All Questions

I'm considering a complex entire function $f$: $\mathbb{C}^n\to \mathbb{C}$. Suppose $f=0$ on $\{(x_1,\cdots,x_n)\in\mathbb{R}^n:\sum_{k=1}^n x_k^2=1\}$. I want to prove $$f=0\textit{ on } M_1=\{(z_1,\...

Let $\mathbb{B}$ be the open unit ball in $\mathbb{C}^n, n\geq 1$ and let $H^\infty (\mathbb{B})$ be the space of all bounded holomorphic functions on $\mathbb{B}$. It is well known that $H^\infty (\...

In theorem 1.2 of Brian Conrad's handout Operations with Pseudo-Riemannian metrics, the author writes Theorem 1.2. Every $C^p$ vector bundle $E\to M$ over a $C^p$ manifold with corners $0\leq p\leq \...

A problem I'm working on requires the application of Cauchy's estimate for the modulus of the coefficients of a holomorphic function's power series representation, but the original functions with ...

It is known that the complement of an analytic set is connected. In general, the complement of a proper complex analytic set in a connected complex manifold is an arcwise connected dense open set. My ...

Assume that $f:\mathbb{C}^n\rightarrow \mathbb{C}$ is a $C^{\infty}$ function such that $f^2$ is (complex) analytic. Then one can show that $f$ is analytic. (Note: Liviu Nicolaescu and Alexandre ...