# Generators of an ideal of $K[[X_1,X_2,X_3]]$

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 $C$?

I shall only propose an outline of the possible proof below, where Claim 1, Claim 2 and Claim 3 have not yet been proved rigorously.

We shall begin the outline. So suppose that every element of $\alpha_1,\ldots,\alpha_e$ contains every parameter of $X_3,\ldots,X_m$ in its presentation of the formal summation. Then, we consider the specialisation defined as follows$\colon$

$$(\lozenge) \quad X_i = S(X_3,\ldots,\check{X_i},\ldots,X_m) \in K[[X_3,\ldots,\check{X_i},\ldots,X_m]].$$

Some sort of Bertini argument, i.e. Bertini irreducibility theorem for the hyperplane section, might justify the following Claim 1$\colon$

${\bf Claim\, 1.}$ After the specialisation, by which I mean simply to insert $(\lozenge)$ into the irreducible polynomial $f$ at ${Q}\,2$ in the above, the specialised polynomial $s(f) \in K[[X_2,\ldots,\check{X_i},\ldots,X_m]][X_1]$ remains still irreducible as long as $S(X_3,\ldots,\check{X_i},\ldots,X_m)$ is sufficiently general.

Moreover, conditions in the question remains unchanged with only the number of variables dropped by one. Thus we can use the induction on $m$ when $m \geq 4$. That is, the ideal $I_1$ of the ring $K[[X_1,\ldots,X_m]]$ turns into another ideal $K_1$ of the ring $K[[X_1,\ldots,\check{X_i},\ldots,X_m]]$ after specialisation. Besides, the number of generators of $K_1$ is bounded by the constant $C$ due to the induction hypothesis.

Finally, we propose the following result$\colon$

${\bf Claim\,2.}$ There exists at least one general specialisation $(\lozenge)$ such that both numbers of generators of ideals $I_1$ and $K_1$ coincide with each other as long as $I_1 \not\supset (X_3,\ldots,X_m)$ (i.e., $I_1$ does not contain every parameter $X_3,\ldots,X_m$).

The assumption of $Q\,2$ ensures that actually $I_1 \not\supset (X_3,\ldots,X_m)$ holds, for otherwise $J_1$ might not be monomial. Now, we proceed as follows. Since the number of generators of $K_1$ is bounded by $C$, Claim 2 ensures that the number of generators of $I_1$ is also bounded by $C$ as desired as long as $m \geq 4$. So eventually, the induction argument obliges us only to show the following claim$\colon$

${\bf Claim\,3.}$ $Q\,1$ is correct.

Wishfully, this outline should be justified rigorously at least in the case where every element $\alpha_i$ of $\alpha_1,\ldots,\alpha_e$ contains every $X_3,\ldots,X_m$ in its presentation of the formal summation, i.e. $\alpha_i = \underset{e_2,\ldots,e_m}{\Sigma} c_{e_2\ldots,e_m \in {\Bbb Z}_{\geq 0}}\,X_2^{e_2} \cdots X_m^{e_m}$.