1
vote
0answers
90 views

Algebraic set given by sequence of polynomials

When working on some problem, I have end up with a following situation. Suppose $P(z)=z^d+a_{d-1}z^{d-1}+\ldots+a_1z+a_0$ is a complex polynomial $d\geq2$ and that $\gamma$ and $\delta$ are non-zero ...
1
vote
0answers
108 views

Automorphism on F_2[[X,S]]

Let us define the automorphism $\sigma$ on ${\Bbb F}_2[[X,S]]$ such that $\sigma \colon S \mapsto S + S^2 + S^3$ $\sigma \colon X \mapsto X + S$. It is easy to see that the ideal $(S)$ is stable ...
3
votes
1answer
298 views

Automorphisms of complete discrete valuation ring

Let ${\Bbb F}_2[[T]]$ be a c.d.v.r over ${\Bbb F}_2$. We consider the automorphism $\sigma$ of ${\Bbb F}_2[[T]]$ such that $\sigma \colon T \mapsto T + c_2T^2 + c_3T^3 + \cdots$ with $c_i \in {\Bbb ...
1
vote
1answer
200 views

Composite families of formal power series over $\mathbb C$ as algebraic variety

I was led to prove that the set of composite families $(f_j)_{j \leq k}$ of germs at $0\in \mathbb C^m$ of a holomorphic function (composite = sharing a common divisor belonging to the maximal ideal) ...
4
votes
0answers
209 views

What is the spectrum of a ring of holomorphic rational power series?

Let $R_\infty$ be the ring of power series in a single variable with rational coefficients that converge in the whole complex plane. Let $R_\rho$ be the subring of $R_\infty$ that defines holomorphic ...
1
vote
0answers
108 views

Is there a standard name for functions of the form $x^\alpha p(x)$, where $p(x)$ is a polynomial?

Is there any existing standard terminology for functions of the form $x^\alpha p(x)$, where $p(x)$ is a polynomial and $\alpha$ is e.g. a complex number? I haven't been able to come up with any good ...
12
votes
3answers
883 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 ...
1
vote
1answer
201 views

finite generation of $G$-equivariant holomorphic maps by polynomials?

Let $V$ and $W$ be two complex vector spaces with an action of a finite group $G$. The $G$-equivariant polynomial maps from $V$ to $W$ are finitely generated as a module over the ring of $G$-invariant ...
18
votes
3answers
714 views

What is the Krull dimension of the ring of holomorphic functions on a complex manifold ?

Consider a connected holomorphic manifold $X$ and its ring of holomorphic functions $\mathcal O(X).$ My general question is simply: in which cases is the Krull dimension $dim \mathcal O(X)$ known? ...
5
votes
2answers
1k views

Algorithm for Weierstrass Preparation Theorem for Formal Power Series

The Weierstrass preparation theorem for formal power series rings guarantees that if a given formal series $f(z) = \sum a_k z^k \in R[[z]]$ where $R$ is a complete local ring with maximal ideal $M$ ...
2
votes
1answer
208 views

Linear independence in the algebraic closure of $\mathbb{C}(z)$

Fix $N>0$. Let $b_i=(b_{i,1}, b_{i,2}, b_{i,3}, b_{i,4})$, $i=1,\ldots, m$, be distinct 4-tuples of integers with with all $0\leq b_{i,j}< N$. (The zero tuple is disallowed.) Define ...
3
votes
1answer
947 views

Amazing examples in complex Algebraic Geometry

Good example teaches sometimes more than couple of theorems. I wonder what are your favourite examples in complex algebraic geometry, the ones that were astonishing for you, the simpler (at least ...