Counting number fields of small degree
Recently, my collaborators Theresa C. Anderson, Ayla Gafni, Kevin Hughes, Robert J. Lemke Oliver, Frank Thorne, Jiuya Wang, Ruixiang Zhang, and I uploaded a preprint to the arxiv called "Improved bounds on number fields of small degree". This collaboration is a continuation1 1though with a few different cast members of our previous work on quantitative Hilbert irreducibility, which will appear in IMRN.
In this paper, we improve the upper bound due to Schmidt for estimates on the number of number fields of degree $6 \leq n \leq 94$. Actually, we improve on Schmidt for all $n \geq 6$, but for $n \geq 95$ Lemke Oliver and Thorne have different, better bounds.
Schmidt proved the following.
For $n \geq 6$, there are $\ll X^{(n+2)/4}$ number fields of degree $n$ and having discriminant bounded by $X$.
We prove a polynomial improvement that decays with the degree.
For $n \geq 6$, there are \begin{equation*} \ll_\epsilon X^{\frac{n + 2}{4} - \frac{1}{4n - 4} + \epsilon} \end{equation*} number fields of degree $n$ and having discriminant bounded by $X$.
Towards the end of this project, we learned that Bhargava, Shankar, and Wang were also producing improvements over Schmidt in this range. On the same day that we posted our paper to the arxiv, they posted their paper, in which they prove the following.
For $n \geq 6$, there are \begin{equation*} \ll_\epsilon X^{\frac{n + 2}{4} - \frac{1}{2n - 2} + \frac{1}{2^{2g}(2n-2)} + \epsilon} \end{equation*} number fields of degree $n$ and having discriminant bounded by $X$, where $g = \lfloor \frac{n-1}{2} \rfloor$.
In both our work and in BSW, the broad strategy is based on Schmidt's approach. For a monic polynomial \begin{equation*} f(x) = x^n + c_1 x^{n-1} + \cdots + c_n, \end{equation*} we define the height $H(f)$ to be \begin{equation*} H(f) := \max( \lvert c_i \rvert^{1/i} ). \end{equation*} Then Schmidt showed that to count number fields of discriminant up to $X$, it suffices to count polynomials of height roughly up to $X^{1/(2n - 2)}$.
The challenge is that most of these polynomials cut out number fields of discriminant much larger than $X$. The challenge is then to count relevant polynomials and to identify irrelevant polynomials.
Remarkably, the broad strategy in out work and in BSW for identifying irrelevant polynomials is similar. For a prototypical polynomial $f$ of degree $n$ and of height $X^{1/(2n-2)}$, we should expect the discriminant of $f$ to be approximately $X^{n/2}$. We should also expect the field cut out by $f$ to have discriminant roughly this size. Recalling that we are counting number fields of discriminant only up to $X$, this means that a relevant polynomial of this height must be exceptional in one of two ways:
- either the discriminant of $f$ is unusually small, or
- the discriminant of the number field cut out by $f$ is much smaller than the discriminant of $f$.
In both out work and in BSW, those $f$ with unusually small discriminant are bounded straightforwardly and lossily.
The heart of the argument is in the latter case. Here, the ratio of the two discriminants is the square of the index $[\mathcal{O}_K : \mathbb{Z}[\alpha]]$, where $\alpha$ is a root of $f$. Thus we bound the number of polynomials whose discriminants have large square divisors.
In establishing bounds for polynomials with particularly squarefull discriminants that our ideas and those in BSW significantly diverge.
In our work, we study the problem locally. That is, we study the behavior of $\psi_{p^{2k}}$, the characteristic function for monic polynomials of degree $n$ over $\mathbb{Z}/p^{2k}\mathbb{Z}$ having discriminant congruent to $0 \bmod p^{2k}$. As in our work on quantitative Hilbert irreducibility, we translate this problem into a sieve problem with local weights coming from Fourier transforms $\widehat{\psi_{p^{2k}}}$ after passing through Poisson summation, and we study the Fourier transforms using a variety of somewhat ad-hoc techniques.
In BSW, they reason differently. They use recent explicit quantitative Hilbert irreducibility work from Castillo and Dietmann to replace the fundamental underlying sieve. To do this, they translate the task of counting relevant polynomials into the task of counting distinguished points in spaces of $n \times n$ symmetric matrices — and then show that Castillo and Dietmann's work bounds these points.
Even though the number field count in BSW is stronger than our number field count, we think that our methods and ideas will have other applications. Further, we've noticed remarkable interactions between local Fourier analysis and discriminants of polynomials.
See also
- See also my note on a description and simplified proofs of many of the ideas in this paper.
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