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There are various results, beginning with van der Waerden, on the asymptotic number of integral polynomials with bounded coefficients, whose Galois group is $S_n$. Some of these results also include $A_n$ in the number of polynomials. In particular, Rainer Dietmann (On the distribution of Galois groups, 2012) shows that the number of polynomials of degree $n$ and height $\leq H$ which do not have Galois group $S_n$ or $A_n$ to be $\ll H^{n-1+e(n)+\varepsilon}$ for a function $e(n)$ which rapidly tends to zero.

On the other hand, Joos Heintz (On polynomials with symmetric Galois group which are easy to compute, 1986) shows that the polynomials over a Hilbertian field $k$, considered as points in $k^{n+1}$, which have Galois group $S_n$ are dense.

Is it known or suspected that the polynomials (over $\mathbb{Z}$, $\mathbb{Q}$, or number fields) with a given Galois group $G \not= S_n,A_n$ are discrete? Is the answer different if instead one takes all polynomials with Galois group different from $S_n,A_n$ or if $A_n$ is dropped from the excluded groups?

There are various results, beginning with van der Waerden, on the asymptotic number of integral polynomials with bounded coefficients, whose Galois group is $S_n$. Some of these results also include $A_n$ in the number of polynomials. In particular, Rainer Dietmann (On the distribution of Galois groups, 2012) shows that the number of polynomials of degree $n$ and height $\leq H$ which do not have Galois group $S_n$ or $A_n$ to be $\ll H^{n-1+e(n)+\varepsilon}$ for a function $e(n)$ which rapidly tends to zero.

On the other hand, Joos Heintz (On polynomials with symmetric Galois group which are easy to compute, 1986) shows that the polynomials over a Hilbertian field $k$, considered as points in $k^{n+1}$, which have Galois group $S_n$ are dense.

Is it known or suspected that the polynomials (over $\mathbb{Z}$, $\mathbb{Q}$, or number fields) with a given Galois group $G \not= S_n,A_n$ are discrete? Is the answer different if instead one takes all polynomials with Galois group different from $S_n,A_n$ or if $A_n$ is dropped from the excluded groups?

EDIT: Certainly it is true over $\mathbb{Z}$ since $\mathbb{Z}^{n+1}$ is discrete. However, I wonder how close two polynomials with Galois group $G \not= S_n,A_n$, or even two distinct groups $\not= S_n,A_n$, can get.

There are various results, beginning with van der Waerden, on the asymptotic number of integral polynomials with bounded coefficients, whose Galois group is $S_n$. Some of these results also include $A_n$ in the number of polynomials. In particular, Rainer Dietmann (On the distribution of Galois groups, 2012) shows that the number of polynomials of degree $n$ and height $\leq H$ which do not have Galois group $S_n$ or $A_n$ to be $\ll H^{n-1+e(n)+\varepsilon}$ for a function $e(n)$ which rapidly tends to zero.

On the other hand, Joos Heintz (On polynomials with symmetric Galois group which are easy to compute, 1986) shows that the polynomials over a Hilbertian field $k$, considered as points in $k^{n+1}$, which have Galois group $S_n$ are dense.

Is it known or suspected that the polynomials (over $\mathbb{Z}$, $\mathbb{Q}$, or number fields) with a given Galois group $G \not= S_n,A_n$ are discrete? Is the answer different if instead one takes all polynomials with Galois group different from $S_n,A_n$ or if $A_n$ is dropped from the excluded groups?

EDIT: Certainly it is false over $\mathbb{Z}$ by taking the neighborhood consisting of just a polynomial with Galois group $S_n$. However, I wonder how close two polynomials with Galois group $\not= S_n,A_n$ can get.

There are various results, beginning with van der Waerden, on the asymptotic number of integral polynomials with bounded coefficients, whose Galois group is $S_n$. Some of these results also include $A_n$ in the number of polynomials. In particular, Rainer Dietmann (On the distribution of Galois groups, 2012) shows that the number of polynomials of degree $n$ and height $\leq H$ which do not have Galois group $S_n$ or $A_n$ to be $\ll H^{n-1+e(n)+\varepsilon}$ for a function $e(n)$ which rapidly tends to zero.

On the other hand, Joos Heintz (On polynomials with symmetric Galois group which are easy to compute, 1986) shows that the polynomials over a Hilbertian field $k$, considered as points in $k^{n+1}$, which have Galois group $S_n$ are dense.

Is it known or suspected that the polynomials (over $\mathbb{Z}$, $\mathbb{Q}$, or number fields) with a given Galois group $G \not= S_n,A_n$ are discrete? Is the answer different if instead one takes all polynomials with Galois group different from $S_n,A_n$ or if $A_n$ is dropped from the excluded groups?

There are various results, beginning with van der Waerden, on the asymptotic number of integral polynomials with bounded coefficients, whose Galois group is $S_n$. Some of these results also include $A_n$ in the number of polynomials. In particular, Rainer Dietmann (On the distribution of Galois groups, 2012) shows that the number of polynomials of degree $n$ and height $\leq H$ which do not have Galois group $S_n$ or $A_n$ to be $\ll H^{n-1+e(n)+\varepsilon}$ for a function $e(n)$ which rapidly tends to zero.

On the other hand, Joos Heintz (On polynomials with symmetric Galois group which are easy to compute, 1986) shows that the polynomials over a Hilbertian field $k$, considered as points in $k^{n+1}$, which have Galois group $S_n$ are dense.

Is it known or suspected that the polynomials (over $\mathbb{Z}$, $\mathbb{Q}$, or number fields) with a given Galois group $G \not= S_n,A_n$ are discrete? Is the answer different if instead one takes all polynomials with Galois group different from $S_n,A_n$ or if $A_n$ is dropped from the excluded groups?

There are various results, beginning with van der Waerden, on the asymptotic number of integral polynomials with bounded coefficients, whose Galois group is $S_n$. Some of these results also include $A_n$ in the number of polynomials. In particular, Rainer Dietmann (On the distribution of Galois groups, 2012) shows that the number of polynomials of degree $n$ and height $\leq H$ which do not have Galois group $S_n$ or $A_n$ to be $\ll H^{n-1+e(n)+\varepsilon}$ for a function $e(n)$ which rapidly tends to zero.

On the other hand, Joos Heintz (On polynomials with symmetric Galois group which are easy to compute, 1986) shows that the polynomials over a Hilbertian field $k$, considered as points in $k^{n+1}$, which have Galois group $S_n$ are dense.

Is it known or suspected that the polynomials (over $\mathbb{Z}$, $\mathbb{Q}$, or number fields) with a given Galois group $G \not= S_n,A_n$ are discrete? Is the answer different if instead one takes all polynomials with Galois group different from $S_n,A_n$ or if $A_n$ is dropped from the excluded groups?

EDIT: Certainly it is false over $\mathbb{Z}$ by taking the neighborhood consisting of just a polynomial with Galois group $S_n$. However, I wonder how close two polynomials with Galois group $\not= S_n,A_n$ can get.