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At some point I needed to prove that some formal (iterative) construction yielded an actual convergent power series. To do so I was led to prove the following lemma, where $\mathcal{B}(\Delta)$ is the …

I'll finaly settle for "a cutout compact set" for want of a better term. But I think this word expresses well the "finitely many" (connected components, non-smooth points) side of the object, which is …

Consider the $\mathbb Z$-module $\mathcal Z$ obtained as the set of sequences of integers $\mathbb Z ^ \mathbb N$ modulo the relation that two sequences are deemed equivalent when their difference is …

Edit: the answer is given by user1952009 in the comments when $\infty$ is locally accessible from $\mathbb{C}\setminus{S}$, but I'll leave my original incomplete answer here for illustration. Note th …

I'm wondering if a compact set $A\subset\mathbb{C}$ satisfying the properties that • $A$ and its complement have finitely many connected components • every connected component of $\partial A$ is the …

There is a paper of Breuer and Simon, "Natural Boundaries and Spectral Theory" (some slides here ). They give, among other things, the definition of "strong natural boundary". This concept relates to …

In general, this question makes sense at a formal level near $\infty$ only, i.e. for power series involving negative powers of $x$. Setting $z:=\frac{1}{x}$, the equation becomes a so-called homologic …

I'll assume all along you're referring to ODE with real-analytic/holomorphic coefficients. You're looking for something called "non-linear differential Galois theory". This is related to this question …

I know that's been a while now that the question has been asked, but as I'm looking more or less into this topic, I think I should share some of my discoveries in the literature. I'm somewhat amazed a …

I don't know offhand the answer of your first question, but I can answer the particular situation you describe afterwards : the holonomy is always trivial. First, notice that a compact leaf $L$ is e …

I've come across a double cover of $\mathbb P_1(\mathbb C)$, ramified at $[1:1]$ and $[-1:1]$ in homogeneous coordinates, given as the quotient by the natural $\mathbb Z/2\mathbb Z$-action generated b …

I'm looking for a reference to cite regarding the property presented in the title: "Closed and bounded sets of a nuclear Fréchet space are compact" Thank you in advance for the help!

Let $\mathcal O(\Omega)$ be the algebra of functions holomorphic on the open set $\Omega\subset\mathbb C$. For $\gamma$ a simple compact curve in $\mathbb C$ consider the linear operator given by path …

It is known that rational functions $f\in \mathbb C(x)$, $0$ not a pole, are the sum of generating series $\sum_{n\geq 0} a_nx^n$ where $(a_n)_n$ is solution of a linear recurrence with constant coeff …

I would like to point out the following very nice reference (unfortunately I cannot plug into MathSciNet right now and only have a preprint version on my computer, but you should be able to track it d …