All Questions
Tagged with ac.commutative-algebra power-series
66 questions
21
votes
1
answer
757
views
Are $\mathbb C$ , $\mathbb C[X]$ definable in $\mathbb C[[X]]$?
Let $L$ be a first-order language and $M$ be an $L$-structure. Let $D \subseteq M^n$ . Let us say $D$ is definable in $M$ if for some finite set (possibly empty) $A=\{a_1,...,a_m\} \subseteq M$ and ...
user111524
19
votes
2
answers
765
views
Is $K[[x_1,x_2,\dots]]$ an $\mathfrak m$-adically complete ring?
I asked this question on Mathematics Stackexchange (link), but got no answer.
Let $K$ be a field, let $x_1,x_2,\dots$ be indeterminates, and form the $K$-algebra $A:=K[[x_1,x_2,\dots]]$.
Recall ...
- 4,416
18
votes
3
answers
8k
views
What are the prime ideals of k[[x,y]]?
Let $k$ be a field. Then $k[[x,y]]$ is a complete local noetherian regular domain of dimension $2$. What are the prime ideals?
I've browsed through the paper "Prime ideals in power series rings" (...
- 63.1k
18
votes
3
answers
2k
views
If a formal power series over the complex numbers satisfies a polynomial identity, does it imply that the power series has a radius of convergence?
Let $ P(z) $ be a $\textit{formal}$ power series in $z$ that a priori may not have a non zero radius of convergence. Assume that $P(0) =0$.
Let $\Phi(w,z)$ be a polynomial in two variables, that ...
- 3,245
12
votes
2
answers
940
views
An analogue of the Bass-Quillen conjecture with power or Laurent series
The famous Quillen-Suslin theorem (formerly known as Serre's problem/conjecture) states that every projective module over $k[x_1,\dots, x_n]$ is free for $k$ a field. Replacing $k$ by a more general ...
- 12.1k
11
votes
1
answer
1k
views
flatness of power series rings
It is known that $A[[X]]$ is flat if $A$ is noetherian (see for example Bourbaki, Algèbre commutative, Ch. III, §3, Cor. 3 p. 146).
What happens if A is not noetherian? Is there an easy counter-...
- 778
11
votes
1
answer
384
views
Derivative of an algebraic power series in positive characteristic
Let $K$ be a field. It is easy to see that if the characteristic of $K$ is $0$ and $f(T)=\sum_{n\ge0}a_nT^n$ is a power series algebraic over $K(T)$, then $f'$ belongs to $K(T)(f)$.
Indeed let $P(X)=\...
- 3,998
10
votes
4
answers
2k
views
Formal power series is Taylor expansion of rational function iff Hankel determinants vanish?
Let $$ u(T)=\sum_{n = 0}^\infty a_nT^n$$ be a formal power series over a field $K$. Then why does $u(T)$ lie in $K(T)$ (i.e. is the Taylor expansion of a rational function) if and only if there is an $...
user97565
10
votes
2
answers
837
views
Principal ideal subrings of formal power series rings
In the formal power series ring $\mathbb{F}[[x]]$ over a field $\mathbb{F}$ of characteristic $p>0$, consider an element of the form $f=\sum_{i=0}^\infty a_ix^{p^i}$. Let $R$ denote the unitary ...
- 630
10
votes
1
answer
818
views
Is $k(\!(x,y)\!)$ a topological field?
More generally, let $(R,m)$ be a Noetherian local domain with fraction field $K$. The $m$-adic topology turns $R$ into a topological ring. When $R$ is a discrete valuation ring, this topology extends ...
- 19.6k
9
votes
1
answer
876
views
A series that is rational?
Let $k=\mathbb F_q(T)$. Can one prove (or disprove) that the series $\sum_{n\ge0}(1-TX^{q^n})Y^{q^n}\in k[[X,Y]]$ belongs to $k(X,Y)$? At first, it looked like it was simple. But in fact, I have no ...
- 3,998
9
votes
2
answers
566
views
Involutions in $\mathbb{F}_p[[x]]$
Question: For a prime $p$, is every involution in $\mathbb{F}_p[[x]]$ with a zero constant term a reduction modulo $p$ of some involution in $\mathbb{Z}[[x]]$?
Here involution in $A[[x]]$ means $f\in ...
- 1,190
9
votes
2
answers
790
views
Algebraic power series of finite order
Apologies if the question is too elementary/something well-known.
I believe it is a well-known fact that the rational formal power series $F(z)=\frac{P(z)}{Q(z)}$ which have finite order under ...
- 24.2k
9
votes
2
answers
1k
views
Modules over Laurent series rings
Let $k[x]$ be the ring of polynomials over a field k in one variable x. A $k[x]$-module is a k-vector space together with a linear endomorphism (the action of x).
The field $k(x)$ of rational ...
- 66.8k
8
votes
1
answer
639
views
A question on algebraic independence
Let $f_1,f_2,\ldots,f_n, g \in \mathbb{F}_q[x_1,...,x_m]$. Assume that $f_1,\ldots,f_n$ vanish at $0$, so that $\mathbb{F}_q[[f_1,...,f_n]]$ is a subring of $\mathbb{F}_q[[x_1,...,x_n]]$. Suppose that ...
- 341
8
votes
1
answer
518
views
A ring of generalized power series
Let $\Bbbk$ be a field; I am interested in the following ring (which I suspect is a field). Its elements are formal expressions that look like
$$ \sum_{n=0}^{\infty} a_n x^{b_n} $$
where $a_n\in \...
- 66.8k
8
votes
0
answers
430
views
name for a degree-like invariant of a power series over a commutative ring
Let $R$ be a commutative ring, and let $f \in R[\![X]\!]$ be a formal power series. Sometimes (and for example, this will always be possible if $R$ is Noetherian), one may write $f$ in the form $$
f =...
- 1,802
7
votes
3
answers
900
views
Groebner bases for power series rings (reference request)
Hello,
Could you help me with a reference to elementary properties of Groebner bases in rings of formal power series over a field? I am especially interested in generic initial ideals.
Thank you in ...
- 1,839
6
votes
1
answer
175
views
Is there an algorithm to find out the number of small solutions to a polynomial equation, when we vary all the coefficients?
Let $\Phi (z,t)$ be a polynomial given by
$$ \Phi(z,t) := z^n + A_{n-1}(t) z^{n-1} + \ldots + A_1(t) z + A_0(t).$$
Assume that $\Phi(0,0) =0$. It is a fact that a solution $z(t)$ of the equation
$$ \...
- 3,245
6
votes
1
answer
336
views
Is the minimal polynomial of an algebraic formal Laurent series always separable?
Let $f(x)\in K((x))$ be an algebraic formal Laurent series and let $P(x,y)\in K(x)[y]$ be its minimal polynomial. Is $P(x,y)$
always separable? An example of non separable polynomial comes
from ...
- 385
6
votes
0
answers
285
views
On the prime spectrum of $R[[X]]$ when the prime spectrum of $R$ is Noetherian
All rings below are commutative with unity.
If $R$ has a.c.c. on radical ideals i.e. if $Spec R$ is Noetherian under Zariski topology, then so is $R[X]$, this is Theorem 2.5 in the following paper ...
- 412
5
votes
1
answer
509
views
how to pass from algebraic power series to the analytic ones
Fix a field of zero characteristic, $k$, e.g. $\Bbb{R}$ or $\Bbb{C}$. Suppose $k$ is normed (and complete for its norm). Consider the ring extensions: $k[x_1,..,x_n]\subset \ k<x_1,..,x_n> \ \...
- 2,232
5
votes
1
answer
428
views
Analytic functions in arbitrary rings?
We have developed a rich theory of analytic functions over $\mathbb{R}^n$ and $\mathbb{C}^n$. This is pretty reasonable, as analyticity here (local representation by power series) is closely linked to ...
- 193
5
votes
0
answers
171
views
Result of Larsen and Lunts on rationality of power series with coefficients in a free abelian group
Let $G$ be a free abelian group (not necessarily finitely-generated) and $F$ be the fraction field of the group ring of $G$. Let $\Theta$ be the set of power series in $F[[t]]$ such that each nonzero ...
- 622
4
votes
2
answers
367
views
Equality in $\mathbb F_q\left(\left(\frac1T\right)\right)$
Can one characterize the $a\in\mathbb F_q\left(\left(\frac1T\right)\right)$ such that $a(T+1)=a(T)$? Although this seems elementary, I did not manage to find a answer.
Thanks in advance for any help.
- 3,998
4
votes
1
answer
434
views
Cancellation problem for Laurent polynomial rings and power series rings
Throughout, let $k$ be an algebraically closed field. For two $k$-algebras $A,B$ let us write $A \cong_k B$ to mean that $A,B$ are isomorphic as $k$-algebras.
It is known that if $A$ is an integral ...
user111524
4
votes
1
answer
343
views
An analogue of rational functions for Hahn series
For any field $k$, we have both the field $k(t)$ of rational functions (formal quotients of polynomials, i.e. the field of fractions of $k[t]$) and the field $k((t))$ of formal Laurent series (which ...
- 66.8k
3
votes
1
answer
354
views
Are the ring of power series and the ring of germs of holomorphic functions catenary?
I am wondering if the following rings are catenary:
If $k$ is a field, is the ring of formal power series $k[[X_1,\dots,X_n]]$ catenary?
Is the ring of complex power series with a non-zero radius of ...
- 529
3
votes
1
answer
579
views
The global (weak) dimension of formal power series rings
Given a commutative ring $R$, what are relations between w.gldim$(R)$ and w.gldim$(R[[x]])$ (gldim$(R)$ and gldim$(R[[x]])$)?
- 199
3
votes
1
answer
319
views
Functions on a field representable by Hahn series?
It is well known (see here for example) that a function over $\mathbb{R}$ is representable by a power series iff its analytic continuation to $\mathbb{C}$ is holomorphic on some open subset of $\...
- 10.1k
3
votes
1
answer
316
views
Polynomials mapping the twisted cubic power series ring to itself
If $f(X) \in \mathbb{F}_2((T))[X]$ and $f(\mathbb{F}_2[[T^2,T^3]]) \subseteq \mathbb{F}_2[[T^2,T^3]]$, then does it follow that $f'(0) \in \frac{1}{T}\mathbb{F}_2[[T]]$?
This might seem like an ...
- 4,985
3
votes
1
answer
155
views
Proving that a morphism between power series rings is regular
Let $k\subset K$ be a separable field extension. As a particular case of M. André Localisation de la lissité formelle one obtains that the natural inclusion of power series rings $k[[X_1,\ldots,X_n]]\...
- 657
3
votes
0
answers
70
views
Degree of an even/odd part of a formal power series over a polynomial ring
Let $K$ be a field with $\operatorname{char}K\ne 2$ (say, $K=\mathbb{R}$ or $\mathbb{C}$) and consider a formal power series $f=f(x)\in K[[x]]$ such that $[K[x,f]:K[x]\,]=d$. Suppose $f_e,f_o\in K[[x]]...
- 1,190
3
votes
0
answers
92
views
Solutions of the vector field $D=A\frac{\partial}{\partial X}+B\frac{\partial}{\partial Y}$ with $A,B\in k[[X,Y]]$
I am trying to understand the article Reduction of Singularities of the Differential Equation $Ady=Bdx$ by Arno van den Essen.
Let $A,B\in k[[X,Y]]$ be formal power series and $D$ the vector field $A\...
- 31
3
votes
0
answers
285
views
Infinite sum in power series ring
Let $R$ be a commutative ring with $1$, $R[[x]]$ be the power series ring over $R$ and $A$ be an (prime) ideal of $R[[x]]$ with $x\not\in A$ and $\{f_i\}_{i=1}^\infty$ be a sequence of element of $A$. ...
- 31
3
votes
0
answers
80
views
Surjective maps between power series rings
Suppose that $A$ is a complete neotherian local ring and that we're given a surjective homomorphism
$f: A[[x_{1}, \ldots, x_{n}]] \rightarrow A[[t_{1}, \ldots, t_{m}]]$.
Can we always find a ...
3
votes
0
answers
256
views
Cohen structure theorem with explicit equations
By Cohen structure theorem, a complete regular equicharacteristic Noetherian local ring is isomorphic to a power series. In particular, this should hold for finite extensions of power series $k[[t]][\...
- 1,142
3
votes
0
answers
457
views
R[[X]] flat as a R[X]-module?
I assume $R[X]\rightarrow R[[X]]$ is not flat in general, but I was wondering if any conditions on a commutative ring $R$ are known such that $R[[X]]$ is flat as a $R[X]$-module.
Would $R$ noetherian ...
- 31
3
votes
0
answers
240
views
Flatness of a homogeneous quasi-coherent sheaf on the formal plane from flatness on lines through the origin?
(This is a follow-up to a previous question; as often happens, I didn't include enough hypotheses, so I'm asking a new, more-likely-to-be-true question).
Consider the formal plane $\operatorname{Spec}...
- 44.7k
2
votes
1
answer
404
views
Formal Cauchy-Riemann equations for formal power series without complex analysis
Consider the ring $\mathbb{C}[[X,Y]]$ and its subring $\mathbb{C}[[X+iY]]$, where $i=\sqrt{-1}$. One can show that $f(X,Y):=u(X,Y)+iv(X,Y)\in \mathbb{C}[[X,Y]]$ lies in $\mathbb{C}[[X+iY]]$ iff $u$ ...
- 7,127
2
votes
1
answer
208
views
elementary question on a completion of a ring
Let $k$ a field, and $k[\epsilon]=k[X]/(X^{2})$ , what is the completion of the ring $k[\epsilon][t]$ with respect to the ideal $(t^{2}+\epsilon)$?
- 3,472
2
votes
4
answers
2k
views
Closed-form for modified formal power series
This question have been driving me crazy for months now. This comes from work on multiple integrals and convolutions but is phrased in terms of formal power series.
We start with a formal power ...
- 195
2
votes
1
answer
707
views
Localisation of $\mathbb{Z}_p[[X]]$ at ideal $(p)$
Let $R=\mathbb{Z}_p[[X]]$ where $\mathbb{Z}_p$ denotes the $p$-adic integers and $p$ is a prime. Then what is $R_{(p)}$ $(R$ localised at the ideal $pR)$ $?$
- 193
2
votes
1
answer
457
views
An element of formal power series over a commutative ring
Let $R$ be a commutative ring with 1 and let $R[[x]]$ be the formal power series ring over $R$. Now let $f\in R[[x]]$ with the property that if $g, h\in R[[x]]$ and $f=g+h$ then either $\langle h \...
- 21
2
votes
1
answer
251
views
Étale fibration for $K[[X_1,...,X_n]]$
Let us consider a formal power series ring $A_n \colon= K[[X_1,\ldots,X_n]]$ with $0 \ll n < \infty$ and we shall consider a prime ideal ${\frak P}$ of $A_n$ such that $1 < {\mathrm{ht}}({\frak ...
- 563
2
votes
1
answer
170
views
Matching power series to infinity
As pointed out by Makoto, on this question about power series rings and the axiom of choice, an idea I had needed the axiom of dependent choice to work. However, the construction raises another ...
- 18.7k
2
votes
0
answers
95
views
Is the positive part of an algebraic bilateral p-adic convergent power series algebraic?
Let $\mathbb{Z}_p \{ X\}$ and $\mathbb{Z}_p \{ X , X^{-1}\}$ be the henselizations of $\mathbb{Z}_p [X]$ and $\mathbb{Z}_p [ X , X^{-1}]$ with respect to the ideals $p\mathbb{Z}_p [X]$ and $p\mathbb{Z}...
- 7,249
2
votes
0
answers
89
views
Orders of certain quotients of power series rings
Let $\Lambda_d := \mathbb{Z}_p[[T_1, \ldots, T_d]]$ denote the ring of formal power series in $d$ variables over the ring of $p$-adic integers. Suppose that $g \in \Lambda_d$ is an irreducible element,...
- 21
1
vote
1
answer
155
views
Convergence of a product in $\mathbb Q_2[[X]]$
I thought it would be very easy to prove, but in fact, I did not manage to prove or disprove this fact:
the sequence of polynomials $$\left(\prod_{j=0}^k\big(1-2^{2^j}X\big)\right)_{k\in\mathbb N}$$
...
- 3,998
1
vote
1
answer
268
views
Flatness on the formal plane from flatness on lines through the origin?
Consider the formal plane $\operatorname{Spec}\mathbb C[[t,h]]$, and let $\mathcal F$ be a quasi-coherent sheaf. Now assume that $\mathcal F$ is flat over infinitely many lines through the origin of ...
- 44.7k