Bounding the number of polynomials whose Galois group is a subgroup of the alternating group

Question

In the article of DAVID ZYWINA https://pdfs.semanticscholar.org/cd50/c2d3fb0ce0c6a66ee629419b69165b30d5bc.pdf. It says that using $n$- dimensional large sieve, we can get the bound

|$\{$$ \ \ f(x)=x^n+a_1x^{n-1}+ \ldots + a_n \in \mathbb{Z}[x] : \max |a_i| \leq B \ \ \ with\ \ \ Gal(f) \subseteq A_n$$\}$| $<< B^{n-1/2}$

(pg.4 after the proof of theorem 1.6)

Equivalently, bound the number of $(a_1, a_2, \ldots,a_n)∈ Z^n$ with $||a|| ≤ B$ for which $∆(a_1, . . . , a_n)$ is a $\textbf{square}$, where $∆(T_1, . . . , T_n) ∈ k[T1, . . . , Tn]$ is the $\textbf{discriminant}$ of $x^n+T_1x^{n-1}+ \ldots + T_n$.

How can I set this up properly to use the large sieve?

Answer 1

The set $$\{ \mathbf{a} =(a_1, \ldots, a_n) \in \mathbb{Z}^n : ||\mathbf{a}|| \leq B, \Delta(a_1,\ldots,a_n) \text{ is a square} \}$$ is a so-called thin set in the sense of Serre.

Serre studies these in detail in the book "Lectures on the Mordell-Weil theorem". He uses the large sieve in Section 13 to show that for any thin set $\Omega \subset \mathbb{Z}^n$ we have $$\# \{ \mathbf{a} \in \Omega : ||\mathbf{a}|| \leq B \} \ll B^{n-1/2}(\log B)^\gamma,$$ for some $\gamma < 1$.

Modulo the logarithm, this gives the bound you want. Possibly you can remove the logarithm in your special case by carefully going through the proof and see if it can saved somewhere.

More results about thin sets and their relationship with Galois groups can be found in Serre's book "Topics in Galois theory".