Taylor series $\sum_{n=0}^\infty a_n(z-z_0)^n$ of an holomorphic function $f$ has one null coefficient for all $z_0$ then $f$ is a polynomial Let $f \in H(\Omega)$, where $\Omega \subset \mathbb{C}$ open.  Suppose that for all $z_0 \in \Omega$ the Taylor series associated to $f$ $$\sum_{n=0}^\infty a_n(z-z_0)^n$$ has one null coefficient $a_N$. How do I prove that $f$ is then a polynomial (without applying Baire's category theorem)?
 A: There is a map $n:\Omega \to \{0,1,...\}$ such that $f^{(n(z))}(z) = 0$ for
each $z \in \Omega$.
Note that $\Omega = \cup_k n^{-1}(k)$. Pick some $z_0$, $r>0$ such that
$\overline{B}(z_0,r) \subset \Omega$. Note that $\overline{B}(z_0,r)$ is
compact.
Since $\overline{B}(z_0,r)= \cup_k (\overline{B}(z_0,r) \cap n^{-1}(k))$ and
$\overline{B}(z_0,r)$ is uncountable, then there is some $k$ such that
$\overline{B}(z_0,r) \cap n^{-1}(k)$ is at least countable.
Hence there is a sequence $z_n \in \overline{B}(z_0,r)$ such that
$f^{(k)}(z_n) = 0$, and since $\overline{B}(z_0,r)$ is
compact, we can assume, by taking a subsequence if necessary,
 that $z_n \to z^*$ for some $z^* \in \Omega$.
A standard result then shows that $f^{(k)}(z) = 0$ for all $z \in B(z_0,r)$.
Hence the Taylor series expansion of $f$ shows that $f(z) = p(z)$ for
some (entire) polynomial $p$ of degree $\partial p <k$, with $z \in B(z_0,r)$.
Since $\Omega $ is open and connected, the same standard result shows that
$f(z) = p(z)$ for all $z \in \Omega$.
A: *

*Show that there exists $n$ such that the set of all $z_0$ with $a_n = 0$ is uncountable.

*Show that this uncountable set has an accumulation point in $\Omega$.

*Use the identity theorem for holomorphic functions to conclude the result.


Tips:


*

*Look at the sets $\Omega_n := \{z_0 \in \Omega \mid a_n = 0\}$ and observe that $\bigcup_{n \in \mathbb{N}} \Omega_n = \Omega$. What do you know about countable unions of countable sets? Can all $\Omega_n$ be countable?

*Take an open ball $B$ in $\Omega$ whose boundary is in $\Omega$. What do you know about $\Omega_n \cap B$? What can you conclude about their cardinality? And what does this say about an accumulation point in $\Omega$?

*Let $g = f^{(n)}$. Then $g = 0$ by the identity theorem. What can you conclude for $f$?
I just saw, that 1 is not necessary to prove. You can skip it.
