Dynamically distinguishing polynomials

A (discrete) dynamical system is a pair \(\left( S,f\right) \) consisting of a set S and a function \(f:S\rightarrow S\). The functional graph of (S, f), which we will denote by \(\Gamma (S,f)\), is the directed graph whose set of vertices is S and whose edges are given by the relation \(s\rightarrow t\) if and only if \(f(s)=t\).

In this paper, we study the suggestion of Konyagin et al. [14] “in reverse”; that is, we fix (integer polynomial) maps and then vary the set upon which they act by reducing these polynomials modulo rational primes. Before stating our results, we introduce a bit of notation. Denote the set of rational primes by \({\mathcal {P}}\). For \(f\in {\mathbb {Z}}[x]\) and \(p\in {\mathcal {P}}\), write

• \(\left[ f\right] _p\) for the polynomial in \({\mathbb {F}}_p[x]\) obtained by reducing the coefficients of f mod p and

• \(\Gamma _{f,p}\) for \(\Gamma \left( {\mathbb {F}}_p,[f]_p\right) \).

We say that a set \({\mathcal {F}}\subseteq {\mathbb {Z}}[x]\) is dynamically distinguishable mod p if for all \(f,g\in {\mathcal {F}}\) with \(f\ne g\). Let \(\mu \) be the natural density on \({\mathcal {P}}\); that is, for any subset \(P\subseteq {\mathcal {P}}\),

$$\begin{aligned} \mu \left( P\right) :=\lim _{X\rightarrow \infty } {\frac{\left| \left\{ p\in {\mathcal {P}} \mid p\le X{\text { and }}p\in P\right\} \right| }{\left| \left\{ p\in {\mathcal {P}}\mid p\le X\right\} \right| }} \quad {\text {(if this limit exists)}}. \end{aligned}$$

In Sect. 4, we prove the following theorem.

Establishing the truth of the suggestion of Konyagin et al. [14] mentioned above would immediately produce the \(k=2\) case of Theorem 1.1 as a weaker corollary.

For any \(f,g\in {\mathbb {Z}}[x]\) and \(p\in {\mathcal {P}}\), the dynamical systems \(\left( [f]_p,{\mathbb {F}}_p\right) \) and \(\left( [g]_p,{\mathbb {F}}_p\right) \) are isomorphic in the category of dynamical systems on the set \({\mathbb {F}}_p\) if and only if f and g are dynamically indistinguishable mod p. In more generality, for any set S and set maps \(f,g:S\rightarrow S\), note that \(\Gamma (S,f)\simeq \Gamma (S,g)\) if and only if there exists a bijective set map \(\varphi :S\rightarrow S\) such that \(\varphi \circ f=g\circ \varphi \). In many settings, researchers study subcategories of the category of dynamical systems on the set S by insisting that the maps f, g, and \(\varphi \) belong to the set of morphisms in an appropriate category containing S as an object. For example, suppose K is a field, \(S={\mathbb {P}}^1(K)\), and \(f,g:S\rightarrow S\) are rational functions. Then in the subcategory of dynamical systems of \({\mathbb {P}}^1(K)\), with the self-maps of \({\mathbb {P}}^1(K)\) restricted to rational maps, the dynamical systems \(({\mathbb {P}}^1(K),f)\) and \(({\mathbb {P}}^1(K),g)\) are isomorphic if and only if there exists a Möbius transformation \(\varphi \) such that \(\varphi \circ f=g\circ \varphi \). Fixing an integer \(d\in {\mathbb {Z}}_{>1}\), setting \({\mathcal {F}}\) to be rational functions of degree d, and studying \(M\left( {\mathbb {P}}^1(K),{\mathcal {F}}\right) \) lead to an interesting moduli space problem, one studied by Silverman [26] using geometric invariant theory. See [3, 8, 17] for further work on this problem and extensions of it.

To prove Theorem 1.1, we will distinguish dynamical systems by their periodic points. If (S, f) is a dynamical system, let \(f^n=(\overbrace{f\circ \cdots \circ f}^{n{\text { times}}})\) for any \(n\in {\mathbb {Z}}_{>0}\). If \(s\in S\) has the property that there is some \(n\in {\mathbb {Z}}_{>0}\) with \(f^n(s)=s\), we say that s is periodic or a periodic point of (S, f). The smallest such n is the period of s. As is standard, we will also refer to points of period one as fixed points. Points of period n are precisely those that lie in cycles of length n in the graph \(\Gamma (S,f)\). Periodic points are a classical object of study in discrete dynamical systems over \({\mathbb {C}}\), going back at least to work of Fatou [9, 10] and Julia [13] in the early twentieth century. Recently there has been much work on statistics of periodic points in families of dynamical systems over finite fields, partially motivated by an attempt started by Bach [1] to make rigorous the heuristic assumptions in Pollard’s “rho method” for integer factorization [23]. For example, in [11], Flynn and Garton prove that for the family of polynomials in \({\mathbb {F}}_q[x]\) of a fixed degree d, the average number of cycles in their associated functional graphs is at least \(\frac{1}{2}\log {q}-4\), as long as \(d\ge \sqrt{q}\). More recently, Bellah et al. [4] develop a heuristic that implies that this average is \(\frac{1}{2}\log {q}+O(1)\) for any d. Burnette and Schmutz  [6] prove, for this same family of polynomials, that if \(d=o\left( \sqrt{q}\right) \) as \(d,q\rightarrow \infty \), then the average “ultimate period” of the associated functional graphs is at least \(\frac{d}{2}\left( 1+o(1)\right) \).

Our proof of Theorem 1.1 relies on the trivial observation that for any \(n\in {\mathbb {Z}}_{>0}\), if one directed graph has a cycle of length n and another does not, then the graphs are not isomorphic. As an illustration of our approach, consider the following example.

Example 1.2

Let \(f=x^2+1\) and \(g=x^2+2\). If \(p\in {\mathcal {P}}\), then \(\Gamma _{f,p}\) has a point of period one if and only if there exists \(\alpha \in {\mathbb {F}}_p\) such that

$$\begin{aligned} 0=[f]_p(\alpha )-\alpha =\alpha ^2+1-\alpha . \end{aligned}$$

Now, such an \(\alpha \) exists if and only if the prime ideal \(\left( p\right) \subseteq {\mathbb {Z}}\) splits (or ramifies) in the splitting field of \(f(x)-x=x^2-x+1\) (over \({\mathbb {Q}}\)). Similarly, \(\Gamma _{g,p}\) has a fixed point if and only if \(\left( p\right) \) splits (or ramifies) in the splitting field of \(g(x)-x\). Let \(K_f\) and \(K_g\) be the splitting fields of \(f(x)-x\) and \(g(x)-x\), respectively. The Frobenius Density Theorem implies that the natural density of primes that split in \(K_f\) and \(K_g\) is the proportion of their Galois groups that fix a root of the polynomials whose roots we adjoin (that is, a root of \(f(x)-x\) and \(g(x)-x\), respectively). Since \({{\mathrm{Gal}}}{\left( K_f/{\mathbb {Q}}\right) }\simeq {{\mathrm{Gal}}}{\left( K_g/{\mathbb {Q}}\right) }\simeq {\mathbb {Z}}/2{\mathbb {Z}}\), the natural density of primes that split in these fields is \(\frac{1}{2}\). Moreover, since \(K_f\) and \(K_g\) are linearly disjoint, we know that \({{\mathrm{Gal}}}{\left( K_fK_g/{\mathbb {Q}}\right) }\simeq {\mathbb {Z}}/2{\mathbb {Z}}\times {\mathbb {Z}}/2{\mathbb {Z}}\); thus, when we apply the theorem to the polynomial \(\left( f(x)-x\right) \left( g(x)-x\right) \), we see that the splitting behavior of prime ideals in these two fields is independent.

That is,

As we intend to distinguish dynamical systems by analyzing their periodic points, we will make use of the theory of dynatomic polynomials (and their Galois groups). See [18, 19, 21] (and the correction in [20]), and  [27, Chap. 4.1] for background in this area. We sketch an introduction, focusing on the aspects of the theory we will use in our results.

The degrees of certain dynatomic polynomials will be important quantities in many computations that follow, so we introduce the following notation.

Note that \(nr_k(n)\) is the degree (in x) of the nth dynatomic polynomial of \(x^k+c\in {\mathbb {Q}}(c)[x]\).

As mentioned in Example 1.2, our proof of Theorem 1.1 relies in part on the knowledge of the structure of the Galois groups of \(\Phi _{f,n}\), where \(n\in {\mathbb {Z}}_{>0}\) and \(f(x)=x^k+m\in {\mathbb {Z}}[x]\) for \(k\in {\mathbb {Z}}_{>1}\) and \(m\in {\mathbb {Z}}\). Moreover, we must find arbitrarily large finite sets of polynomials of this form that have the property that the splitting fields of their dynatomic polynomials are linearly disjoint. For a specific polynomial \(f\in {\mathbb {Z}}[x]\) of this form and any large n, it is difficult to compute the Galois group of \(\Phi _{f,n}\), since the degree of \(\Phi _{f,n}\) is so large, but—thanks to work of Morton [19, Theorem D]—the Galois groups of \(\Phi _{f,n}\) for \(f(x)=x^k+c\in {\mathbb {Q}}(c)[x]\) are known. The remainder of this section addresses the question of linear disjointness in the function field setting.

We will need the following elementary lemma of field theory.

The next theorem generalizes the first part of Theorem D in [19].

The corollary below follows immediately from Theorem 2.3 and by work of Morton. It will be crucial in the proof of Theorem 1.1.

In this section, we analyze some statistics of a certain family of wreath products. As these groups appear as Galois groups of dynatomic polynomials, these statistics are a vital component of our proof of Theorem 1.1. We begin with some definitions.

Now, the Galois groups in the conclusion of Corollary 2.4 are isomorphic to direct sums of the wreath products defined above. With this in mind, we need a bit more notation before proceeding—notation whose purpose will become clear in the proof of Theorem 1.1.

We turn to computing \(P_{r,k}\) for general \(r\in {\mathbb {Z}}_{>0}\) and \(k\in {\mathbb {Z}}_{>1}\). To do so, we recall the rencontres numbers from combinatorics. For any \(r\in {\mathbb {Z}}_{>0}\) and \(i\in \left\{ 0,\ldots ,r\right\} \), we will denote the (r, i)th rencontres number by \(D_{r,i}\); that is, \(D_{r,i}\) is the number of permutations of \(\left\{ 1,\ldots ,r\right\} \) with exactly i fixed points. In particular, the number of derangements of \(\left\{ 1,\ldots ,r\right\} \) is \(D_{r,0}\). For convenience, we set \(D_{0,0}=1\). We now record some basic identities involving rencontres numbers, which we will use in the proof of Theorem 3.5.

We now prove an important estimate on \(P_{r,n}\) for all wreath products defined above (that is, a larger class of wreath products than those which arise as Galois groups of dynatomic polynomials).

We record a simple bound we will use in our study of fixed-point proportions. The goal is to prove that \(P_k(n)(1-P_k(n))\) is close enough to \(\frac{1}{n}\) to satisfy the hypotheses of Lemma 3.7, so the exact error bound does not matter much.

Before proving the corollary we will use in the proof of Theorem 1.1, we prove a short lemma about a certain class of recurrence relations.

Putting together the results in this section, we can now prove Corollary 3.3.

In this section, for any polynomial \(f(c,x)\in {\mathbb {Q}}[c][x]\) and any \(a\in {\mathbb {Q}}\), we will write \(f_a\) for the specialization of f at \(c=a\); that is, \(f_a=f_a(x)=f(a,x)\in {\mathbb {Q}}[x]\). Below is a version of the Hilbert irreducibility theorem, one which we will apply in the proof of Theorem 1.1.

Next, we recall a case of the Frobenius density theorem (See [28] for more details).

In light of the Frobenius density theorem, one might hope that given \(f\in {\mathbb {Z}}[x]\) and \(p\in {\mathcal {P}}\), the roots of \(\left[ \Phi _{f,n}\right] _p\) are precisely the points of \(\left( {\mathbb {F}}_p,[f]_p\right) \) of period n, but—as mentioned in Sect. 1—this hope would be in vain. Indeed, even before reducing mod p, if \(\alpha \in \overline{{\mathbb {Q}}}\) is a point of period n in \(\left( \overline{{\mathbb {Q}}},f\right) \), then \(\Phi _{f,n}(\alpha )=0\), but the converse is not always true—that is, there are examples of (K, f), n, \(\alpha \), and d, where \(d<n\), \(\alpha \) is a point of period d, but \(\Phi _{f,n}(\alpha )=0\), see [27, Example 4.2]. In general, if \(\alpha \in \overline{{\mathbb {Q}}}\) and \(\Phi _{f,n}(\alpha )=0\), then \(\alpha \) is of period d for some \(d\le n\), and \(d<n\) is possible only if the polynomial derivative of \(f^d\) evaluated at \(\alpha \) is a root of unity; this quantity is known as the multiplier of \(\alpha \). The way in which the period depends on the multiplier is the content of the following theorem [27, Theorem 4.5].

Luckily, given \(f\in {\mathbb {Z}}[x]\) and \(n\in {\mathbb {Z}}_{>0}\), the following corollary provides a sufficient condition that ensures that for all but finitely many primes \(p\in {\mathcal {P}}\), the dynamical system \(\left( {\mathbb {F}}_p,[f]_p\right) \) has a point of period n if and only if \(\left[ \Phi _{f,n}\right] _p\) has a root. In the proof of Theorem 1.1, we will use the work in Sect. 2 to ensure that the polynomials obtained by applying the Hilbert irreducibility theorem satisfy this sufficient condition.

Finally, we can apply Corollaries 2.4, 3.3, and the results mentioned above to prove Theorem 1.1.