Dynamically distinguishing polynomials
A polynomial with integer coefficients yields a family of dynamical systems indexed by primes as follows: for any prime , reduce its coefficients mod and consider its action on the field . We say a subset of is dynamically distinguishable mod if the associated mod dynamical systems are pairwise non-isomorphic. For any , we prove that there are infinitely many sets of integers of size such that is dynamically distinguishable mod for most (in the sense of natural density). Our proof uses the Galois theory of dynatomic polynomials largely developed by Morton, who proved that the Galois groups of these polynomials are often isomorphic to a particular family of wreath products. In the course of proving our result, we generalize Morton’s work and compute statistics of these wreath products.
Key words and phrases:Arithmetic Dynamics, Finite Fields, Galois Theory, Wreath Products
2010 Mathematics Subject Classification:Primary 37P05; Secondary 37P25, 11R32, 20B35
A (discrete) dynamical system is a pair consisting of a set and a function . The functional graph of , which we will denote by , is the directed graph whose set of vertices is and whose edges are given by the relation if and only if .
Recently there has been interest in the following problem: given a set and a family of self-maps of , describe or enumerate the set , where for two directed graphs and , we write if they are isomorphic as directed graphs. For example, for any and prime power , Bach and the first author [BB13] bound the size of , where and is the set of affine-linear transformations from to itself. Konyagin et. al. [KLM16] give nontrivial upper and lower bounds on . Similarly, Ostafe and Sha [OS16] give bounds on for certain families of rational functions and “sparse” polynomials. A special case of Theorem 2.8 of [KLM16] proves that
as increases amongst odd prime powers. Moreover, the authors suggest that it is “most likely” that for any rational prime with ,
However, they also state that “proving [this suggestion] may be difficult… as there is no intrinsic reason for this to be true.”
In this paper, we study the suggestion of [KLM16] “in reverse”; that is, we fix (integer polynomial) maps, 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 . For and , write
for the polynomial in obtained by reducing the coefficients of mod and
We say that a set is dynamically distinguishable mod if for all with . Let be the natural density on ; that is, for any subset ,
In Section 4, we prove the following theorem.
Let be an integer. For any and any , there exist infinitely many sets of integers of size such that
For any and , the dynamical systems and are isomorphic in the category of dynamical systems on the set if and only if and are dynamically indistinguishable mod . In more generality, for any set and set maps , note that if and only if there exists a bijective set map such that . In many settings, researchers study subcategories of the category of dynamical systems on the set by insisting that the maps , and belong to the set of morphisms in an appropriate category containing as an object. For example, suppose is a field, , and are rational functions. Then in the subcategory of dynamical systems of , with the self-maps of restricted to rational maps, the dynamical systems and are isomorphic if and only if there exists a Möbius transformation such that . Fixing an integer , setting to be rational functions of degree , and studing leads to an interesting moduli space problem, one studied by Silverman in [Sil98] using geometric invariant theory. See [BCE15], [DeM07], and [Lev11] 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 is a dynamical system, let for any . If has the property that there is some with , we say that is periodic or a periodic point of . The smallest such is the period of . As is standard, we will also refer to points of period one as fixed points. Points of period are precisely those that lie in cycles of length in the graph . Periodic points are a classical object of study in discrete dynamical systems over , going back at least to work of Fatou [Fat19, Fat20] and Julia [Jul18] in the early 20th 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 [Bac91] to make rigorous the heuristic assumptions in Pollard’s “rho method” for integer factorization [Pol75]. For example, in [FG14], Flynn and the second author prove that for the family of polynomials in of a fixed degree , the average number of cycles in their associated functional graphs is at least , as long as . More recently, Bellah, the second author, et. al. [BGTW16] develop a heuristic that implies that this average is for any . In [BS15], Burnette and Schmutz prove, for this same family of polynomials, that if as , then the average “ultimate period” of the associated functional graphs is at least .
Our proof of Theorem 1.1 relies on the trivial observation that for any , if one directed graph has a cycle of length and another does not, then the graphs are not isomorphic. As an illustration of our approach, consider the following example.
Let and . If , then has a point of period one if and only if there exists such that
Now, such an exists if and only if the prime ideal splits (or ramifies) in the splitting field of (over ). Similarly, has a fixed point if and only if splits (or ramifies) in the splitting field of . Let and be the splitting fields of and , respectively. The 4 implies that the natural density of primes that split in and is the proportion of their Galois groups that fix a root of the polynomials whose roots we adjoin (that is, a root of and , respectively). Since , the natural density of primes that split in these fields is . Moreover, since and are linearly disjoint, we know that ; thus, when we apply the theorem to the polynomial , we see that the splitting behavior of prime ideals in these two fields is independent. That is,
The goal of this paper is to generalize this argument to points of period greater than one. However, to produce polynomials in and apply the 4, as in Example 1.2, we must prove several theorems to overcome various obstacles. Before describing them, we introduce the notational conventions we will use throughout the rest of the paper. If is a field and , we will write to denote the Galois group of the splitting field of over . Additionally, if is a finite subset of , say with splitting fields , then we will write for the splitting field of . (Of course, if we choose an algebraic closure of , then is isomorphic to the compositum of the images of the embeddings of the s in that algebraic closure.) Similarly, for any family of groups , we will write for their direct product (if for a positive integer , we will write for this group, and if there is some group such that for all , we will write .) The following fact, which we will use often, relates these conventions: if is a field and is finite subset of , say with splitting fields , then the members of are pairwise -linearly disjoint if and only if
Now, if is a group and is the symmetric group on letters, we write to mean the wreath product . That is, , where acts on by permuting coordinates. In particular, we note that . See [Isa08, Chapter 3A] for background on the wreath product. (In Section 3, we introduce and analyze the aspects of the wreath product that we require for this paper.)
With these notations in hand, we can now describe the path to generalizing Example 1.2.
If is a field and , then is a fixed point in if and only if is a root of . To generalize the argument of Example 1.2, we review the famous “dynatomic polynomials of ” in Section 2, which we will denote by for any . These polynomials have the property that for any , every point of period in is a root of (in particular, ). When is the rational function field , Morton [Mor98, Theorem D] proved that if for some , then for any with , the splitting fields of and are linearly disjoint. In Theorem 2.3, we generalize Morton’s theorem to prove that for any , there exist infinitely many sets of integers of size such that for any and with , the splitting fields of and are linearly disjoint. We point out that this includes the case where , which is quite important for our applications.
In Example 1.2, we set , and applied the 4 to . In general, the Galois groups of dynatomic polynomials are quite often wreath products of the form for . To apply the 4, we must study the action of these wreath products on the roots of dynatomic polynomials. In Theorem 3.5, we prove that for any , the proportion of the group (considered with its natural action on ) that acts with a fixed point is approximately .
In Example 1.2, with , we used the fact that for any , the polynomial has a root if and only if has a fixed point. Unfortunately, the picture is not quite so clear for points of period greater than one. For example, if we let , then has exactly one root (with multiplicity two), which happens to have period one in . In Corollary 4.3, we provide a sufficient condition on and that ensures that has a root in if and only if has a point of period for all but finitely many primes .
2. Galois groups of dynatomic polynomials
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 [MP94], [Mor96] (and the correction in [Mor11]), [Mor98], and [Sil07, Chapter 4.1] for background in this area. We sketch an introduction, focusing on the aspects of the theory we will use in our results.
Let be a field, , and . The points of period of the dynamical system are certainly roots of the polynomial . However, if and , then this polynomial vanishes on points of period as well (for example, if is a fixed point of , i.e. , then for all ). In an attempt to sieve out the points of lower period, one defines the th dynatomic polynomial of for any :
where is the usual Möbius function. The fact that
follows quickly by applying the Möbius inversion formula. As usual, we omit “” from the notation “”; we will always specify the set of coefficients of , so that the field will be clear from context. As indicated by its name, the th dynatomic polynomial is analogous to the th cyclotomic polynomial, which vanishes precisely on primitive th roots of unity. (As mentioned in the discussion following Example 1.2, it turns out that may occasionally vanish on points of period for : see [Sil07, Example 4.2]. In Corollary 4.3, we address this inconvenience.) We should mention that it is not a priori obvious that is a polynomial. See [MP94, Theorem 2.5] for a proof that . (In particular, if and is monic, then by Gauss’s Lemma.) The degrees of certain dynatomic polynomials will be important quantities in many computations that follow, so we introduce the following notation.
For any and , let
Note that is the degree (in ) of the th dynatomic polynomial of .
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 , where and for and . 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 of this form and any large , it is difficult to compute the Galois group of , since the degree of is so large, but—thanks to work of Morton [Mor98, Theorem D]—the Galois groups of for 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.
Let be a field and let . Let be an irreducible polynomial, and let be the polynomial in obtained by applying to each of the coefficients of . Let be the splitting fields of , respectively. Then and are isomorphic as fields. In particular,
if is the fraction field of a Dedekind domain and is a prime of , then
ramifies in if and only if ramifies in .
Let be an algebraic closure of containing both and . Then we can extend to some automorphism [Lan02, Theorem V.2.2.8]. It is easy to see that furnishes a one-to-one correspondence between the roots of and the roots of ; thus is an isomorphism. Statement (1) follows immediately, and the map from to is given by
For (2), if the prime of ramifies in , there is a prime of with , and
so ramifies in . Replacing by its inverse shows that the converse holds as well. ∎
For the rest of this section, we will work with polynomials . For any , let
denote the splitting field of , and
denote the splitting field of .
These splitting fields will be defined over or , depending on context. There should be no ambiguity about which definition is intended. Note that in either case, is the compositum of the fields for all positive integers dividing .
The next theorem generalizes the first part of Theorem D in [Mor98].
Let be an integer and . Suppose that . Then there exist infinitely many -tuples of integers such that
Following the proof of Theorem 10 in [Mor98], for any , there exists a polynomial such that the finite primes in that ramify in have the form , where satisfies . The roots of are the roots of the hyperbolic components of the degree- Multibrot set, which is the famous Mandelbrot set when . It is a consequence of the structure of the Multibrot set that and have no roots in common if (closures of hyperbolic components of different periods may only intersect at a root of the component of higher period, see [Bra89] and [Sch94].) For any , consider the unique defined by . Then in the notation of Lemma 2.2, so the primes that ramify in have the form , where satisfies .
With the above facts in mind, let be the (finite) set
then choose such that the sets are pairwise disjoint. As is a finite set, there are infinitely many such choices. For any , let
Recall that for any and , we have . Thus
By our choice of the s, these two fields have no finite ramified primes in common, so they are linearly disjoint over . Therefore the fields are linearly disjoint over . The result now follows by elementary Galois theory.∎
Theorem 9 in [Mor98] shows that satisfies the assumptions of Theorem B in the same paper, which proves that for any , both and are isomorphic to . Applying Lemma 2.2, with , we see that the same is true of the Galois group of for any .
Let be any of the (infinitely many) -tuples that satisfy the conclusion of Theorem 2.3. From the proof of Theorem 2.3, we know that if are distinct integers in and is a positive integer divisor of , then and are linearly disjoint over . Thus
Let . By Theorem B from [Mor98] again, we know is isomorphic to a subgroup of . Conversely, since contains , we see that is isomorphic to a subgroup of , so the proof is complete. ∎
3. Fixed point proportions in wreath products
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.
Suppose that . Recall the definition of from the end of Section 1. Let denote . The group acts on the set ; concretely, for any , this action is
For any , define
then we set
In many cases, this action matches the action of the Galois groups of dynatomic polynomials on the roots of those polynomials, so we make the following definition.
For any and , let
where as in Definition 2.1.
When we apply the results of this section in the proof of Theorem 1.1, the groups will be isomorphic to the groups in a setting where and the roots of are exactly the points of period in . In this setting, we can identify with the union of the cycles of length in in such a way that the permutation action of on the roots of is precisely the action of on described above (see Section 4 of [MP94] for details). In particular, in the proof of Theorem 1.1, we will exploit the fact that
for the polynomials and integers under consideration.
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.
If are groups acting on sets , say with actions , respectively, define the product action of on to be the action
Suppose and let be any increasing arithmetic progression of positive integers. For any , define
so that acts on with the product action defined above. Next, for any , let
once again, acts on with the product action induced from the action of the s on the s. In the proof of Theorem 1.1, we require knowledge of the proportion of these groups that act with a fixed point. To begin specifying the quantity we need, we first set, for any ,
Let . Define
The main technical result of this section is Corollary 3.3, which exhibits a recurrence relation on the terms of sequences of the form and computes the limit of this sequence; the recurrence relation uses the quantities , for —these quantities were defined in in Definition 3.1. We defer the proof until the end of the section, after establishing some estimates on fixed-point proportions in wreath products.
If and is any increasing arithmetic progression of positive integers, then for any ,
We turn to computing for general and . To do so, we recall the rencontres numbers from combinatorics. For any and , we will denote the th rencontres number by ; that is, is the number of permutations of with exactly fixed points. In particular, the number of derangements of is . For convenience, we set . We now record some basic identities involving rencontres numbers, which we will use in the proof of Theorem 3.5, below.
For all ,
For (1), note that a permutation of with precisely fixed points is completely determined by choosing its fixed points and specifying its action on the remaining non-fixed points. For (2), observe that , as each permutation in contributes to exactly one term in the sum, then apply (1). ∎
We now prove an important estimate on for all wreath products defined above (that is, a larger class of wreath products than those which arise as Galois groups of dynatomic polynomials).
Suppose that . Then
We begin by noting that if , then is a multiple of . This follows from the fact that if fixes any , then it must fix each for all . Now, if , , and , then , acting on , has at least fixed points. Moreover, there is a subset of the fixed points of such that
if is a fixed point of , then if and only if .
In other words, if , , and , then if and only if there exists with and for all , if and only if . Using this fact, and enumerating permutations by their number of fixed points, note that
Using Lemma 3.4, we see that