Theresa Anderson, Ayla Gafni, Robert Lemke Oliver, George Shakan, Ruixiang Zhang, and I have just uploaded a preprint to the arXiv called *Quantitative Hilbert Irreducibility and Almost Prime Values of Polynomial Discriminants.*

George has also written about this paper on his site.

This project began at an AIM workshop on Fourier analysis, arithmetic statistics, and discrete restriction.

Our guiding question was very open. For some *nice* local polynomial conditions, can we make sense of the Fourier transforms of these local conditions well enough to have arithmetic application?

This is partly inspired from *Orbital exponential sums for prehomogeneous vector spaces* by Takashi Taniguchi and Frank Thorne (preprint available on the arXiv). In this paper, Frank and Takashi algebraically compute Fourier transforms of a couple arithmetically interesting functions on prehomogeneous vector spaces over finite fields. It turns out that one can, for example, explicitly and completely compute the Fourier transform of the characteristic function of singular binary cubic forms over $\mathbb{F}_{q}$.

In a companion paper, Takashi and Frank combine those computations with sieves to prove that there are $\gg X / \log X$ cubic fields whose discriminant is squarefree, bounded above by $X$, and has at most $3$ prime factors. They also show there are $\gg X / \log X$ quartic fields whose discriminant is squarefree, bounded above by $X$, and has at most $8$ prime factors.

## Results

We have two classes of result. Both rely on similar types of analysis, and are each centered on a study of a particular indicator-type function, its Fourier transform, and a sieve.

First, we prove a bound on the number of polynomials whose Galois group is a subgroup of $A_n$. For $H > 1$, define \begin{equation*} V_n(H) = \{ f \in \mathbb{Z}[x] : \mathrm{ht}(f) \leq H \} \end{equation*} and \begin{equation*} E_n(H, A_n) := \# \{ f \in V_n(H) : \mathrm{Gal}(f) \subseteq A_n \}. \end{equation*} We show that \begin{equation} E_n(H, A_n) \ll H^{n – \frac{2}{3} + O(1/n)}. \end{equation} This is an improvement on progress towards a conjecture of Van der Waerden and is a quantitative form of Hilbert’s Irreducibility Theorem, which shows (among other applications) that most monic irreducibile polynomials have full Galois group.

However I should note that Bhargava has announced a proof of a (slightly weakened form of) Van der Waerden’s conjecture, and his result is strictly stronger than our result.

Secondly, we prove that for any $n \geq 3$ and $r \geq 2n – 3$, we have \begin{equation} \# \{ f \in \mathbb{Z}[x] : \mathrm{ht}(f) \leq H, f \, \text{monic }, \omega(\mathrm{Disc}(f)) \leq r \} \gg_{n, r} \frac{H^n}{\log H}, \end{equation} where $\omega(\cdot)$ denotes the number of distinct prime divisors. Qualitatively, this says that there are lots of polynomials with almost prime discriminants.

As a corollary of this second result, we prove that for $n \geq 3$ and $r \geq 2n – 3$, \begin{equation} \# \{ F / \mathbb{Q} : [F \colon Q] = n, \mathrm{Disc}(F) \leq X, \omega(\mathrm{Disc}(F)) \leq r \} \gg_{n, r, \epsilon} X^{\frac{1}{2} + \delta_n – \epsilon} \end{equation} for explicit $\delta_n > 0$ and any $\epsilon > 0$. This shows that there are at least $X^{1/2}$ cubic fields whose discriminants are divisible by at most $3$ primes, or at least $X^{1/2}$ quartic fields whose discriminants are divisible by at most $5$ primes, for example. We guarantee fewer fields than Taniguchi and Thorne, but we guarantee fields with fewer prime factors and cover all degrees.

In the remainder of this post, I’ll describe a line of thinking that went towards proving our first result.

## Odd polynomials

We initially studied the Fourier transform of the *odd-polynomial* indicator function. We call a function $f(x) \in \mathbb{F}_p[x]$ *odd* if it has no repeated roots and the factorization type of $f$ corresponds to an odd permutation in the Galois group. That is, we can write $f$ as \begin{equation*} f(x) = f_1(x) f_2(x) \cdots f_r(x) \bmod p, \end{equation*} and there will be an element of the Galois group with cycle type $(\deg f_1) (\deg f_2) \cdots (\deg f_r)$. For *odd* $f$, this cycle must be an odd permutation.

A more convenient description of *oddness* is in terms of the Möbius function on $\mathbb{F}_p[x]$. A degree $n$ polynomial $f$ is odd precisely if $\mu_p(f) = (-1)^{n+1}$. Define $1^p_{sf}(f)$ to be the squarefree indicator function on $\mathbb{F}_p[x]$, and define $1^p_{odd, n}$ to be the odd indicator function on degree $n$ polynomials on $\mathbb{F}_p[x]$. Then \begin{equation*} 1^p_{odd, n}(f) = 1^p_n(f)\frac{(-1)^{n+1}\mu_p(f) + 1^p_{sf}(f)}{2}. \end{equation*} (Here, $1^p_n(f)$ keeps only the degree $n$ polynomials).

## Fourier transform of odd indicator function: a first approach

We then studied the Fourier transform of $1^p_{odd, n}$. Identifying the vector space of polynomials of degree at most $n$ over $\mathbb{F}_p[x]$, which we denote at $V_n(\mathbb{Z}/p\mathbb{Z})$, as $(\mathbb{Z}/p\mathbb{Z})^{n+1}$, we can study the Fourier transform of a function $\psi:V_n(\mathbb{Z}/p\mathbb{Z}) \longrightarrow \mathbb{C}$, \begin{equation*} \widehat{\psi}(\mathbf{u}) = \frac{1}{p^{n+1}} \sum_{f \in V_n(\mathbb{Z}/p\mathbb{Z})} \psi(f) e_p(\langle f, \mathbf{u} \rangle). \end{equation*} Here, $e_p(x) = e^{2 \pi i x / p}$.

It is possible to understand this Fourier transform using ideas similar to those of Takashi and Thorne. $\mathrm{GL}(2)$ acts on these polynomials in a similar way as it acts on quadratic forms, *and* $1^p_{odd, n}$ is invariant under this action. As in Takashi and Thorne, one can study the sizes of the Fourier transform on each orbit. This leads to several classical polynomial counting problems.

But unlike the prehomogeneous vector space context of Takashi and Thorne, we can’t *completely* determine the Fourier transform. For general degree, there are too many other terms.

Ultimately, we intend to use the knowledge of this Fourier transform as an ingredient in a sieve. An old theorem of Dedekind shows that if $\mathrm{Gal}(f) \subseteq A_n$, then $f$ is never *odd* mod any prime $p$.

We could use a Selberg sieve in the following form. For a nonnegative weight function $\phi: V_n(\mathbb{R}) \longrightarrow \mathbb{R}$ (roughly supported on the box $[-1, 1]^{n+1}$). Then consider \begin{equation}\label{eq:basic_sieve} \sum_{f \in V_n(\mathbb{Z})} \phi(f/H) \Big(\sum_{d: f \bmod p \text{ is odd } \forall p \mid d} \lambda_d \Big)^2 \geq 0 \end{equation} for some real weights $\lambda_d$ to be chosen later, but where $\lambda_1 = 1$.

For $f$ with $\mathrm{Gal}(f) \subseteq A_n$, $f$ is never odd. Thus the sum of weights $\lambda_d$ is exactly $\lambda_1 = 1$ for those $f$, and we get that \eqref{eq:basic_sieve} is bounded below by \begin{equation}\label{eq:basic_sieve_LHS} \sum_{\substack{f \in V_n(\mathbb{Z}) \\\\ \mathrm{Gal}(f) \subseteq A_n}} \phi(f/H). \end{equation} On the other hand, \eqref{eq:basic_sieve} is equal to \begin{equation}\label{eq:basic_sieve_RHS} \sum_{d_1, d_2} \lambda_{d_1} \lambda_{d_2} \sum_{f \in V_n(\mathbb{Z})} \phi(f / H) \prod_{p \mid [d_1, d_2]} 1^p_{odd, n}(f). \end{equation} Thus we have that \eqref{eq:basic_sieve_LHS} $\leq$ \eqref{eq:basic_sieve_RHS}. To bound \eqref{eq:basic_sieve_RHS}, we use Poisson summation to transform the sum of $\phi 1^p_{odd, n}$ into a dualized sum of $\widehat{\phi} \widehat{1}^p_{odd, n}$ and use our understanding of the Fourier transform $1^p_{odd, n}$ to (try to) get good bounds. Then one plays a game of optimizing over the weights $\lambda_d$.

### Problem

There is a major problem with this approach. As we’re unable to completely determine the Fourier transform, it’s necessary to determine where it’s large and small and to handle the regions where it’s large well. Let’s look again at the expression \begin{equation*} 1^p_{odd, n}(f) = 1^p_n(f)\frac{(-1)^{n+1}\mu_p(f) + 1^p_{sf}(f)}{2}. \end{equation*} The Fourier transform of $\mu_p$ is expected to behave very well away from $0$. But the Fourier transform of $1^p_{sf}$ can be shown to have large Fourier coefficients away from $0$, strongly affecting the resulting bounds.

## Graded indicator function: a second approach

Instead of studying the indicator function $1^p_{odd, n}$, we chose to study a sort of *graded* indicator function \begin{equation*} \psi_p(f) = \frac{(-1)^{n+1}1^p_n(f)\mu_p(f) + 1}{2}. \end{equation*} This is $1$ if $f$ is odd and squarefree, $0$ if $f$ is squarefree and even, and $1/2$ if $f$ is not squarefree.

On the Fourier transform side, we completely understand the Fourier transform of $1$ and we can hope to have good understanding of the Möbius function. So we should expect much better bounds.

But on the other side, this is not as clean of an indicator function as $1^p_{odd, n}$. In comparison to the basic sieve inequality \eqref{eq:basic_sieve_LHS} $\leq$ \eqref{eq:basic_sieve_RHS}, the product of indicator functions on the right hand side now becomes much messier, and the basic setup no longer applies.

Instead, in \eqref{eq:basic_sieve}, we replace $\big( \sum \lambda_d \big)^2$ by a positive semidefinite quadratic form in $\lambda_{d_1}, \lambda_{d_2}$ to get a modified Selberg sieve inequality similar to \eqref{eq:basic_sieve_LHS} $\leq$ \eqref{eq:basic_sieve_RHS}. The tail of the argument remains largely the same. Instead of bounding \eqref{eq:basic_sieve_RHS}, we bound

\begin{equation*} \sum_{d_1, d_2} \lambda_{d_1} \lambda_{d_2} \sum_{f \in V_n(\mathbb{Z})} \phi(f / H) \prod_{p \mid [d_1, d_2]} \psi_p(f). \end{equation*}

After Poisson summation, the goal becomes controlling $\widehat{\psi_p}(f)$, which essentially boils down to understanding $\widehat{\mu_p}(f)$.

In explicit coordinates, this is the task of understanding \begin{equation*} \widehat{\mu_p}(u_0, \ldots, u_n) = \frac{1}{p^{n+1}} \sum_{t_i \in \mathbb{F}_p} \mu_p(t_n x^n + \cdots + t_0) e_p(u_n t_n + \cdots + u_0 t_0). \end{equation*} This is a $\mathbb{F}_p[x]$-analogue of the classical question of bounding \begin{equation*} \sum_{n \leq x} \mu(n) e(n\theta) \end{equation*} for some real $\theta$. Baker and Harman have proved that GRH implies that\begin{equation*} \Big \lvert \sum_{n \leq x} \mu(n) e(n\theta) \Big \rvert \ll x^{\frac{3}{4} + \epsilon}, \end{equation*} and Porritt has proved the analogous result holds over function fields (where RH is known).

Applying this bound in our modified form of the Selberg sieve is what allows us to prove our first theorem.

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