I've just started studying complex analytic geometry and I'm confused at almost everything for now. I got some exercises to solve and my major problem with them is that the lecture started as the composition of complex analysis and differential geometry (I had both, but they weren't my fav's :) ) and I don't have any intuition whether the solution (or at least the idea) should be focused on analytic methods strictly or where I should use the, say, 'differential geometry tool'. Here we go.
Let $\mathcal{O}_m$ denote the ring of holomorphic germs at $0\in\mathbb{C}^m$. A polynomial $P\in\mathcal{O}_m[t]$ is said to be distinguished, if it's unitary and its coefficients vanish at the origin.
Now I recall the definition of the germ:
$\overline{\mathcal{O}_m}=\left\{f \ : \ \exists\ U -neighbourhood \ of \ 0\in\mathbb{C}^m, such\ that\ f\in\mathcal{O}(U) \right\}$
and
$\sim$ - equiv. relation: for $f,g\in\overline{\mathcal{O}_m}$ $$f\sim g \iff \exists\ V\ neighbourhood\ of\ 0\in\mathbb{C}^m: f=g\ on\ V .$$
And then $\mathcal{O}_m:=\overline{\mathcal{O}_m}/\sim$.
Now, let $a_0=1$, we take $a_1,\ldots ,a_d$, and let define $$P(x,t)=t^d+a_1(x)t^{d-1}+\ldots +a_d(x), $$ and $a_j(0)=0$.
I have to show that if $\forall \varepsilon>0\exists \delta>0$ such that $$max_{j=1}^{m} |x_j|<\delta$$ and $P(x,t)=0$, this implies that $|t|<\varepsilon$.
The statement seems analytic, but there is too much information for me :(
Would be grateful for any hints.