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.



Laurent series with analytic coefficients
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.


Can you use TeX in your writing?


