Dynamically distinguishing polynomials
 Andrew Bridy^{1}Email authorView ORCID ID profile and
 Derek Garton^{2}
https://doi.org/10.1186/s4068701701033
© The Author(s) 2017
Received: 13 October 2016
Accepted: 9 March 2017
Published: 10 July 2017
Abstract
A polynomial with integer coefficients yields a family of dynamical systems indexed by primes as follows: For any prime p, reduce its coefficients mod p and consider its action on the field \({\mathbb {F}}_p\). We say a subset of \({\mathbb {Z}}[x]\) is dynamically distinguishable mod p if the associated mod p dynamical systems are pairwise nonisomorphic. For any \(k,M\in {\mathbb {Z}}_{>1}\), we prove that there are infinitely many sets of integers \({\mathcal {M}}\) of size M such that \(\left\{ x^k+m\mid m\in {\mathcal {M}}\right\} \) is dynamically distinguishable mod p for most p (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.
Keywords
Mathematics Subject Classification
1 Introduction
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 \(f(s)=t\).

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

\(\Gamma _{f,p}\) for \(\Gamma \left( {\mathbb {F}}_p,[f]_p\right) \).
Theorem 1.1
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 selfmaps 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

If K is a field and \(f\in K[x]\), then \(\alpha \in K\) is a fixed point in \(\left( K,f\right) \) if and only if \(\alpha \) is a root of \(f(x)x\). To generalize the argument of Example 1.2, we review the famous “dynatomic polynomials of f” in Sect. 2, which we will denote by \(\Phi _{f,n}\) for any \(n\in {\mathbb {Z}}_{>0}\). These polynomials have the property that for any \(n\in {\mathbb {Z}}_{>0}\), every point of period n in \(\left( K,f\right) \) is a root of \(\Phi _{f,n}\) (in particular, \(\Phi _{f,1}=f(x)x\)). When K is the rational function field \({\mathbb {Q}}(c)\), Morton [19, Theorem D] proved that if \(f(x)=x^k+c\) for some \(k\in {\mathbb {Z}}_{>1}\), then for any \(n,n^\prime \in {\mathbb {Z}}_{>0}\) with \(n\ne n^\prime \), the splitting fields of \(\Phi _{f,n}\) and \(\Phi _{f,n^\prime }\) are linearly disjoint. In Theorem 2.3, we generalize Morton’s theorem to prove that for any \(k,M,N\in {\mathbb {Z}}_{>1}\), there exist infinitely many sets of integers \({\mathcal {M}}\) of size M such that for any \(f,g\in \left\{ x^k+(c+m)\mid m\in {\mathcal {M}}\right\} \subseteq {\mathbb {Q}}(c)[x]\) and \(n,n^\prime \) with \(n,n^\prime \le N\), the splitting fields of \(\Phi _{f,n}\) and \(\Phi _{g,n^\prime }\) are linearly disjoint. We point out that this includes the case where \(n=n^\prime \), which is quite important for our applications.

In Example 1.2, we set \(f(x)=x^2+1\) and applied the Frobenius density theorem to \({{\mathrm{Gal}}}{\left( \Phi _{f,1}/{\mathbb {Q}}\right) }\simeq {\mathbb {Z}}/2{\mathbb {Z}}\). In general, the Galois groups of dynatomic polynomials are quite often wreath products of the form \({\mathbb {Z}}/n{\mathbb {Z}}\wr S_r\) for \(n,r\in {\mathbb {Z}}_{>0}\). To apply the Frobenius density theorem, we must study the action of these wreath products on the roots of dynatomic polynomials. In Theorem 3.5, we prove that for any \(n,r\in {\mathbb {Z}}_{>0}\), the proportion of the group \({\mathbb {Z}}/n{\mathbb {Z}}\wr S_r\) (considered with its natural action on \({\mathbb {Z}}/n{\mathbb {Z}}\times \left\{ 1,\ldots ,r\right\} \)) that acts with a fixed point is approximately \(1e^{\frac{1}{n}}\).

In Example 1.2, with \(f(x)=x^2+1\), we used the fact that for any \(p\in {\mathcal {P}}\), the polynomial \(\left[ f(x)x\right] _p\) has a root if and only if \(\left( {\mathbb {F}}_p,[f]_p\right) \) has a fixed point. Unfortunately, the picture is not quite so clear for points of period greater than one. For example, if we let \(g(x)=x^2+3\), then \(\left[ \Phi _{g,2}\right] _5\) has exactly one root (with multiplicity two), which happens to have period one in \(\left( {\mathbb {F}}_5,[g]_5\right) \). In Corollary 4.3, we provide a sufficient condition on \(f\in {\mathbb {Z}}[x]\) and \(n\in {\mathbb {Z}}_{>0}\) that ensures that \(\left[ \Phi _{f,n}\right] _p\) has a root in \({\mathbb {F}}_p\) if and only if \(\left( {\mathbb {F}}_p,[f]_p\right) \) has a point of period n for all but finitely many primes p.

Finally, in Sect. 4, we apply the Hilbert irreducibility theorem to the polynomials produced in Theorem 2.3 to prove Theorem 1.1.
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 [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.
Definition 2.1
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.
Lemma 2.2
 (1)
\({{\mathrm{Gal}}}(f/K)\simeq {{\mathrm{Gal}}}(f^\sigma /K)\), and
 (2)if K is the fraction field of a Dedekind domain and \({\mathfrak {p}}\) is a prime of K, then$$\begin{aligned} {\mathfrak {p}} {\text { ramifies in }}L {\text { if and only if }}\sigma ({\mathfrak {p}}){\text { ramifies in }}L^\sigma . \end{aligned}$$
Proof

\(\Sigma _{f,n}\) denote the splitting field of \(\Phi _{f,n}\), and

\(K_{f,n}\) denote the splitting field of \(f^n(x)x\).
The next theorem generalizes the first part of Theorem D in [19].
Theorem 2.3
Proof
Following the proof of Theorem 10 in [19], for any \(n\in {\mathbb {Z}}_{>0}\), there exists a polynomial \(\delta _n(x)\in {\mathbb {Z}}[x]\) such that the finite primes in \(\overline{{\mathbb {Q}}}(c)\) that ramify in \(\Sigma _{f,n}\) have the form \(cb\), where \(b\in \overline{{\mathbb {Q}}}\) satisfies \(\delta _n(b)=0\). The roots of \(\delta _n(x)\) are the roots of the hyperbolic components of the degreek Multibrot set, which is the famous Mandelbrot set when \(k=2\). It is a consequence of the structure of the Multibrot set that \(\delta _n(x)\) and \(\delta _d(x)\) have no roots in common if \(d\ne n\). (Closures of hyperbolic components of different periods may only intersect at a root of the component of higher period, see [5, 24].) For any \(m\in {\mathbb {Z}}\), consider the unique \(\sigma \in {{\mathrm{Aut}}}(\overline{{\mathbb {Q}}}(c)/\overline{{\mathbb {Q}}})\) defined by \(\sigma (c)=c+m\). Then \(f+m=f^\sigma \) in the notation of Lemma 2.2, so the primes that ramify in \(\Sigma _{f+m,n}\) have the form \(c\left( bm\right) \), where \(b\in \overline{{\mathbb {Q}}}\) satisfies \(\delta _n(b)=0\).
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.
Corollary 2.4

any field in \({\mathcal {F}}\left( {\mathbf {m}}\right) \) is linearly disjoint from the compositum of the others,

if \(\Sigma _{f+m_i,d}\in {\mathcal {F}}({\mathbf {m}})\), then \({{\mathrm{Gal}}}{\left( \Sigma _{f+m_i,d}/{\mathbb {Q}}(c)\right) }\simeq {{\mathrm{Gal}}}{\left( \Sigma _{f+m_i,d}/\overline{{\mathbb {Q}}}(c)\right) }\simeq \left( {\mathbb {Z}}/d{\mathbb {Z}}\wr S_{r_k(d)}\right) \), and

\({{\mathrm{Gal}}}{\left( \left( \prod _{i=1}^M{K_{f+m_i, N}}\right) \Big /{\mathbb {Q}}(c)\right) }\simeq \prod _{i=1}^N{\prod _{d\mid N}{\left( {\mathbb {Z}}/d{\mathbb {Z}}\wr S_{r_k(d)}\right) }}\).
Proof
Theorem 9 in [19] shows that \(f(x)=x^k+c\in {\mathbb {Q}}(c)[x]\) satisfies the assumptions of Theorem B in the same paper, which proves that for any \(n\in {\mathbb {Z}}_{>0}\), both \({{\mathrm{Gal}}}(\Phi _{f,n}/{\mathbb {Q}}(c))\) and \({{\mathrm{Gal}}}(\Phi _{f,n}/\overline{{\mathbb {Q}}}(c))\) are isomorphic to \({\mathbb {Z}}/d{\mathbb {Z}}\wr S_{r_k(d)}\). Applying Lemma 2.2, with \(\sigma :c\mapsto c+m\), we see that the same is true of the Galois group of \(\Phi _{f+m,n}\) for any \(m\in {\mathbb {Q}}\).
3 Fixedpoint 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.
Definition 3.1
Remark 3.2
When we apply the results of this section in the proof of Theorem 1.1, the groups \({\mathbb {Z}}/n{\mathbb {Z}}\wr S_{r_k(n)}\) will be isomorphic to the groups \({{\mathrm{Gal}}}(\Phi _{f,n}/{\mathbb {Q}})\) in a setting where \(f\in {\mathbb {Z}}[x]\) and the roots of \(\Phi _{f,n}\) are exactly the \(nr_k(n)\) points of period n in \(\left( \overline{{\mathbb {Q}}},f\right) \). In this setting, we can identify \(B\left( n,r_k(n)\right) \) with the union of the \(r_k(n)\) cycles of length n in \((\overline{{\mathbb {Q}}},f)\) in such a way that the permutation action of \({{\mathrm{Gal}}}{\left( \Phi _{f,n}/{\mathbb {Q}}\right) }\) on the roots of \(\Phi _{f,n}\) is precisely the action of \({\mathbb {Z}}/n{\mathbb {Z}}\wr S_{r_k(n)}\) on B(n, r) described above (see Sect. 4 of [21] for details).
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.
Corollary 3.3
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.
Lemma 3.4
 (1)
\(D_{r,i}=\left( {\begin{array}{c}r\\ i\end{array}}\right) D_{ri,0}\) and
 (2)
\(\sum _{i=1}^r{\left( {\begin{array}{c}r\\ i\end{array}}\right) D_{ri,0}}=r!D_{r,0}\).
Proof
For (1), note that a permutation of \(\left\{ 1,\ldots ,r\right\} \) with precisely i fixed points is completely determined by choosing its i fixed points and specifying its action on the \(ri\) remaining nonfixed points. For (2), observe that \(\sum _{i=0}^r D_{r,i} = S_r=r!\), as each permutation in \(S_r\) contributes to exactly one term in the sum, then apply (1). \(\square \)
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).
Theorem 3.5
Proof

\(\left R\right =j\) and

if \(i^\prime \in \left\{ 1,\ldots ,r\right\} \) is a fixed point of \(\pi \), then \(i^\prime \in R\) if and only if \(\overline{a_{i^\prime }}=0\).
We record a simple bound we will use in our study of fixedpoint proportions. The goal is to prove that \(P_k(n)(1P_k(n))\) is close enough to \(\frac{1}{n}\) to satisfy the hypotheses of Lemma 3.7, so the exact error bound does not matter much.
Theorem 3.6
Proof
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.
Lemma 3.7
Proof
Putting together the results in this section, we can now prove Corollary 3.3.
Proof of Corollary 3.3
4 Applying the Hilbert Irreducibility and the Frobenius Density Theorems
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.
Remark 4.1
The Hilbert irreducibility theorem is normally stated for irreducible polynomials (as in [25]). To obtain the version stated above, let g(c, x) be the minimal polynomial of a primitive element of \(K/{\mathbb {Q}}(c)\), which is irreducible over \({\mathbb {Q}}(c)\). Then specialize g(c, x) instead of f(c, x). Moreover, if f(c, x) has no repeated roots in \(\overline{{\mathbb {Q}}(c)}\), then there are only finitely many \(a\in {\mathbb {Q}}\) for which \(f_a(x)\) has a repeated root in \(\overline{{\mathbb {Q}}}\) (these are precisely the a for which the discriminant \({{\mathrm{Disc}}}{f(c,x)}\) vanishes under the specialization at \(c=a\)). For more on the connection between the Hilbert irreducibility theorem and Galois theory, see, for example, [7], [15, Chap. VIII], and [30, Chap. 1].
Next, we recall a case of the Frobenius density theorem (See [28] for more details).
Remark 4.2
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].
 (1)
\(n=m\),
 (2)
\(n=mj\), when \(\lambda \) is a primitive jth root of unity, or
 (3)
\(n=mjp^e\), with \(e\in {\mathbb {Z}}_{>0}\), when \(\lambda \) is a primitive jth root of unity and \({{\mathrm{char}}}{K}=p>0\).
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.
Corollary 4.3
Proof
Let \(p\in {\mathcal {P}}\), and suppose that \(\left[ \Phi _{f,n}\right] _p\) has a root. Let K be the splitting field of \(\Phi _{f,n}\), choose any prime \({\mathfrak {p}}\) lying over p, and denote by \(\overline{\cdot }\) the reduction \({\mathcal {O}}_K\rightarrow {\mathcal {O}}_K/{\mathfrak {p}}\). Since \({\mathcal {O}}_K/{\mathfrak {p}}\) is an extension of \({\mathbb {F}}_p\) and \(\left[ \Phi _{f,n}\right] _p\) has a root, there exists \(a\in {\mathbb {Z}}\subseteq {\mathcal {O}}_K\) such that \(\left[ \Phi _{f,n}\right] _p\left( \overline{a}\right) =0\). Since \(\Phi _{f,n}\) splits in K, we know that \(\left[ \Phi _{f,n}\right] _p\) splits in \({\mathcal {O}}_K/{\mathfrak {p}}\) and the roots of \(\Phi _{f,n}\) map onto the roots of \(\left[ \Phi _{f,n}\right] _p\) under \(\overline{\cdot }\); choose any \(\alpha \in K\) such that \(\Phi _{f,n}(\alpha )=0\) and \(\overline{\alpha }=\overline{a}\).
Now suppose that \([f]_p\) has a point of period n in \(\left( {\mathbb {F}}_p,[f]_p\right) \). It is easy to see that \([f^n]_p=\left( [f]_p\right) ^n\), so \(\left[ \Phi _{f,n}\right] _p= \Phi _{[f]_p,n}\). By the roots and multipliers theorem, with \(K={\mathbb {F}}_p\), we know that \(\left[ \Phi _{f,n}\right] _p\) has a root in \({\mathbb {F}}_p\). \(\square \)
Finally, we can apply Corollaries 2.4, 3.3, and the results mentioned above to prove Theorem 1.1.
Proof of Theorem 1.1

any field in \(\bigcup _{j=1}^M{{\mathcal {F}}\left( m_j\right) }\) is linearly disjoint from the compositum of the others,

if \(\Sigma _{f_{m_j},d}\in \bigcup _{j=1}^M{{\mathcal {F}}\left( m_j\right) }\), then \({{\mathrm{Gal}}}{\left( \Sigma _{f_{m_j},d}/{\mathbb {Q}}\right) }\simeq \left( {\mathbb {Z}}/d{\mathbb {Z}}\wr S_{r_k(d)}\right) \), and

for any \(j\in \left\{ 1,\ldots ,M\right\} \), we know \(\left( f_{m_j}\right) ^N(x)x\) has no repeated roots.
Acknowledgements
We would like to thank the referee for a careful reading of the paper and very helpful comments on the presentation. We also thank Patrick Morton for helpful comments about the proofs in his series of papers on dynatomic polynomials, Rafe Jones for pointing us to Morton’s work, and Robert Lemke Oliver for many constructive conversations regarding the topics in this paper.
Declarations
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