What is the probability that a random sequence of polynomials is regular?

Question

Let $k$ be a finite field or a field with a height function, such as a number field. Consider the ring $k[[x_1,\dots, x_n]]$ and let $\mathfrak{m}$ be its maximal ideal.

What is the asymptotic probability that a random sequence $s_1,\dots, s_n$ of polynomials in $\mathfrak{m}$ is regular?

By asymptotic probability I mean the following.
Let $N>0$. There are finitely many sequences $s_1,\dots, s_n$ in $\mathfrak{m}$ of length $n$, for which $s_i$ are polynomials of $\deg\leq N$ and for which height of every coefficient of $s_i \leq N$. (When $k$ is finite, we don't need to worry about height.) Let $P_N$ be the proportion of regular sequences among such sequences. Then the question is: Does the limit $$\lim_{N\to\infty} P_N$$ exist, and if it does, what does it equal to? It seems reasonable to expect that it equals $1$.

Answer 1

This is not an answer, but an elaborated comment. The claim should follow from the following two observations.

1. Let $\mathfrak{q}$ be a proper ideal of $\mathfrak{m}$ of codimension $< n$. Take its primary decomposition $\mathfrak{q} = \cap_{j=1}^k \mathfrak{p}_j$. Then $\lim_{N \to \infty}\frac{|\mathfrak{m}_N\setminus \cup_{j=1}^k \mathfrak{p}_j|}{|\mathfrak{m}_N|} = 1$.

2. If $(f_1, \ldots, f_m)$ is a regular sequence for some $m < n$, then let $\mathfrak{q}_{\vec f}$ be the ideal generated by $(f_1, \ldots, f_m)$ with primary decomposition $\mathfrak{q}_{\vec f} = \cap_{j=1}^{k_{\vec f}} \mathfrak{p}_{\vec f,j}$. Then $(f_1, \ldots, f_m, f)$ is a regular sequence iff $f \in \mathfrak{m} \setminus \cup_{j=1}^{k_{\vec f}} \mathfrak{p}_{\vec f,j}$.

Observation 2 is true (follows e.g. from Atiyah-Macdonald, Proposition 4.7). Observation 1 seems 'obvious', but I don't see how to prove it in a straightforward manner. And, of course one has to be careful about taking the limits. I hope it helps.

Answer 2

For finite fields, observation (1) of auniket is straightforward. It is equivalent to show, for any prime ideal $\mathfrak{p}$ of codimension $<n$, $$\lim_{N \to \infty} \frac{|\mathfrak{m}_N \cap \mathfrak{p}|}{|\mathfrak{m}_N|} = 0.$$ Let $V_N$ be the image of $\mathfrak{m}_N$ in $R/\mathfrak{p}$. Over finite fields, we have the exact sequence of $k$-vector spaces $$0 \to \mathfrak{m}_N \cap \mathfrak{p} \to \mathfrak{m}_N \to V_N \to 0$$ so we are studying $$\lim_{N \to \infty} \frac{1}{|V_N|} = 0.$$

Since $R/\mathfrak{p}$ has positive Krull dimension we have $\dim_k R/\mathfrak{p} = \infty$. Since $R/\mathfrak{p}$ is the ascending union $\bigcup V_N$, this shows that $\lim_{N \to \infty} |V_N|=\infty$, as desired.

I haven't thought through how to put all the steps together here.