Small factors in random polynomials over a finite field

Previously I mentioned very briefly Granville’s The Anatomy of Integers and Permutations, which explores an analogy between prime factorizations of integers and cycle decompositions of permutations. Today’s post is a record of the observation that this analogy factors through an analogy to prime factorizations of polynomials over finite fields in the following sense.

Theorem: Let Image may be NSFW.
Clik here to view. be a prime power, let Image may be NSFW.
Clik here to view. be a positive integer, and consider the distribution of irreducible factors of degree Image may be NSFW.
Clik here to view. 1, 2, ... k in a random monic polynomial of degree Image may be NSFW.
Clik here to view. over Image may be NSFW.
Clik here to view. $\mathbb{F}_q$ . Then, as Image may be NSFW.
Clik here to view. $q \to \infty$ , this distribution is asymptotically the distribution of cycles of length Image may be NSFW.
Clik here to view. 1, 2, ... k in a random permutation of Image may be NSFW.
Clik here to view. elements.

One can even name what this random permutation ought to be: namely, it is the Frobenius map Image may be NSFW.
Clik here to view. $x \mapsto x^q$ acting on the roots of a random polynomial Image may be NSFW.
Clik here to view., whose cycles of length Image may be NSFW.
Clik here to view. are precisely the factors of degree Image may be NSFW.
Clik here to view. of Image may be NSFW.
Clik here to view..

Combined with our previous result, we conclude that as Image may be NSFW.
Clik here to view. $q, n \to \infty$ (with Image may be NSFW.
Clik here to view. tending to infinity sufficiently quickly relative to Image may be NSFW.
Clik here to view.), the distribution of irreducible factors of degree Image may be NSFW.
Clik here to view. 1, 2, ... k is asymptotically independent Poisson with parameters Image may be NSFW.
Clik here to view. $1, \frac{1}{2}, ... \frac{1}{k}$ .

Proof modulo some analytic details

By the cyclotomic identity, we have

Image may be NSFW.
Clik here to view. $\displaystyle \sum_f t^{\deg f} = \frac{1}{1 - qt} = \prod_P \left( \frac{1}{1 - t^{\deg P}} \right) = \prod_{d \ge 1} \left( \frac{1}{1 - t^d} \right)^{M(q, d)}$

where the sum runs over all monic polynomials over Image may be NSFW.
Clik here to view. $\mathbb{F}_q$ , the product runs over over all monic irreducible polynomials over Image may be NSFW.
Clik here to view. $\mathbb{F}_q$ , and

Image may be NSFW.
Clik here to view. $\displaystyle M(q, d) = \frac{1}{d} \sum_{r | d} \mu(r) q^{d/r}$

is the number of monic irreducible polynomials of degree Image may be NSFW.
Clik here to view. over Image may be NSFW.
Clik here to view. $\mathbb{F}_q$ . This is Euler product for the zeta function for Image may be NSFW.
Clik here to view. $\mathbb{F}_q[x]$ . To get the results we want we need an analogue for polynomials over finite fields of the exponential formula which keeps track of how many factors of a given degree a polynomial has. Letting Image may be NSFW.
Clik here to view. c_i(f) denote the number of irreducible factors of degree Image may be NSFW.
Clik here to view. of Image may be NSFW.
Clik here to view., this is given by

Image may be NSFW.
Clik here to view. $\displaystyle \sum_f t^{\deg f} z_1^{c_1(f)} z_2^{c_2(f)} ... = \prod_P \left( \frac{1}{1 - z_{\deg P} t^{\deg P}} \right) = \prod_{d \ge 1} \left( \frac{1}{1 - z_d t^d} \right)^{M(q, d)}$ .

To get a generating function describing the joint moment generating function of the variables Image may be NSFW.
Clik here to view. c_1, c_2, ... we will also need to divide the terms corresponding to polynomials of degree Image may be NSFW.
Clik here to view. by the number of such polynomials, which is Image may be NSFW.
Clik here to view. q^n . This is equivalent to replacing Image may be NSFW.
Clik here to view. with Image may be NSFW.
Clik here to view. $\frac{t}{q}$ , which gives

Image may be NSFW.
Clik here to view. $\displaystyle \sum_f \frac{t^{\deg f}}{q^{\deg f}} z_1^{c_1(f)} z_2^{c_2(f)} ... = \prod_{d \ge 1} \left( \frac{1}{1 - \frac{z_d t^d}{q^d} } \right) ^{ M(q, d) }$ .

We are now in a position to take the limit Image may be NSFW.
Clik here to view. $q \to \infty$ . We will do so somewhat cavalierly: since Image may be NSFW.
Clik here to view. $M(q, d) \sim \frac{q^d}{d}$ as Image may be NSFW.
Clik here to view. $q \to \infty$ and Image may be NSFW.
Clik here to view. is fixed, we expect that

Image may be NSFW.
Clik here to view. $\displaystyle \left( \frac{1}{1 - \frac{z_d t^d}{q^d}} \right)^{M(q, d)} \to \exp \left( \frac{z_d t^d}{d} \right)$

as Image may be NSFW.
Clik here to view. $q \to \infty$ , hence the above generating in some sense approaches

Image may be NSFW.
Clik here to view. $\displaystyle \exp \left( z_1 t + z_2 \frac{t^2}{2} + z_3 \frac{t^3}{3} + ... \right)$

as Image may be NSFW.
Clik here to view. $q \to \infty$ , at least in the sense that their lower-order terms should match up reasonably well (where “lower-order” depends on Image may be NSFW.
Clik here to view.). But this is precisely

Image may be NSFW.
Clik here to view. $\displaystyle \sum_{n \ge 0} \frac{t^n}{n!} \sum_{\sigma \in S_n} z_1^{c_1(\sigma)} z_2^{c_2(\sigma)} ...$

by the exponential formula.

Some comments on Granville’s analogy

The tripartite analogy between cycle decomposition of permutations, prime factorization of polynomials over a finite field, and prime factorization of integers can be made more specific as follows. There is a lot to say on this subject, but we will content ourselves with two sets of analogies.

First, the following three statements are analogous.

A random permutation of Image may be NSFW.
Clik here to view. elements is a Image may be NSFW.
Clik here to view.-cycle with probability Image may be NSFW.
Clik here to view. $\frac{1}{d}$ .
A random monic polynomial of degree Image may be NSFW.
Clik here to view. over Image may be NSFW.
Clik here to view. $\mathbb{F}_q$ is irreducible with probability asymptotically Image may be NSFW.
Clik here to view. $\frac{1}{d}$ (as Image may be NSFW.
Clik here to view. $q \to \infty$ ).
An integer of size approximately Image may be NSFW.
Clik here to view. is prime with probability approximately Image may be NSFW.
Clik here to view. $\frac{1}{\log n}$ (the prime number theorem).

This suggests that the correct analogue of the degree of a positive integer Image may be NSFW.
Clik here to view. is Image may be NSFW.
Clik here to view. $\log n$ , an idea which is also suggested by the fact that the norm of a polynomial Image may be NSFW.
Clik here to view. of degree Image may be NSFW.
Clik here to view. over Image may be NSFW.
Clik here to view. $\mathbb{F}_q$ is Image may be NSFW.
Clik here to view. q^d , since this is the size of Image may be NSFW.
Clik here to view. $\mathbb{F}_q[x]/f(x)$ , whereas the norm of a positive integer Image may be NSFW.
Clik here to view. is Image may be NSFW.
Clik here to view..

Second, the following three statements are analogous.

The distribution of cycles of length Image may be NSFW.
Clik here to view. of a random permutation of Image may be NSFW.
Clik here to view. elements is asymptotically (as Image may be NSFW.
Clik here to view. $n \to \infty$ ) independent Poisson with parameters Image may be NSFW.
Clik here to view. $1, \frac{1}{2}, ... \frac{1}{k}$ .
The distribution of irreducible factors of degree Image may be NSFW.
Clik here to view. of a random monic polynomial over Image may be NSFW.
Clik here to view. $\mathbb{F}_q$ of degree Image may be NSFW.
Clik here to view. is asymptotically (as Image may be NSFW.
Clik here to view. $q, n \to \infty$ with Image may be NSFW.
Clik here to view. growing sufficiently fast relative to Image may be NSFW.
Clik here to view.) independent Poisson with parameters Image may be NSFW.
Clik here to view. $1, \frac{1}{2}, ... \frac{1}{k}$ .
The distribution of prime factors in the intervals Image may be NSFW.
Clik here to view. $[1, e], [e, e^2], ... [e^{k-1}, e^k]$ of a random integer in the interval Image may be NSFW.
Clik here to view. $[e^{n-1}, e^n]$ is asymptotically (as Image may be NSFW.
Clik here to view. $n \to \infty$ ) independent Poisson with parameters asymptotic to the sequence Image may be NSFW.
Clik here to view. $1, \frac{1}{2}, ... \frac{1}{k}$ .

The third statement in this analogy is false as stated; for example, the distribution of prime factors in the interval Image may be NSFW.
Clik here to view. [1, e] is asymptotically geometric, not Poisson. It may be true if one takes Image may be NSFW.
Clik here to view. $k \to \infty$ in addition to Image may be NSFW.
Clik here to view. in a suitable way while throwing out small primes in a way dependent on Image may be NSFW.
Clik here to view. and while changing “independent Poisson” to “independent asymptotically Poisson,” where this second use of “asymptotically” depends on Image may be NSFW.
Clik here to view..

I offer as weak evidence the following heuristic calculation. First, the probability that a large positive integer Image may be NSFW.
Clik here to view. is divisible by some Image may be NSFW.
Clik here to view. is Image may be NSFW.
Clik here to view. $\frac{1}{m}$ . If Image may be NSFW.
Clik here to view. is a random variable which takes on non-negative integer values, then a straightforward telescoping argument shows that

Image may be NSFW.
Clik here to view. $\displaystyle \mathbb{E}(X) = \sum_{d \ge 1} \mathbb{P}(X \ge d)$ .

If Image may be NSFW.
Clik here to view. is the exponent of a prime Image may be NSFW.
Clik here to view. in the prime factorization of Image may be NSFW.
Clik here to view., then heuristically Image may be NSFW.
Clik here to view. $\mathbb{P}(X \ge d) = \frac{1}{p^d}$ , so

Image may be NSFW.
Clik here to view. $\displaystyle \mathbb{E}(X) = \sum_{d \ge 1} \frac{1}{p^d} = \frac{1}{p-1}$ .

There is a well-known asymptotic for the sum of the reciprocals of the primes which implies that

Image may be NSFW.
Clik here to view. $\displaystyle \sum_{p \le n} \frac{1}{p-1} = \log \log n + O(1)$

and consequently the expected number of prime factors (counted with multiplicity) in the interval Image may be NSFW.
Clik here to view. $[e^{k-1}, e^k]$ of a large positive integer is heuristically asymptotically Image may be NSFW.
Clik here to view. $\log k - \log (k-1) \sim \frac{1}{k}$ .

(Edit, 11/11/12:) The heuristic calculation can be taken further as follows. Letting Image may be NSFW.
Clik here to view. c_p denote the random variable describing the exponent of Image may be NSFW.
Clik here to view. in a random prime factorization, we have heuristically Image may be NSFW.
Clik here to view. $\mathbb{P}(c_p = k) = \left( 1 - \frac{1}{p} \right) \frac{1}{p^k}$ and therefore heuristically the moment generating function

Image may be NSFW.
Clik here to view. $\displaystyle \mathbb{E}(e^{tc_p}) = \left( 1 - \frac{1}{p} \right) \sum_{k \ge 0} \frac{e^{tk}}{p^k} = \frac{1 - \frac{1}{p}}{1 - \frac{e^t}{p}}$ .

For prime factorizations of large positive integers we expect the variables Image may be NSFW.
Clik here to view. $c_p, p \in [e^{k-1}, e^k]$ to heuristically be asymptotically independent, so heuristically their sum has moment generating function

Image may be NSFW.
Clik here to view. $\displaystyle \prod_{p \in [e^{k-1}, e^k]} \frac{1 - \frac{1}{p}}{1 - \frac{e^t}{p}}$ .

By the PNT, there are approximately Image may be NSFW.
Clik here to view. $\frac{e^k - e^{k-1}}{k}$ primes in this interval. Let us blithely pretend that they all have absolute value approximately Image may be NSFW.
Clik here to view. $e^k - e^{k-1}$ , because this is somewhere between Image may be NSFW.
Clik here to view. $e^{k-1}$ and Image may be NSFW.
Clik here to view. e^k and gives the expected value we computed above. Then we are reduced to a calculation very similar to the calculation over finite fields. Namely, we get a moment generating function

Image may be NSFW.
Clik here to view. $\displaystyle \left( 1 - \frac{1}{e^k - e^{k-1}} \right)^{ \frac{e^k - e^{k-1}}{k} } \left( 1 - \frac{e^t}{e^k - e^{k-1}} \right)^{ - \frac{e^k - e^{k-1}}{k} }$

and as Image may be NSFW.
Clik here to view. $k \to \infty$ this is asymptotic to

Image may be NSFW.
Clik here to view. $\displaystyle \exp \left( \frac{1}{k} (e^t - 1) \right)$

which is the moment generating function of a Poisson random variable with parameter Image may be NSFW.
Clik here to view. $\frac{1}{k}$ as expected.

Small factors in random polynomials over a finite field

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