchy product.
The ...
2
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Ring of invariants of symmetric group acting on ring of formal power series
Let $k$ be a field and $R=k[[x_1, \dots, x_n]]$ be the ring of formal power series in $n$ variables. Let $S_n$ be the symmetric group of order $n!$. Then $S_n$ acts on $R$ by $k$-automorphisms by ...
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Size of the set of formal power series?
I just need to verify this for self-study:
Claim: The set of all power series in one variable with rational coefficients is not countable.
Proof: That set is isomorphic to $\mathbb{Q}^\mathbb{N}$ ...
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