# 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.

371 questions
Filter by
Sorted by
Tagged with
1answer
36 views

### Quotient of formal power series ring

Let $k$ be an algebraically closed field of characteristic zero, and let us consider the formal power series ring $k[[x]]$. What is the quotient $k[[x]]/x^{n}$ ?
2answers
126 views

### What is the universal property of the algebra of formal power series over a commutative ring?

Let $A$ be a commutative ring. For every set $I$, $A^{\mathbb{N}^{(I)}}$ is the algebra of formal power series. Suppose $\sigma:I\rightarrow J$ is a bijection. Ignoring topology, what is the canonical ...
0answers
34 views

### Extended ideal in formal power series ring is always equal to kernel of canonical reduction homomorphism $R[[X]] \rightarrow (R/I)[[X]]$?

Let $R$ be a commutative ring with unit element, let $I$ be an ideal in $R$. Let $A = R[[X]]$, the ring of formal power series with coefficients in $R$. Let $I_A$ be the ideal generated by elements of ...
1answer
31 views

### What is the formal power series in Shermann-Morrison formula?

I'm learning about the Shermann-Morrison formula and the way to find the inverse matrix uses a formal power series. My question is how is this formal power series done? I read in some websites and I ...
0answers
22 views

### Can you use the inverse Z transform to compute a power series?

I know one way to compute a power series is to use a Taylor Series and compute derivatives. I'm wondering if you can also do it using the Inverse Z Transform? It looks like yes, but I'm getting stuck ...
0answers
24 views

### Generating Function where $x^n$ is replaced with $x^n/(1-x^n)$

Suppose $\phi(x) = \sum_{n=1}^\infty a_n x^n$. If we know the closed form expression for $\sum_{n=1}^\infty a_n \frac{x^n}{1-x^n}$ is there a way to find the closed form expression for $\phi(x)$?
1answer
72 views

### Power Series where partial sums are irreducible polynomials?

I want to show that there exists some formal power series, $f(x)\in\mathbb{Z}[[x]]$, such that each consecutive partial sum is irreducible in $\mathbb{Z}[x]$. Rewording this in terms of polynomials, I ...
0answers
22 views

### structure of ideals in formal power series

Let be $R$ a commutative unitary ring. If $R$ is a field then the ideals of $R[[t]]$ are only the ideals generated by $t^{n}$. If $R$ is not a field, but for example a $\mathbb{C}$ algebra?
1answer
72 views

### Division in $A[[x]]$ [duplicate]

I was looking for a division in a ring of formal power series. Specifically, let be $A$ a commutative ring with unit. Take $A[[x]]$ and $f\in A[x]$ a monic polynomial not invertible in $A[[x]]$ is so ...
1answer
62 views

0answers
46 views

### Is $X+Y+XY$ and $X+Y-XY$ is isomorphic formal group law over integer ring?

I would like to check whether $X+Y+XY$ and $X+Y-XY$ is isomorphic formal group law over integer ring $\Bbb{Z}$ or not. It is known that it is isomorphic over rational field. But what about in $\Bbb{Z}$...
1answer
150 views

### How to confirm $\phi(F_1(x,y))＝F_2(\phi(x),\phi(y))$,where $F_1$ and $F_2$ are formal group law of elliptic curve $E_1$, $E_2$.

This question is from Silverman's 'the arithmetic of elliptic curves',$p134$. Let $K$ be a field of characteristic $p > 0$, let $E_1/ K$ and $E_2/K$ be elliptic curves, and let $\phi : E_1 \to E_2$...
0answers
40 views

### Exercise about tensor product

Let be $R$ a $\mathbb{C}-$ algebra and take $R\otimes_{\mathbb{C}}\mathbb{C}[[t]]$, Is this isomorphic as ring to $R[[t]]?$ I think that I can prove it using the quozient of these ring and take the ...
0answers
166 views

0answers
52 views

### Inverse of formal multiplicative group

I am reading about the formal multiplicative group, with addition given by $F(x,y)=x+y+xy$, and I am wondering if there is a nice way to describe the inverse of an element. So if I let $x+y+xy=0$, ...
1answer
38 views

0answers
31 views

### torsion element of group associated to formal group

Let $R$ be complete local ring $M$ be the maximal ideal of $R$ $F$ be a formal group defined over $R$, with group law $F（X,Y）$. According Silverman's book 'the arithmetic of elliptic curves', example ...
0answers
79 views

### Is formal group a pair wise concept?

In Silverman's 'the arithmetic of elliptic curves', Formal group is defined as a power series which satisfies some conditions. But the book also reads the formal group like a pair, $(\mathrm{F},F)$. ...
0answers
31 views

0answers
38 views

### What is the relation between formal group and formal scheme

What is the relation between formal group and formal scheme? Formal group is power series, which behaves like ' a group law without any group elements'. Is former one is special case of latter one?
0answers
33 views

### Sum of convergent power series with increasing degrees is convergent.

I'm contemplating Exercise 1.1.3 from Greuel, Lossen and Shustin's Introduction to Singularities and Deformations. It states Let $(f_k)_{k \in \mathbb N}$ be a sequence of convergent power series ...
0answers
37 views

1answer
63 views

### An analytic function on $\Bbb{C}$ has infinite radius of convergence about any point

I may be overlooking a small detail, but this question has me stumped. Suppose $f:\Bbb{C}\to \Bbb{C}$ is analytic. Then, by definition of analyticity, for every $z_0\in \Bbb{C}$, there exists a formal ...
0answers
50 views

1answer
85 views

### Problem with a Arithmetico-Geometric Series

Good afternoon to everyone, I have the following question: What does the arithmetico-geometric series: $$S = \sum^{\infty}_{n=1} ne^{-nrt}$$ Converge to? ($r > 1$, $t > 1$) I tried to break it ...
1answer
58 views

2answers
63 views

### Where do the powers go?

I have this weird identity between power series. It's kind of like the relation between a geometric series and $\frac{1}{1-x}$. I was wondering if there was some theory developed along these lines I ...