6
$\begingroup$

For motivation and related questions, see below.

Rough sketch of the question. View $\bigsqcup_{p \text{ prime}} (\mathbb{Z}/p\mathbb{Z})$ as a ‘subset’ of the unit circle, via $a\pmod{p} \mapsto e^{2\pi i \cdot a/p}$. (This is not injective for $0 \pmod{p}$, but I wish to ignore that for the moment.)

Let $f \in \mathbb{Z}[X]$ be a polynomial. If $a \pmod{p}$ is a root of multiplicity $m$ of the polynomial $f \pmod{p}$, then we attach a ‘weight’ $m$ to $e^{2\pi i \cdot a/p}$.

Question 1: Are these weights uniformly distributed over the unit circle?

Several remarks.

  1. I leave it as an exercise to the reader to make the question precise; using limits over circle segments, and primes going to infinity.
  2. If $\deg(f) \le 1$, the answer to question 1 is no.
  3. I have collected data for 1000 monic irreducible polynomials (with $2 \le \deg(f) \le 10$), and their roots modulo the first 10000 primes. Upon calculating some statistics I think there is some evidence for question 1. For almost all polynomials in my dataset, the first 3 moments of the samples are $< 3\%$ from the expected value for a uniform distribution.
  4. I am not an expert in statistics. But it might be the case that the correct distribution to look at is the circular uniform distribution. I have not yet calculated the circular moments for the dataset described above.

Motivation. The main motivation comes from thinking about densities of places of finitely generated fields.

Let $S$ be a subset of $\bigsqcup_{p \text{ prime}} \mathbb{F}_{p} = \{ \text{closed points of degree $1$ in $\mathbb{A}^{1}_{\mathbb{Z}}$}\}$ with positive Dirichlet density.

Question 2: Is there a closed point of $\mathbb{A}^{1}_{\mathbb{Q}}$ whose closure in $\mathbb{A}^{1}_{\mathbb{Z}}$ intersects $S$
(a) infinitely often, or (b) with positive density?

Such a closed point of $\mathbb{A}^{1}_{\mathbb{Q}}$ corresponds with a finite field extension of $\mathbb{Q}$, and thus with a polynomial in $\mathbb{Z}[X]$. Every circle segment of the unit circle gives rise to a positive density subset of $\bigsqcup_{p \text{ prime}} \mathbb{F}_{p}$. This provides a link between question 1 and question 2, although I do not see a logical implication from one to another.

These questions are somewhat similar to statements of the Sato–Tate conjecture (or some generalisation of it). I do not see a direct link, but if someone sees how to connect this to statements about the distribution of eigenvalues of a certain Frobenius operator, please let me know. If I understand things correctly, the $0$-dimensional case of the Sato–Tate conjecture is Chebotarev's density theorem. Of course this describes at how many primes one expects a root of $f$, but it does not describe “where” the root in $\mathbb{F}_{p}$ will be.

$\endgroup$
  • $\begingroup$ Did you look closely at $f=x^2+1$? $\endgroup$ – Uri Bader Sep 20 '16 at 9:23
  • $\begingroup$ @UriBader — Yes. In your example $f$ has no roots if $p \cong 3 \pmod{4}$. But if you lump together all the roots that it has modulo all the primes, then you get a picture that looks very much like a uniform distribution. All those roots will be of the form $a \pmod{p}$ with $p \cong 1 \pmod{4}$, but that does not matter; I think. (Also see my remark about Chebotarev's density theorem, at the end of my question.) $\endgroup$ – user98708 Sep 20 '16 at 9:28
  • $\begingroup$ I tried to imply two things: (1) Seems natural that you have different limits along "arithmetically define" subsequences, such as 1 mod 4, 3 mod 4, and (2) I would start by asking for a solution for this specific case, which might be well known or a not-so-hard interpretation of a well known theorem. $\endgroup$ – Uri Bader Sep 20 '16 at 9:37
  • 7
    $\begingroup$ It's conjectured (though I'm not sure by whom originally) that if $f$ is irreducible of degree $\geq 2$, then one gets equidistribution . The only non-trivial case known is degree $2$, by Duke-Friedlander-Iwaniec (negative discriminant, "Equidistribution of roots of a quadratic congruence to prime moduli", Annals of Math., 1995) and Toth (positive discriminant, "Roots of quadratic congruences", IMRN, 2000). $\endgroup$ – Denis Chaperon de Lauzières Sep 20 '16 at 9:49
  • 2
    $\begingroup$ Related problems are discussed in my paper, Polynomial congruences and density, Mathematics Magazine 80 (2007) 299-302, maa.org/sites/default/files/Myerson10-200716952.pdf $\endgroup$ – Gerry Myerson Sep 20 '16 at 12:42
9
$\begingroup$

In my paper, Polynomial congruences and density, Mathematics Magazine 80 (2007) 299-302, I cite the result of Christopher Hooley, On the distribution of roots of polynomial congruences, Mathematika 11 (1964) 39-49, MR 29 #1173, to the effect that if $f$ is an irreducible polynomial of degree at least 2, with integer coefficients, then the sequence formed by ordering $S_f$ by increasing denominator is uniformly distributed in $[0,1)$. Here, for positive integer $m$, $$S_f(m)=\{\,r/m:0\le r\le m-1,\,\gcd(r,m)=1,\,m\mid f(r)\,\}$$ and $S_f$ is the union of the sets $S_f(m)$ over all positive integers $m$. I then prove, without reference to Hooley's result, a theorem with weaker hypotheses, and a weaker conclusion:

Let $f(t)=t^eg(t)$ where $e$ is a nonnegative integer, $g$ is a polynomial of degree at least 2 with integer coefficients, and $g(0)\ne0$. Define $T_f$ by $$T_f=\{\,r/m:\gcd(r,m)=1,\,m\mid f(r)\,\}$$ Then $T_f$ is dense in the reals.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.