Laurent series with analytic coefficients

Question

Let $A=H(D(0,1))$ the ring of holomorphic functions on the open unity disc.

I consider the function $f$:

$$f (t)=\sum f_{i}t^{i} \in A[[t]]$$

I suppose that the $t$-adic valuation of it is less or equal than $r$, so the first $r$ functions, $f_{1},f_{2},...,f_{r}$ do not simultaneously vanish.

Do I have that $t^{r}$ belongs to the ideal generated by $f$ in $A[[t]]$?

The question is also valid if I dont take holomorphic functions, but polynomial functions.

Answer 1

The answer is no. Let $r=2$, $f_0,f_1$ never simultaneously vanish. You want $(f_0+f_1t+...)(g_0+g_1t+...)=t^2$. For this we need $g_0=g_1=0$ and $f_0g_2=1$, which is impossible if $f_0$ vanish somewhere.

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