Let us consider an irreducible polynomial $$f \colon= \alpha_e + \alpha_{e-1}X_1 + ... + \alpha_1X_1^{e-1} + X_1^e \in K[[X_2,X_3]][X_1]$$ and set $$\iota_1 \colon K[[X_2,X_3]] \hookrightarrow K[[X_1,X_2,X_3]]/(f).$$

Let us also consider an irreducible polynomial $$g \colon= \beta_e + \beta_{e-1}X_1 + ... + \beta_1X_1^{e-1} + X_1^e \in K[[X_2]][X_1]$$ such that $$g \equiv f\pmod{X_3}.$$ We set $$\iota_2 \colon K[[X_2]] \hookrightarrow K[[X_1,X_2]]/(g).$$

Once and for all, we shall fix the ideal $I_2$ of $K[[X_1,X_2]]$ such that $g \in I_2$, and denote by $J_2$ the intersection $K[[X_2]] \cap I_2$ via embedding $\iota_2$. Set

$$ J_2 = (\gamma) \subset K[[X_2]]. $$

Now we shall consider the ideal $I_1$ of $K[[X_1,X_2,X_3]]$ such that $f \in I_1$ and $I_2 \equiv I_1$ mod $(X_3)$. We denote by $J_1$ the intersection $K[[X_2,X_3]] \cap I_1$ via embedding $\iota_1$.

Suppose that the ideal $J_1$ is monomial; viz.

$$ J_1 = (\delta) \subset K[[X_2,X_3]]. $$

**Question 1**. Is there a constant $C >0$ which depends only on $I_2$ and $g$ such that the number of generators of $I_1$ is bounded by $C$?

**Question 2**. What if for a general $m \geq 3$?

That is, for an arbitrary integer $m \geq 3$ suppose that we have an irreducible polynomial $$f \colon= \alpha_e + \alpha_{e-1}X_1 + ... + \alpha_1X_1^{e-1} + X_1^e \in K[[X_2,\ldots,X_m]][X_1]$$ and an ideal $I_1$ of $K[[X_1,\ldots,X_m]]$ such that $$f \in I_1, \qquad g \equiv f \pmod{X_3,\ldots,X_m}, \qquad I_2 \equiv I_1 \pmod{X_3,\ldots,X_m},$$ and $$J_1 \colon= K[[X_2,\ldots,X_m]] \cap I_1 \subset K[[X_1,\ldots,X_m]]/(f)$$ is a monomial ideal of $K[[X_2,\ldots,X_m]]$. Then, is there a constant $C$, depending only on $I_2$ and $g$, such that the number of generators of $I_1$ is bounded by $C$?