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 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} de belongs to ideal generated by f in A[[t]]?

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

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.