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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Lang's proof that $k[[X_1,...,X_n]]$ is UFD

I have trouble understanding the proof of Chapter IV Theorem 9.3 in Lang's Algebra, 3rd edition, where he proves that if $k$ is a field, then the ring of formal power series in $n$ variables is a UFD, ...
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$n$-th partial sum -- generating functions

Consider a finite series $S_n:=\sum_{i=0}^n i\cdot2^i$. The generating function for infinite formal series $\sum i\cdot x^i$ is $$a(x)=x\left(\frac{1}{1-x}\right)'=\frac{x}{(1-x)^2},$$ hence $$\frac{a(...
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  • 759
1 vote
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55 views

Finding the kernel of a given map

Let $\mathbb K$ be an algebraic closed field and let $f,g\in\mathbb K[X,Y]$ be two polynomials so that $V_{\mathbb K}(f)$ and $V_{\mathbb K}(g)$ don't have common irreductible components and $(0,0)\in ...
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1 vote
0 answers
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Equality of two free modules of the same finite rank under strong hypothesis.

So basically the question is the one of the title of the post, but let me show you the context: Let $\mathbb K$ be an algebraic closed field and let $f\in \mathbb K[X,Y]$ be a polynomial such that is ...
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2 votes
1 answer
60 views

Counterexample : composition of power series

I'm looking for some counterexample for the following situation : let $S$ et $T$ be two power series, with respective positive radius $R_S$ and $R_T$, with $T(0)=0$. Therefore there is $\rho>0$ ...
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2 votes
1 answer
89 views

Convergence of formal power series

So I've just come across formal power series and was somewhat interested, but still can't seems to understand them. I was reading this explanation about the convergence of a formal power series and ...
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-1 votes
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Multiplying divergent integrals using Hardy fields approach

So, I wonder if the following makes sense. Suppose we want to multiply $\int_0^\infty e^x dx\cdot\int_0^\infty e^x dx$. The partial sums of these improper integrals are $\int_0^x e^x dx=e^x-1$. Now we ...
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4 votes
1 answer
81 views

Closed-form solution of the recurrence relation $f(m,n)=f(m-1,n)+f(m,n-1)+f(m-1,n-1)$

I'm working with the recurrence relation $f(m,n)=f(m-1,n)+f(m,n-1)+f(m-1,n-1)$, with the boundary condition $f(m,0)=f(0,n)=1$. After some work, it is not hard to show the generating function is $F(x,y)...
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  • 641
2 votes
1 answer
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Formal power series vs. convergent power series

In Richard Borcherds' excellent online commutative algebra lecture series he talks about the Weierstrass preparation theorem in lecture 10 and uses it to prove that the ring of formal power series is ...
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1 vote
1 answer
57 views

Degree of extension concerning p-adic numbers.

We define $$\mathbb{Z}[[X]]_{conti}:=\{t:\mathbb{R}_{\geq 0}\rightarrow \mathbb{Z}\ (t\in \mathbb{Z}^{\mathbb{R}_{\geq 0}})|\forall M\geq 0\ \{r\in\mathbb{R}_{\geq0}|r\leq M\land t(r)\neq0\}\ are \ ...
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6 votes
1 answer
135 views

Group of units of $\mathbb{Z}_3[[x]]$

I am trying to calculate the group of units of the power series ring $\mathbb{Z}_3[[x]]$. I know that all the unit elements are of the form $u+\sum_1^{\infty} a_nx^n$ where $u$ is a unit in $\mathbb{Z}...
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  • 5,894
0 votes
0 answers
28 views

An elegant computation of series

I try to compute the following series, $$S(\omega,k)=\sum_{p=0}^{\infty}\frac{(-k)^p(p+k)^2}{(p+k)^2+\omega^2}\frac{1}{p!},$$ where $\omega>0$, $k>0$. To be honest, I have no idea how to compute ...
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3 votes
1 answer
105 views

Finitely generated projective $\mathbb{Z}[[x]]$-modules are free?

$\newcommand{\Z}{\mathbb{Z}}$ Let $A = \Z[[x]]$ be the ring of power series. We know that this is not a PID. In particular a submodule of the free $A$-module $\bigoplus_{i=1}^nA$ does not have to be ...
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0 votes
1 answer
35 views

power series with real powers

Let $0<\varepsilon<1$. Consider the functions $f(x)=x-x^{\varepsilon}$, $g(x)=x^2-x^{2\varepsilon}$, and $h(x)=x^3-x^{3\varepsilon}$ in $[0,1]$. It can be shown that f(x)-g(x), and f(x)-h(x) ...
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Solve the equation $\exp(F)=1$ in the Laurent formal power series ring

Consider the Laurent formal power series in $n$ variables $ R=\mathbb C[[x_1^\pm,\dots, x_n^\pm]] $. We aim to solve the equation $$ \exp(F)\equiv 1, \qquad F\in R $$ There is an obvious solution $F\...
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2 votes
0 answers
62 views

Zero divisors of the formal power series ring $A[[x]]$

In class we started studying the formal power series ring $A[[x]]$ of a ring $A$ and I've been all day trying to find how the zero divisors of this formal power series ring $A[[x]]$ should be, if $A$ ...
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2 votes
1 answer
98 views

Is it possible to get positive radius of convergence for composition of two formal power series, if none of them has positive RoC. [closed]

I get counter example/proof for all other possibilities. But this one I couldn't do.
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1 vote
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Quotient ring of formal power series

Let $R$ be a ring and $f \in R[[x]]$. Do we know anything about the ring $R[[x]]/(f)$. I understand that by making $R$ local and complete and adding some requirements on $f$, we can say a lot by using ...
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0 votes
0 answers
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More about the formal power series; for a given field $F$, what is $F[[x]]$?

If $F$ is a field, I know $F[x]$ is ED(hence PID and hence UFD). Then what happens to $F[[x]]$? I Know every ideal of $F[[x]]$ is principal so $F[[x]]$ is at least PID. (I Know $F[[x]]$ is UFD) And $\...
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4 votes
2 answers
83 views

Is evaluation of formal power series compatible with composition over nonarchimedean complete fields?

In algebraic number theory, one may want to consider a $p$-adic local field and consider the $p$-adic logarithm and $p$-adic exponential function on it. These form inverse homomorphism between a ...
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2 votes
2 answers
64 views

Difference between $R[x]$ and $R[[x]]$ [closed]

I am watching Richard E. BORCHERDS lectures on Rings and Modules 21 Formal power series and have a question about relationship between $R[x]$ and $R[[x]]$. Richard just say $R[[x]]$ as formal power ...
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2 votes
1 answer
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Is $k[[x]]$ ever a finitely generated $k[x]_{(x)}$ module?

For $k$ a field, the localization $k[x]_{(x)}$ naturally includes into $k[[x]].$ I can prove that if $k = \mathbb{C},$ then this inclusion is not surjective, and $k[[x]]$ is not even finitely ...
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  • 651
1 vote
0 answers
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Formal Laurent series in topological rings

If $R$ is just a ring, then $R((t))$, the ring of formal Laurent series with coefficients in $R$, is given by elements of the form $\sum r_kt^k$ such that $r_k=0$ for $k$ smaller than some constant. ...
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  • 190
0 votes
1 answer
44 views

differential equation to power to power series

What is the power series, (or summation form) for the following equation? I know the first couple of terms, but am unable to write it as a power series. The equation is dy/dx +x*y = x^2 There is also ...
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  • 23
0 votes
0 answers
18 views

Composition of Power Series Convergence

This is question is regarding Proposition 5.1 on page 22-23 of Elementary Theory of Analytic Functions of One or Several Complex Variables by Cartan. Here is the proposition: Proposition 5.1. Suppose ...
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5 votes
2 answers
121 views

Can the field of Laurent series be made into an Archimedean ordered field?

In today's analysis class, my professor introduced the field of formal Laurent series $\Bbb R((x))$. He also talked about the dictionary order on $\Bbb R((x))$ and why it is not an Archimedean ordered ...
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  • 365
1 vote
1 answer
57 views

Root continuity principle in $\overline{\mathbb{C}((t))}[T]$.

is anyone aware of an extension of the usual argument for root continuity for polynomials with complex coefficients to the case where the base field is the Puiseux series field over the complex ? Here,...
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0 votes
0 answers
14 views

Is there a natural, operations-preserving bijection between Levi-Civita field and (a subset of) divergent integrals?

For instance, would the following definition of multiplication on divergent integrals correspond to the multiplication in Levi-Civita field? $\int_0^\infty f(x)dx \cdot \int_0^\infty g(x)dx =\int_0^{\...
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0 votes
0 answers
35 views

Formal Taylor Series Expansion

In this paper, the authors consider how to convert a system of $N$ ODEs to a PDE where $N \rightarrow \infty$ in some appropriate sense. In what follows $u_{n+1}$, the solution of the $(n+1)$th ODE, ...
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  • 3,173
1 vote
0 answers
64 views

Is $\frac{1}{x(1+x)}$ a well-defined generating function?

Is this a well-defined generating function? $$\frac{1}{x(1+x)}$$ We know that $\:\frac{1}{(1+x)} = \sum_{n \ge0}(-1)^nx^n$,$\:$ hence the notation $\frac{1}{x}\sum_{n \ge0}(-1)^nx^n \:$ would act as ...
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1 vote
0 answers
35 views

Solving a system of nonlinear algebraic equations with power series

In a physics problem I am working on, I need to solve a system of nonlinear algebraic equations arising from truncating Taylor polynomials. I will outline the physical background for context, but my ...
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4 votes
1 answer
74 views

An infinite family of Artin-Schreier polynomials which all split in $\mathbf{F}_q(\!(\theta)\!)$

Let $\mathbf{F}_q$ be a finite field with $q$ elements and let $K$ denote the local function field $\mathbf{F}_q(\!(\theta)\!)$. Let $R$ be its valuation ring $\mathbf{F}_q[\![\theta]\!]$. Let $u$ be ...
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  • 1,350
0 votes
1 answer
57 views

Taking the logarithm of a formal power series definition

I came across with the logarithm of a formal power series in a paper i am reading, but i have not found what is the definition of this 'operator(?)'. Can someone please let me know the definition and ...
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1 vote
0 answers
42 views

Substitution homomorphism in formal power series ring being a ring isomorphism [duplicate]

I am currently studying the ring of formal power series over an arbitrary characteristic field K, and I'm trying to prove the following property: Let $f = (f_{1},\cdots,f_{m}) \in \mathbb{K}[[x_{1},\...
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0 answers
32 views

About formal power series substitution homomorphism and Jacobian determinants. [duplicate]

I'm currently studying the ring of formal power series over an arbitrary characteristic field K, and I'm trying to prove the following property: Let $f = (f_{1},\cdots,f_{m}) \in \mathbb{K}[x_{1},\...
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2 votes
0 answers
87 views

Silverman AEC: Lemma IV.6.3 [on the convergence of formal logarithm]

In Silverman's Arithmetic of Elliptic Curves, Lemma IV.6.3(a) states that Let $R$ be a ring of characteristic $0$, complete with respect to a discrete valuation $v$, and let $p\in\mathbb{Z}$ be a ...
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1 vote
1 answer
80 views

Silverman AEC: Corollary IV.4.4.

In Silverman's Arithmetic of Elliptic Curves, Corollary IV.4.4 states that (for an arbitrary ring $R$), Let $\mathcal{F}/R$ be a formal group and let $p\in \mathbb{Z}$ be a prime. There are power ...
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  • 1,032
0 votes
1 answer
42 views

Generating function from $\frac{bx}{1-ax}$

I know that $\displaystyle\frac{1}{1-x}=\sum_{n=0}^{\infty} x^n$ (also generates the sequence of 1's). It is possible to prove from this that $\displaystyle\frac{bx}{1-ax}=\sum_{n=1}^{\infty} ba^{n-1} ...
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0 votes
0 answers
66 views

Ideals in the ring of formal power series

It is known that, if $R$ is a commutative Noetherian ring, the ring of formal power series $R[[x_1,...,x_n]]$ is a flat $R$-algebra. This gives us a flat ring homomorphism $\phi:R \rightarrow R[[x_1,.....
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1 vote
0 answers
46 views

Generating function of the succession $\{\frac{n^{3}}{n!}\}_{n\in\mathbb{N}}$

I'm having serious trouble with this question. What is the generating function of $\{\frac{n^{3}}{n!}\}_{n\in\mathbb{N}}$? I have thought first that I know some things related to this. Firstly, I know ...
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0 votes
1 answer
45 views

Understanding the structure of $\mathbb{Z}[[x]]/(x-x^2)$

I'm currently trying to understand the structure of quotients of power series rings, and found a particular example I'm confused about. Let $f = x-x^2$ be a polynomial in $\mathbb{Z}[[x]]$, and ...
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0 votes
1 answer
85 views

Order of a real formal power series

I am struggling to understand what the order of a real formal power series is. I am working through the book "Topology: An Introduction" by Stefan Waldmann. There he describes the "...
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  • 223
3 votes
2 answers
76 views

Solving an implicit equation to find the exponential generating function of a specific sequence

Suppose we have a set $A$ with $n$ elements. First we partition $A$ in at least two blocks. Then, we partition each block in the previous partition that is not of size 1 in at least two blocks. We ...
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  • 136
6 votes
1 answer
127 views

Generating function of a polynomial

Suppose I want to find a simple formula for the generating function of a general polynomial sequence $a_n=P(n)$. Obviously it is enough to find the generating function of the sequence $a_n=n^k$ for ...
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  • 2,360
1 vote
1 answer
101 views

Find the closed form for the generating fuction

I am given $g_0 = g_1 = \frac{1}{2}$ and $g_n + (n+1)g_{n+1} = \frac{1}{n!}$ where $n\geq2$. I need to find the closed form for the generating function $g(z)$ and the closed form for $g_n$. I'm not ...
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  • 103
2 votes
1 answer
64 views

Division in the ring of formal power series

I was wondering if the following simplification is allowed in the ring of formal power series: If I have $ \displaystyle\frac{1+x}{1+2x+x^2}$ , can I simplify it so that it is equal to $\displaystyle\...
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7 votes
2 answers
322 views

Difference between generating functions and formal power series

So I was reading about generating functions and formal power series, and it seems that these two concepts are used interchangeably. Can someone please tell me the difference between them? Is ...
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9 votes
1 answer
171 views

Replacing ideal generators in $R[[X]]$ by polynomials

Consider a ring $R$ and the ring of formal power series $R[[X]]$ over $R$. Note that $R[X]$ naturally embeds into $R[[X]]$. Now let $I$ be a finitely generated ideal of $R[[X]]$, say, $I=\langle f_1,\...
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  • 15.2k
0 votes
0 answers
44 views

A smooth monotone increasing function $g:\mathbb R^+\to\mathbb R^+$ satisfying $g(g(x))=4 x^2+x$ (half-iterate)

Suppose $f(x)$ is a real-valued function of a real variable that is continuous, smooth and monotone increasing at least on $x\in[0,\infty)$, satisfying the boundary condition $f(0)=0$. We will pick $f(...
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1 vote
1 answer
64 views

Special case where $I\subseteq J$ implies $R/J\subseteq R/I$ seems to hold

Consider a ring $R$ and two ideals $I,J$ such that $I\subseteq J$. There is a natural projection $R/I\to R/J$ by the isomorphism theorem. There is, in general, no homomorphism going the other way ...
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