Let ${\Bbb C}[X_1,\ldots,X_n]$ be a $n$-variable polynomial ring over a complex number field ${\Bbb C}$. For its maximal ideal $(X_1,\ldots,X_n)$, we define the geometric regular local ring as

$R \colon= {\Bbb C}[X_1,\ldots,X_n]_{(X_1,\ldots,X_n)}$,

which is the localisation of ${\Bbb C}[X_1,\ldots,X_n]$ at $(X_1,\ldots,X_n)$. $R$ has the unique maximal ideal $(X_1,\ldots,X_n)$.

We shall consider the polynomial ring $R[X]$ over $R$ and choose $m$ Weierstrass polynomials

\begin{align}
&
f_1(X) = c_{1,0} + c_{1,1}X + \ldots + c_{1,e_1}X^{e_1} \\
&
{\phantom{AAA}} ... \\
&
f_m(X) = c_{m,0} + c_{m,1}X + \ldots + c_{m,e_{m}}X^{e_m},
\end{align}
where $c_{1,0},\ldots,c_{i,j},\ldots,c_{m,e_m} \in (X_1,\ldots,X_n)$.

Suppose that ${\mathrm{GCD}}(f_1(X),\ldots,f_m(X)) = 1$.

Now for a function $F(X) \in R[X]$, we consider the following condition$\colon$

$(\sharp)$ $\quad F(\alpha) \in (f_1(\alpha),\ldots,f_m(\alpha))\phantom{A}$ for any $\alpha \in {\Bbb C}$,

where $(f_1(\alpha),\ldots,f_m(\alpha))$ is the ideal of $R$.

## Q. Does $F(X) \in (f_1(X),\ldots,f_m(X))$?