Valentin Huguin
email: valentin.huguin@math.univ-toulouse.fr

UNICRITICAL POLYNOMIAL MAPS WITH RATIONAL MULTIPLIERS

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Unicritical polynomial maps with rational multipliers

Introduction

Given a polynomial map f : C → C and a point z 0 ∈ C, we study the sequence (f •n (z 0 )) n≥0 of iterates of f at z 0 . The set {f •n (z 0 ) : n ≥ 0} is called the forward orbit of z 0 under f .

The point z 0 is said to be periodic for f if there exists an integer n ≥ 1 such that f •n (z 0 ) = z 0 ; the least such integer n is called the period of z 0 . The forward orbit of z 0 , which has cardinality n, is said to be a cycle for f . The multiplier of f at z 0 is the derivative of f •n at z 0 ; equivalently, it is the product of the derivatives of f along the cycle. In particular, f has the same multiplier at each point of the cycle.

The multiplier is invariant under conjugacy: if f and g are two polynomial maps, φ is an invertible affine map such that φ • f = g • φ and z 0 is a periodic point for f , then φ (z 0 ) is a periodic point for g with the same period and the same multiplier.

In this paper, we wish to examine the polynomial maps that have only integeror rational -multipliers. Definition 1. A polynomial map f : C → C of degree d ≥ 2 is said to be a power map if it is affinely conjugate to z → z d .

For every d ≥ 2, there exists a unique polynomial T d ∈ C[z] such that

T d z + z -1 = z d + z -d .
The polynomial T d is monic of degree d and is called the dth Chebyshev polynomial.

Example 2. We have T 2 (z) = z 2 -2 and T 3 (z) = z 3 -3z. Definition 3. A polynomial map f : C → C of degree d ≥ 2 is said to be a Chebyshev map if it is affinely conjugate to ±T d .

Remark 4. For every d ≥ 2, the polynomials -T d and T d are affinely conjugate if and only if d is even.

These conjugacy classes of polynomials share the following well-known property: Proposition 5. Suppose that f : C → C is a power map or a Chebyshev map. Then f has only integer multipliers.

2010 Mathematics Subject Classification. Primary 37P05, 37P35; Secondary 37F10, 37F45. 1 1.1. Statement of the results. We are interested in the converse of Proposition 5. More precisely, we wish to show that every polynomial map that has only integeror rational -multipliers is either a power map or a Chebyshev map.

We restrict ourselves to unicritical polynomial maps -that is, polynomial maps of degree d ≥ 2 that have a unique critical point in the complex plane. Theorem 6. Assume that f : C → C is a unicritical polynomial map that has only rational multipliers. Then f is either a power map or a Chebyshev map.

Remark 7. For every d ≥ 2, the polynomial T d has exactly d-1 critical points given by 2 cos πj d for j ∈ {1, . . . , d -1}. In particular, a Chebyshev map is unicritical if and only if it has degree 2.

Using similar arguments, we also obtain a result concerning cubic polynomial maps with symmetries -that is, cubic polynomial maps that commute with a nontrivial invertible affine map.

Theorem 8. Assume that f : C → C is a cubic polynomial map with symmetries that has only integer multipliers. Then f is a power map or a Chebyshev map.

1.2. Motivation. In a more general setting, Milnor conjectured in [START_REF]Dynamics on the Riemann sphere[END_REF] that power maps, Chebyshev maps and flexible Lattès maps are the only rational maps whose multipliers are all integers. We may even extend his question as follows: Question 9. Let K be a number field, and denote by O K its ring of integers. Assume that f : C → C is a rational map whose multipliers all lie in O K -or K. Is f necessarily a finite quotient of an affine map -that is, either a power map, a Chebyshev map or a Lattès map?

We give here a positive answer in the case of rational numbers and unicritical polynomial maps. To the author's knowledge, this question has not been studied before. This one can be viewed as an analog of questions concerning rational preperiodic points for a rational map, which have received a lot of attention (see [BIJ + 19] and [START_REF] Joseph | The arithmetic of dynamical systems[END_REF]).

In [START_REF] Eremenko | Rational maps with real multipliers[END_REF], Eremenko and van Strien investigated the rational maps that have only real multipliers: they proved that, if f : C → C is such a map, then either f is a Lattès map or its Julia set J f is contained in a circle; they also gave a description of these maps.

1.3. Organization of the paper. In Section 2, we prove some stronger versions of Theorem 6. More precisely, given an integer d ≥ 2, the conjugacy classes of unicritical polynomials of degree d are parameterized by a one-parameter family (f c ) c∈C , and we determine the parameters c ∈ C for which the multiplier polynomials of f c have only rational roots. Using the same strategy, we also prove Theorem 8.

In Section 3, we study the periodic points and the multipliers of a polynomial map by means of polynomials associated with its dynamics. More precisely, we present certain results about the dynatomic polynomials and the multiplier polynomials of a monic polynomial, emphasizing the case of unicritical polynomials.

Finally, in Section A, we prove that the Diophantine equation that arose in our proof of Theorem 6 has no nontrivial solution.

Acknowledgments. The author would like to thank his Ph.D. advisors, Xavier Buff and Jasmin Raissy, for all their suggestions and encouragements.

Proofs of the results

We shall prove here Theorem 6 and Theorem 8. Our proofs rely on the result below, which will be presented in greater detail in Section 3 and is an immediate consequence of Proposition 30, Proposition 38 and Corollary 42.

Fix an integer d ≥ 2. Since every polynomial map f : C → C is affinely conjugate to a monic polynomial map and the multiplier is invariant under conjugacy, we may restrict our attention to monic polynomials.

Proposition 10. Assume that f : C → C is a monic polynomial map of degree d. Then

• there exists a unique sequence Φ f n n≥1 of elements of C[z] such that, for every n ≥ 1, we have

f •n (z) -z = k|n Φ f k (z) ; • for every n ≥ 1, there is a unique monic polynomial M f n ∈ C[λ] such that M f n (λ) n = res z Φ f n (z), λ -(f •n ) (z) ,
where res z denotes the resultant with respect to z; • given a subring R of C and n ≥ 1, the multipliers of f at its cycles with period n all lie in R if and only if

M f n splits into linear factors of R[λ].
Definition 11. Suppose that f : C → C is a monic polynomial map of degree d.

For n ≥ 1, the polynomial Φ f n is called the nth dynatomic polynomial of f and the polynomial M f n is called the nth multiplier polynomial of f .

For c ∈ C, let f c : C → C be the polynomial map

f c : z → z d + c .
For every c ∈ C, the map f c is unicritical with critical point 0 and critical value c. Furthermore, if f : C → C is a unicritical polynomial map of degree d, then there exists a parameter c ∈ C -which is unique up to multiplication by a (d -1)th root of unity -such that f is affinely conjugate to f c . Consequently, to prove Theorem 6, we are reduced to determining the parameters c ∈ C for which the polynomials

M fc n ∈ C[λ], with n ≥ 1, have only rational roots. Note that, if c ∈ C is such a parameter, then, for every n ≥ 1, the polynomial M fc n lies in Q[λ] and its discriminant ∆ n (c) = disc M fc n
is the square of a rational number. In fact, we shall see that, to prove Theorem 6, it suffices to examine the polynomials M fc n for only a few small values of n.

2.1. Quadratic polynomial maps. Let us examine here the quadratic polynomials that have only integer -or rational -multipliers. Suppose that d = 2. Then, for every c ∈ C, the map f c is a power map if and only if c = 0 and is a Chebyshev map if and only if c = -2. Using the software SageMath, we can compute M fc n and ∆ n (c) for c ∈ C and small values of n.

Example 12. For every c ∈ C, we have

M fc 1 (λ) = λ 2 -2λ + 4c , M fc 2 (λ) = λ -4c -4 , M fc 3 (λ) = λ 2 + (-8c -16)λ + 64c 3 + 128c 2 + 64c + 64 , M fc 4 (λ) = λ 3 + 16c 2 -48 λ 2 + -256c 4 -256c 3 + 256c 2 + 768 λ -4096c 6 -12288c 5 -12288c 4 -12288c 3 -8192c 2 -4096 .
Remark 13. Observe that, for n ∈ {1, . . . , 4}, the coefficients of M fc n are polynomials in 4c with integer coefficients. As we shall see in Section 3, this is true for all n ≥ 1 (compare [Bou14, Lemma 1]).

Example 14. For every c ∈ C, we have

∆ 1 (c) = -2 2 (4c -1) , ∆ 2 (c) = 1 , ∆ 3 (c) = -2 6 (4c + 7)c 2 , ∆ 4 (c) = -2 24 64c 3 + 144c 2 + 108c + 135 (c + 2) 2 c 6 .
Remark 15. We shall see in Section 3 that, for every n ≥ 1, the roots of ∆ n that have an odd multiplicity are precisely the parameters c 0 ∈ C for which the map f c0 has a cycle with period n and multiplier 1 (see [START_REF] Morton | On certain algebraic curves related to polynomial maps[END_REF]Proposition 9]).

First, let us examine the quadratic polynomials whose multipliers are integers. By Proposition 10, for every c ∈ C, the map f c has an integer multiplier at each cycle with period 1 or 2 if and only if the polynomials M fc 1 and M fc 2 split into linear factors of Z[λ], which occurs if and only if there exists m ∈ Z such that c = 1-m 2 4 . In particular, there exist infinitely many such parameters c ∈ C. In contrast, by considering also the multipliers at the cycles with period 3, we obtain the following: Proposition 16. Assume that f : C → C is a quadratic polynomial map that has an integer multiplier at each cycle with period less than or equal to 3. Then f is either a power map or a Chebyshev map.

Proof. There exists a parameter c ∈ C such that f is affinely conjugate to f c . By Proposition 10, the polynomials M fc n , with n ∈ {1, 2, 3}, split into linear factors of Z[λ], and hence 4c is an integer and ∆ 1 (c) = -2 2 (4c -1) and ∆ 3 (c) = -2 6 (4c + 7)c 2 are the squares of integers. Therefore, either c = 0 or there exist a, b ∈ Z ≥0 such that -(4c -1) = a 2 and -(4c + 7) = b 2 . In the latter case, we have (a -b)(a + b) = 8, and hence

a -b = 1 a + b = 8 or a -b = 2 a + b = 4 ,
which yields (a, b) = (3, 1) and c = -2. Thus, the proposition is proved.

Let us now study the quadratic polynomial maps whose multipliers are rational. There exist infinitely many parameters c ∈ C for which the map f c has a rational multiplier at each cycle with period less than or equal to 3. More precisely, a parameter c ∈ C has this property if and only if c is rational and ∆ 1 (c) and ∆ 3 (c) are the squares of rational numbers, which occurs if and only if c = 0 or there

exists r ∈ Q =0 such that c = -(r 4 +3r 2 +4) 4r 2
. In contrast, by considering also the multipliers at the cycles with period 4, we are led to examine the rational points on a certain elliptic curve and we obtain the following result, which is a stronger version of Theorem 6 in the case of quadratic polynomials: Proposition 17. Assume that f : C → C is a quadratic polynomial map that has a rational multiplier at each cycle with period less than or equal to 4. Then f is either a power map or a Chebyshev map.

Proof. There exists a parameter c ∈ C such that f is affinely conjugate to f c . By Proposition 10, the polynomials M fc n , with n ∈ {1, . . . , 4}, split into linear factors of Q[λ], and hence c is rational and

∆ 4 (c) = -2 24 64c 3 + 144c 2 + 108c + 135 (c + 2) 2 c 6
is the square of a rational number. Note that, if there exists r ∈ Q such that

-64c 3 + 144c 2 + 108c + 135 = r 2 , then c = -3
4 and the rational numbers a = r-18 3(4c+3) and b = -(r+18) 3(4c+3) satisfy

a 3 + b 3 -4 = -4 64c 3 + 144c 2 + 108c + 135 + r 2 (4c + 3) 3 = 0 ,
which contradicts the fact that the Diophantine equation x 3 + y 3 = 4z 3 has no solution (x, y, z) ∈ Z 3 with z = 0 by Lemma 56. Therefore, we have c ∈ {-2, 0}. Thus, the proposition is proved.

Remark 18. It follows from [EvS11, Theorem 1] that, for every c ∈ C, the map f c has a real multiplier at each cycle if and only if c ∈ (-∞, -2] ∪ {0}. In particular, the property of having only real multipliers does not characterize power maps and Chebyshev maps among the quadratic polynomials.

2.2. Unicritical polynomial maps of degree at least 3. We shall see here that, unlike in the case of quadratic polynomials, power maps are the only unicritical polynomial maps of degree at least 3 that have only real multipliers. Note that, for every c ∈ C, the map f c is a power map if and only if c = 0. First, suppose that d = 3. Using the software SageMath, we can compute M fc n and ∆ n (c) for c ∈ C and n ∈ {1, 2}.

Example 19. For every c ∈ C, we have

M fc 1 (λ) = λ 3 -6λ 2 + 9λ -27c 2 and M fc 2 (λ) = λ 3 -27λ 2 + 162c 2 + 243 λ -729c 4 -1458c 2 -729 .
Remark 20. We shall see in Section 3 that, for every n ≥ 1, the coefficients of M fc n are polynomials in 27c 2 with integer coefficients (compare [Mil14, Theorem 1.1]).

Example 21. For every c ∈ C, we have ∆ 1 (c) = -3 6 27c 2 -4 c 2 and ∆ 2 (c) = -3 12 27c 2 + 32 c 6 . It follows from Proposition 10 that, for every c ∈ C, the map f c has a real multiplier at each fixed point if and only if c 2 is real and ∆ 1 (c) ≥ 0, which occurs if and only if c 2 ∈ 0, 4 27 . In particular, power maps are not the only cubic unicritical polynomial maps whose multiplier at each fixed point is real.

Remark 22. There also exist infinitely many parameters c ∈ C for which the map f c has a rational multiplier at each fixed point. More precisely, a parameter c ∈ C has this property if and only if the polynomial M fc 1 has a rational root and its discriminant ∆ 1 (c) is the square of a rational number, which occurs if and only if

there exists r ∈ Q such that c 2 = 4(r 2 -1) 2 (r 2 +3) 3 .
In contrast, by considering also the multipliers at the cycles with period 2, we obtain the result below, which immediately implies Theorem 6 in the case of cubic unicritical polynomials.

Proposition 23. Assume that f : C → C is a cubic unicritical polynomial map that has a real multiplier at each cycle with period 1 or 2. Then f is a power map.

Proof. There exists a parameter c ∈ C such that f is affinely conjugate to f c . By Proposition 10, the polynomials M fc 1 and M fc 2 split into linear factors of R[λ], and hence c 2 is real and

∆ 1 (c) = -3 6 27c 2 -4 c 2 ≥ 0 and ∆ 2 (c) = -3 12 27c 2 + 32 c 6 ≥ 0 .
Therefore, we have

c 2 ∈ -32 27 , 0 ∩ 0, 4 27 = {0} .
Thus, the proposition is proved.

Let us now investigate the unicritical polynomial maps of degree at least 4 whose multipliers are real. We shall see that, unlike in the case of cubic unicritical polynomials, the property of having a real multiplier at each fixed point characterizes here power maps. Our result relies on the calculation of M fc 1 for c ∈ C. Example 24. Suppose that d ≥ 2 and c ∈ C. Then we have

M fc 1 (λ) = res z z d -z + c, λ -dz d-1 = (-d) d d-1 j=1 z d j -z j + c , where z 1 , . . . , z d-1 are the roots of dz d-1 -λ ∈ C[z]. It follows that M fc 1 (λ) = (-d) d d-1 j=1 d -1 (λ -d)z j + c = (-d) d   c d-1 + d-1 j=1 d -j (λ -d) j σ j c d-1-j   ,
where σ 1 , . . . , σ d-1 are the elementary symmetric functions of z 1 , . . . , z d-1 . Therefore, by the relations between roots and coefficients of a polynomial, we have

M fc 1 (λ) = λ(λ -d) d-1 + (-d) d c d-1
. The following result is a stronger version of Theorem 6 in the case of unicritical polynomials of degree at least 4.

Proposition 25. Assume that f : C → C is a unicritical polynomial map of degree d ≥ 4 that has a real multiplier at each fixed point. Then f is a power map.

Proof. There exists a parameter c ∈ C such that f is affinely conjugate to f c . By Proposition 10, the polynomial

M fc 1 (λ) = λ(λ -d) d-1 + (-d) d c d-1 splits into linear factors of R[λ],
and hence the same is true of the polynomial

L(λ) = λ d M fc 1 λ -1 + d = (-d) d c d-1 λ d + dλ + 1
and, by Rolle's theorem, of its derivative

L (λ) = (-1) d d d+1 c d-1 λ d-1 + d .
Therefore, we have c = 0 since the set of roots of L is invariant under multiplication by a (d -1)th root of unity and d ≥ 4. Thus, the proposition is proved.

Finally, we have proved Theorem 6, which follows immediately from Proposition 17, Proposition 23 and Proposition 25.

2.3. Cubic polynomial maps with symmetries. We shall use here the same strategy to study the cubic polynomial maps with symmetries whose multipliers are integers and prove Theorem 8.

For a ∈ C, let g a : C → C be the cubic polynomial map

g a : z → z 3 + az .
For every a ∈ C, the map g a fixes 0 with multiplier a and commutes with z → -z. Furthermore, if f : C → C is a cubic polynomial map with symmetries, then there exists a unique parameter a ∈ C such that f is affinely conjugate to g a . Unlike the family of cubic unicritical polynomial maps, the family of cubic polynomial maps with symmetries contains both power maps and Chebyshev maps. More precisely, for every a ∈ C, the map g a is a power map if and only if a = 0 and is a Chebyshev map if and only if a = ±3.

Using the software SageMath, we can compute M ga n for a ∈ C and n ∈ {1, 2, 3}. Example 26. For every a ∈ C, we have

M ga 1 (λ) = (λ -a)(λ + 2a -3) 2 , M ga 2 (λ) = λ -4a 2 -12a -9 λ + 2a 2 -9 2 , M ga 3 (λ) = N 3 (a, λ) 2 , where N 3 ∈ Z[a, λ] is given by N 3 (a, λ) = λ 4 + 2a 3 + 12a 2 -18a -108 λ 3 + -48a 6 -72a 5 + 396a 4 + 486a 3 -324a 2 + 1458a + 4374 λ 2 + 32a 9 -792a 7 -432a 6 + 5832a 5 + 5832a 4 -7290a 3 -8748a 2 -39366a -78732 λ + 256a 12 + 384a 11 -4608a 10 -6912a 9 + 24624a 8 + 36936a 7 -23328a 6 -34992a 5 -131220a 4 -196830a 3 + 236196a 2 + 354294a + 531441 .
Moreover, we have

disc λ N 3 (a, λ) = 2 12 3 12 D 3 (a) 4a 3 + 12a 2 -3a -27 2 (a -3) 4 (a + 3) 4 a 12 ,
where disc λ denotes the discriminant with respect to λ and D 3 ∈ Z[a] is given by D 3 (a) = 4a 8 + 16a 7 -35a 6 -206a 5 -113a 4 + 376a 3 + 715a 2 + 1690a + 2197 .

It follows from Proposition 10 that, for every a ∈ C, the map g a has an integer multiplier at each cycle with period 1 or 2 if and only if a is an integer. By considering also the multipliers at the cycles with period 3, we obtain the following stronger version of Theorem 8: Proposition 27. Assume that f : C → C is a cubic polynomial map with symmetries that has an integer multiplier at each cycle with period less than or equal to 3. Then f is either a power map or a Chebyshev map.

Proof. There exists a parameter a ∈ C such that f is affinely conjugate to g a . By Proposition 10, the polynomials M ga n , with n ∈ {1, 2, 3}, split into linear factors of Z[λ], and hence a is an integer and

disc λ N 3 (a, λ) = 2 12 3 12 D 3 (a) 4a 3 + 12a 2 -3a -27 2 (a -3) 4 (a + 3) 4 a 12
is the square of an integer. Now, note that, if

D 3 (a) = 4a 8 + 16a 7 -35a 6 -206a 5 -113a 4 + 376a 3 + 715a 2 + 1690a + 2197
is the square of an integer, then its residue class in Z/32Z is a square, and hence a ≡ 1 (mod 8). Moreover, observe that D 3 (1 + 8b) is not the square of an integer whenever b ∈ {-7, . . . , 13} and we have

L(b) 2 < D 3 (1 + 8b) < (L(b) + 1)
2 for all b ∈ Z \ {-7, . . . , 13}, where

L(b) = 8192a 4 + 6144a 3 + 720a 2 -252a -50 .
Therefore, D 3 (a) is not the square of an integer, and hence a ∈ {-3, 0, 3}. Thus, the proposition is proved.

Using the software SageMath, we obtain that D 3 (a) is not the square of a rational number whenever a is a rational number with height at most 10 4 . Thus, it seems likely that the question below has a negative answer, which would imply that every cubic polynomial map with symmetries that has a rational multiplier at each cycle with period less than or equal to 3 is either a power map or a Chebyshev map.

Question 28. Does the hyperelliptic curve of genus 3 over Q given by b 2 = D 3 (a) have a rational point other than the two points at infinity? Remark 29. Note that the curve of genus 1 given by N 3 (a, λ) = 0 together with the point 9 2 , 1647 4 defines an elliptic curve E over Q. Using the software Magma, we obtain that its group of rational points E(Q) is a free abelian group of rank 1. In particular, there exist infinitely many parameters a ∈ C for which the map g a has a rational multiplier at each cycle with period 1 or 2 and at a cycle with period 3. Another approach to proving that power maps and Chebyshev maps are the only cubic polynomial maps with symmetries that have a rational multiplier at each cycle with period less than or equal to 3 could be to show that the group E(Q) does not contain 4 distinct points with the same a-coordinate.

Certain polynomials related to the dynamics of a polynomial map

We study here the dynamics of a polynomial map from an algebraic point of view. More precisely, we present certain results about the dynatomic polynomials and the multiplier polynomials of a monic polynomial. In particular, we shall prove here Proposition 10, which is the key point in our proofs of Theorem 6 and Theorem 8. The results presented in this section are mostly well known.

Fix an integer d ≥ 2. Although we are interested in the dynamics of complex polynomial maps, we shall consider monic polynomials over an arbitrary integral domain in order to derive information about the coefficients of the dynatomic polynomials and the multiplier polynomials.

3.1. Dynatomic polynomials. Let us present here the dynatomic polynomials of a monic polynomial, which are related to its periodic points.

If R is a commutative ring and f ∈ R[z] is a monic polynomial of degree d, then, for every n ≥ 1, the roots of the polynomial f •n (z) -z are precisely the periodic points for the map f : R → R with period dividing n. Thus, it is natural to try to factor these polynomials in order to separate the periodic points for f : R → R according to their periods.

In the particular case where a 0 , . . . , a d-1 are indeterminates over Z,

R = Z [a 0 , . . . , a d-1 ] and f (z) = z d + d-1 j=0 a j z j ∈ R[z] ,
the polynomials f •n (z) -z, with n ≥ 1, are separable, which allows us to factor them over R according to the periods of their roots. By a specialization argument, this provides a factorization of the polynomials f •n (z) -z, with n ≥ 1, in the general case of a commutative ring R and a monic polynomial f ∈ R[z] of degree d (compare [MP94, Section 2]). More precisely, we have the following result, where µ : Z ≥1 → {-1, 0, 1} denotes the Möbius function:

Proposition 30. Suppose that R is a commutative ring and f ∈ R[z] is a monic polynomial of degree d. Then there exists a unique sequence Φ f n n≥1 of elements of R[z] such that, for every n ≥ 1, we have

f •n (z) -z = k|n Φ f k (z) .
Furthermore, for every n ≥ 1, the polynomial Φ f n is monic of degree ν(n), where

ν(n) = k|n µ n k d k .
Definition 31. Suppose that R is a commutative ring and f ∈ R[z] is a monic polynomial of degree d. For n ≥ 1, the polynomial Φ f n is called the nth dynatomic polynomial of f . Remark 32. If R is an integral domain and f ∈ R[z] is a monic polynomial of degree d, then it follows from the Möbius inversion formula that, for every n ≥ 1, we have

Φ f n (z) = k|n f •k (z) -z µ( n k ) .
Remark 33. If R is a commutative ring and f ∈ R[z] is a nonmonic polynomial of degree d, then the existence of a sequence Φ f n n≥1 as in Proposition 30 holds but the uniqueness may fail when R is not an integral domain. For example, if

R = Z/4Z and f (z) = 2z 2 + z ∈ R[z] ,
then we have

f •n (z) = 2z 2 + z if n is odd z if n is even
for all n ≥ 1, and hence every sequence

Φ f n n≥1 of elements of R[z] satisfying Φ f 1 (z) = 2z 2 , Φ f 2 ∈ 2R[z] and Φ f n (z) = 1 for n ≥ 3 is such that, for every n ≥ 1, we have f •n (z) -z = k|n Φ f k (z) .
Let us now describe the roots of the dynatomic polynomials. If R is a commutative ring, f ∈ R[z] is a monic polynomial of degree d and n ≥ 1, then each root of the polynomial Φ f n is a periodic point for the map f : R → R with period dividing n since it is also a root of the polynomial f •n (z) -z. Conversely, if R is an integral domain, then each periodic point for f : R → R with period n is a root of Φ f n . However, it may occur that roots of Φ f n have period less than n. More precisely, we have the following result:

Proposition 34 ([MS95, Proposition 3.2]). Assume that R is an integral domain, f ∈ R[z] is a monic polynomial of degree d and n ≥ 1. Then z 0 ∈ R is a root of the polynomial Φ f n if and only if z 0 is a periodic point for f : R → R with period k ≥ 1 and multiplier λ 0 ∈ R that satisfy

• k = n,
• or there exists an integer l ≥ 1 such that n = kl and λ 0 is a primitive lth root of unity, • or R has characteristic p > 0 and there exist integers l, m ≥ 1 such that n = klp m and λ 0 is a primitive lth root of unity.

Remark 35. If R is a commutative ring that is not an integral domain, f ∈ R[z] is a monic polynomial of degree d and n ≥ 1, then it may occur that periodic points for f : R → R with period n are not roots of the polynomial Φ f n . For example, if

R = Z/4Z and f (z) = z 2 + 3z + 2 ∈ R[z] ,
then 0 ∈ R is a periodic point for f with period 2 but is not a root of the polynomial

Φ f 2 (z) = z 2 + 2 ∈ R[z]
. Now, let us focus on the family (f c ) c∈C of unicritical polynomial maps given by

f c (z) = z d + c .
Given a parameter c ∈ C, every cycle of parabolic basins for f c contains the unique critical point 0 ∈ C, which allows us to determine the multiplicities of the roots of the polynomials Φ fc n , with n ≥ 1. Thus, we have the result below, which is a stronger version of Proposition 34 in the case of the polynomial f c .

Proposition 36 ([BL14, Proposition 2.2]). Assume that c ∈ C and n ≥ 1. Then z 0 ∈ C is a root of the polynomial Φ fc n if and only if • either z 0 is a periodic point for f c with period n and multiplier different from 1, in which case ord z0 Φ fc n = 1, • or z 0 is a periodic point for f c with period n and multiplier 1, in which case ord z0 Φ fc n = 2, • or z 0 is a periodic point for f c with period a proper divisor k of n and multiplier a primitive n k th root of unity, in which case ord z0 Φ fc n = n k . Now, view c as an indeterminate over Z and consider

R = Z[c] and f (z) = z d + c ∈ R[z] . For n ≥ 1, define Φ n ∈ Z[c, z] to be the image of Φ f n under the canonical ring isomorphism from R[z] to Z[c, z].
By the uniqueness in Proposition 30, for every c ∈ C and every n ≥ 1, we have

Φ fc n (z) = Φ n (c, z) ∈ C[z] .
In particular, the coefficients of the polynomials Φ fc n , with n ≥ 1, are polynomials in c with integer coefficients.

Finally, let us state the result below due to Bousch. It has also been proved with different approaches by Buff Proposition 37 ([Bou92, Chapitre 3, Théorème 1]). For every n ≥ 1, the polynomial Φ n is irreducible over C.

Multiplier polynomials.

We shall now present the multiplier polynomials of a monic polynomial, which play a crucial role in our proofs of Theorem 6 and Theorem 8.

Note that, if a 0 , . . . , a d-1 are indeterminates over Z,

R = Z [a 0 , . . . , a d-1 ] and f (z) = z d + d-1 j=0 a j z j ∈ R[z] ,
then, for every n ≥ 1, we have

res z Φ f n (z), λ -(f •n ) (z) = ν(n) j=1 λ -(f •n ) (z j ) ,
where z 1 , . . . , z ν(n) are the roots of the polynomial Φ f n in an algebraic closure K of the field of fractions of R. Since the map f : K → K has the same multiplier at each point of a cycle and the roots of Φ f n in K are simple and are precisely the periodic points for f : K → K with period n, it follows that the polynomial

res z Φ f n (z), λ -(f •n ) (z) ∈ R[λ]
is the nth power of some monic polynomial in R[λ]. By a specialization argument, the same is true in the general case of a commutative ring R and a monic polynomial f ∈ R[z] of degree d (compare [MP94, Section 5]). More precisely, we have the following result: Proposition 38. Suppose that R is an integral domain and f ∈ R[z] is a monic polynomial of degree d. Then, for every n ≥ 1, there exists a unique monic polynomial

M f n ∈ R[λ] that satisfies M f n (λ) n = res z Φ f n (z), λ -(f •n ) (z)
. Furthermore, the polynomial M f n has degree ν(n) n . Definition 39. Suppose that R is an integral domain and f ∈ R[z] is a monic polynomial of degree d. For n ≥ 1, the polynomial M f n is called the nth multiplier polynomial of f . Remark 40. If R is a commutative ring that is not an integral domain and f ∈ R[z] is a monic polynomial of degree d, then, for every n ≥ 1, the existence of a polynomial M f n as in Proposition 38 holds but the uniqueness may fail. For example, if

R = Z/4Z and f (z) = z 2 ∈ R[z] ,
then we have f •2 (z) = 0, and hence

res z Φ f 2 (z), λ -f •2 (z) = λ 2 = (λ + 2) 2 .
If K is an algebraically closed field, f ∈ K[z] is a monic polynomial of degree d and n ≥ 1, then we have

M f n (λ) n = ν(n) j=1 λ -(f •n ) (z j ) ,
where z 1 , . . . , z ν(n) are the -not necessarily distinct -roots of the polynomial Φ f n . Note that, if z 0 ∈ K is a periodic point for f with period a proper divisor k of n and multiplier a n k th root of unity, then we have (f •n ) (z 0 ) = 1 by the chain rule. Therefore, by Proposition 34, we have the result below, which gives the connection between the multipliers of a monic polynomial map and its multiplier polynomials.

Proposition 41. Assume that K is an algebraically closed field, f ∈ K[z] is a monic polynomial of degree d and n ≥ 1. Then λ 0 ∈ K is a root of the polynomial M f n if and only if • λ 0 is the multiplier of f at a cycle with period n,

• or λ 0 equals 1 and there exist integers k, l ≥ 1 such that n = kl and f has a cycle with period k and multiplier a primitive lth root of unity, • or λ 0 equals 1, the field K has characteristic p > 0 and there exist integers k, l, m ≥ 1 such that n = klp m and f has a cycle with period k and multiplier a primitive lth root of unity.

An immediate consequence of Proposition 41 is the following result, which is the key ingredient in our proofs of Theorem 6 and Theorem 8: Let us now focus on the multiplier polynomials associated with the family (f c ) c∈C . We may determine the multiplicities of their roots to obtain the following stronger version of Proposition 41: Proposition 43. Assume that c ∈ C and n ≥ 1. Then λ 0 ∈ C is a root of the polynomial M fc n if and only if • either λ 0 is not 1 and is the multiplier of f c at a cycle with period n, in which case ord λ0 M fc n equals the number of cycles for f c with period n and multiplier λ 0 , • or λ 0 equals 1 and is the multiplier of f c at a cycle with period n, in which case ord λ0 M fc n = 2, • or λ 0 equals 1 and f c has a cycle with period a proper divisor k of n and multiplier a primitive n k th root of unity, in which case ord λ0 M fc n = 1. Proof. Since the polynomial Φ fc n is monic, we have

Corollary 42. Assume that K is an algebraically closed field, R is a subring of K, f ∈ K[z]
M fc n (λ) n = ν(n) j=1 λ -(f •n c ) (z j ) ,
where z 1 , . . . , z ν(n) are the -not necessarily distinct -roots of the polynomial Φ fc n . By Proposition 36, it follows that

M fc n (λ) = (λ -1) 2p+q r j=1 λ -(f •n c ) (w j ) ,
where p is the number of cycles for f c with period n and multiplier 1, q is the number of cycles for f c with period a proper divisor k of n and multiplier a primitive n k th root of unity and w 1 , . . . , w r are representatives for the cycles for f c with period n and multiplier different from 1. Since every cycle of parabolic basins for f c contains the unique critical point 0 ∈ C, we have p + q ∈ {0, 1}. This completes the proof of the proposition. Now, view c as an indeterminate over Z and recall that

R = Z[c] and f (z) = z d + c ∈ R[z] . For n ≥ 1, define M n ∈ Z[c, λ] to be the image of M f n under the canonical ring isomorphism from R[λ] to Z[c, λ].
By the uniqueness in Proposition 38, for every c ∈ C and every n ≥ 1, we have

M fc n (λ) = M n (c, λ) ∈ C[λ] .
In particular, the coefficients of the polynomials M fc n , with n ≥ 1, are polynomials in c with integer coefficients. In fact, the following result shows that more is true, as observed in Remark 13 and Remark 20.

Proposition 44. For every n ≥ 1, the polynomial M n lies in Z d d c d-1 , λ , has leading coefficient ±d ν(n) and degree (d-1)ν(n) d in c and is monic in λ of degree ν(n) n . Proof. We have M n (c, λ) = λ ν(n) n + ν(n) n j=1 (-1) j σ j (c)λ ν(n) n -j ,
where σ 1 , . . . , σ ν(n) n ∈ R are the elementary symmetric functions of the -not necessarily distinct -roots of the polynomial M f n in an algebraic closure K of the field of fractions of R. Moreover, for every parameter c ∈ C, since we have

M fc n (λ) = M n (c, λ) ∈ C[λ] ,
σ 1 (c), . . . , σ ν(n) n (c) are the elementary symmetric functions of the -not necessarily distinct -roots of the polynomial M fc n . First, let us prove that, for every j ∈ 1, . . . , ν(n) n , we have

σ j ∈ Z d d c d-1 .
Choose a primitive (d -1)th root of unity ω ∈ C. For every parameter c ∈ C, since the maps f c and f ωc are affinely conjugate, it follows from Proposition 43 that the polynomials M fc n and M fωc n have the same roots -counting multiplicities. Therefore, for every j ∈ 1, . . . , ν(n) n , we have σ j (c) = σ j (ωc) for all c ∈ C, and hence σ j lies in the subring

Z c d-1 of R. Since Z d d c d-1 is integrally closed in Z c d-1 and each root of M f n in K is of the form (f •n ) (z 0 ) = n-1 j=0 f f •j (z 0 ) ,
where z 0 ∈ K is a periodic point for f : K → K, it suffices to prove the fact below (compare [Mil14, Theorem 1.1]) to conclude that

σ j ∈ Z d d c d-1 for all j ∈ 1, . . . , ν(n) n . Thus, the polynomial M n lies in Z d d c d-1 , λ . Claim 45. If z 0 ∈ K is a periodic point for f , then f (z 0 ) is integral over Z d d c d-1 . Proof of Claim 45. Choose a primitive (d -1)th root d 1 d-1 of d in K, and, for m ∈ Z, define d m d-1 = d 1 d-1 m . For every k ≥ 1, we have d d k d-1 f •k z d 1 d-1 = d d k-1 d-1 f •(k-1) z d 1 d-1 d + d d k d-1 c .
It follows by induction that, for every k ≥ 0, the polynomial d

d k d-1 f •k z d 1 d-1 is monic in z of degree d k with coefficients in Z d d d-1 c . Therefore, d 1 d-1 z 0 is integral over Z d d c d-1 since it is a root of the monic polynomial d d k d-1 f •k z d 1 d-1 -d d k -1 d-1 z ∈ Z d d d-1 c [z] ,
where k is the period of z 0 , and

d d d-1 c is integral over Z d d c d-1 ,
and hence the same is true of f (z 0 ) = dz d-1 0 . Thus, the claim is proved.

It remains to examine the leading term of the polynomial M n in c. To do this, let us first prove the following fact:

Claim 46. If c ∈ C and z 0 ∈ C is a periodic point for f c , then |z 0 | ≤ 1 + |c| 1 d . Proof of Claim 46. The map ψ c : x → x d -x -|c| is strictly increasing on [1, +∞) and satisfies ψ c 1 + |c| 1 d ≥ 0. Therefore, whenever |z| > 1 + |c| 1 d , we have |f c (z)| ≥ |z| d -|c| > |z| . It follows that, if z 0 ∈ C satisfies |z 0 | > 1 + |c| 1 d , then f •k
c (z 0 ) k≥0 diverges to ∞, and hence z 0 is not periodic for f c . Thus, the claim is proved. Now, note that, if c ∈ C and λ 0 ∈ C is a root of the polynomial M fc n , then there exists a periodic point z 0 ∈ C for f c such that

λ 0 = (f •n c ) (z 0 ) = d n n-1 j=0 f •j (z 0 ) d-1 . If c ∈ C and z 0 ∈ C is a periodic point for f c ,
|z 0 | -|c| 1 d ≤ 1 + |c| 1 d 1 d .
It follows that there exists a map η : C → R >0 that satisfies

η(c) = O |c| (d-1)n d -d-1 d 2 as c → ∞ and |λ 0 | -d n |c| (d-1)n d ≤ η(c) whenever c ∈ C and λ 0 ∈ C is a root of M fc n . Therefore, for every j ∈ 1, . . . , ν(n) n , the polynomial σ j ∈ Z[c] has degree at most j(d-1)n d
, with equality and leading coefficient ±d

ν(n) if j = ν(n)
n . This completes the proof of the proposition. Now, let us consider the multibrot set M = {c ∈ C : 0 has bounded forward orbit under f c } .

For n ≥ 1, we call hyperbolic component of M with period n a component W of the set of parameters c ∈ C for which the map f c has an attracting cycle with period n. Given a hyperbolic component W of M and a parameter c ∈ W , we denote by λ W (c) the multiplier of f c at its unique attracting cycle. For every hyperbolic component W of M, the map λ W : W → D(0, 1) is a branched (d -1)-sheeted holomorphic covering map, its unique critical point is the unique parameter c W ∈ W for which the point 0 is periodic for f c W and λ W extends to a unique map λ W : W → D(0, 1) that induces a covering map from W \ {c W } onto D(0, 1) \ {0}. Furthermore, for every n ≥ 1, the hyperbolic components of M with period n have pairwise disjoint closures (see [DH85, Exposé XIV and Exposé XIX] or [Mil00, Theorem 6.5]). Using these facts, we obtain the following result (compare [MV95, Proposition 3.2]):

Proposition 47. Assume that n ≥ 1. Then, for every λ 0 ∈ C that satisfies 0 < |λ 0 | ≤ 1, the polynomial M n (c, λ 0 ) ∈ C[c] is separable. Furthermore, we have M n (c, 0) = ±d ν(n) Φ n (c, 0) d-1 and the polynomial Φ n (c, 0) ∈ Z[c] is separable. Suppose that n ≥ 1. The polynomial Φ f
n is irreducible over C(c) by Proposition 37, and hence its roots in an algebraic closure of C(c) are Galois conjugates of each other over C(c). It follows that the same is true of the polynomial M f n , which shows that the polynomial M n is the power of some irreducible polynomial in C[c, λ]. Therefore, since the polynomial M n (c, 1) ∈ Z[c] is separable by Proposition 47, we have the following:

Proposition 48 ([Mor96, Corollary 1]). For every n ≥ 1, the polynomial M n is irreducible over C.

Discriminants of the multiplier polynomials.

Finally, let us study here the discriminants of the multiplier polynomials associated with the family (f c ) c∈C . Their expressions, for d ∈ {2, 3} and small values of n, are an essential ingredient in our proofs of Proposition 17 and Proposition 23.

For

n ≥ 1, define ∆ n (c) = disc λ M n (c, λ) ∈ Z[c] .
In fact, for every n ≥ 1, the polynomial ∆ n lies in Z d d c d-1 by Proposition 44. Furthermore, for every c ∈ C and every n ≥ 1, we have

∆ n (c) = disc M fc n ∈ C . Example 49. By Example 24, we have M 1 (c, λ) = λ(λ -d) d-1 + (-d) d c d-1 and ∂M 1 ∂λ (c, λ) = d(λ -1)(λ -d) d-2 .
Therefore, we have

∆ 1 (c) = (-1) d(d-1) 2 res λ M 1 (c, λ), ∂M 1 ∂λ (c, λ) = (-1) d(d-1) 2 d d M 1 (c, 1)M 1 (c, d) d-2 = (-1) d(d-1) 2 d d(d-1) c (d-1)(d-2) d d c d-1 -(d -1) d-1 .
Suppose that n ≥ 1. By Proposition 43, the roots of the polynomial ∆ n are precisely the parameters c 0 ∈ C for which f c0 has a cycle with period n and multiplier 1 or f c0 has two distinct cycles with period n and the same multiplier. Thus, in order to factor ∆ n , it is natural to try to define a polynomial that vanishes precisely at the parameters c 0 ∈ C for which the map f c0 has a cycle with period n and multiplier 1. Note that, by Proposition 43, the roots of the polynomial M n (c, 1) ∈ Z[c] are precisely the parameters c 0 ∈ C for which either f c0 has a cycle with period n and multiplier 1 or f c0 has a cycle with period a proper divisor k of n and multiplier a primitive n k th root of unity, which suggests factoring this polynomial. For k ≥ 1 and l ≥ 2, define

P k,l (c) = res λ (C l (λ), M k (c, λ)) ∈ Z[c] ,
where C l ∈ Z[λ] denotes the lth cyclotomic polynomial. We have

P k,l (c) = ϕ(l) j=1 M k (c, ω j ) ,
where ϕ : Z ≥1 → Z ≥1 denotes Euler's totient function and ω 1 , . . . , ω ϕ(l) are the primitive lth roots of unity in C. By Proposition 43 and Proposition 44, it follows that the polynomial P k,l lies in Z d d c d-1 , has leading coefficient ±1 in d d c d-1 and its roots are precisely the parameters c 0 ∈ C for which the map f c0 has a cycle with period k and multiplier a primitive lth root of unity. Furthermore, by Proposition 47 and since every cycle of parabolic basins contains a critical point, the polynomial P k,l is separable and, for every k ≥ 1 and every l ≥ 2 such that (k, l) = (k , l ), the polynomials P k,l and P k ,l have no common roots.

Remark 50. For every k ≥ 1 and every l ≥ 2, the parameters c 0 ∈ C for which the map f c0 has a cycle with period k and multiplier a primitive lth root of unity are precisely the parameters at which a cycle with period kl degenerates to a cycle with period k. These are the parameters that lie in the intersections of the closures of hyperbolic components of M with period k with the closures of hyperbolic components of M with period kl. Remark 51. For every n ≥ 1, the parameters c 0 ∈ C for which the map f c0 has a cycle with period n and multiplier 1 are precisely the parameters at which two distinct cycles with period n collide. These are the parameters that occur at the cusps of the hyperbolic components of M with period n.

Suppose that

Since the polynomials Q n and ∆ n lie in Z d d c d-1 and the polynomial Q n has leading coefficient ±1 in d d c d-1 and its roots are simple and are also roots of ∆ n , the polynomial Q n divides ∆ n in Z d d c d-1 . In fact, the result below shows that more is true, as observed in Remark 15 (compare [Mor96, Proposition 9]).

Proposition 52. For every n ≥ 1, there exist a unique squarefree integer a n and a unique polynomial R n ∈ Z[c] with positive leading coefficient that satisfy

∆ n = a n Q n R 2
n . Furthermore, the polynomial R n lies in Z d d c d-1 and its roots are precisely the parameters c 0 ∈ C for which the map f c0 has two distinct cycles with period n and the same multiplier.

Proof. Since the polynomials Q n and ∆ n lie in Z d d c d-1 and the polynomial Q n has leading coefficient ±1 in d d c d-1 , is separable and its roots are also roots of ∆ n , it suffices to prove that, for every parameter c 0 ∈ C, we have

ord c0 ∆ n = ε n (c 0 ) + 2κ n (c 0 ) ,
where ε n (c 0 ) ∈ {0, 1} equals 1 if and only if f c0 has a cycle with period n and multiplier 1 and κ n (c 0 ) ∈ Z ≥0 is positive if and only if f c0 has two distinct cycles with period n and the same multiplier.

Suppose that c 0 ∈ C. Choose representatives z 1 , . . . , z r for the cycles for f c0 with period n and multiplier different from 1. For every j ∈ {1, . . . , r}, we have ord zj Φ For every j ∈ {1, . . . , r}, we have Φ f n (ζ j ) = 0, and hence ζ j is a periodic point for the map f : S → S with period a divisor of n. Since the points ζ j (c 0 ) = z j , with j ∈ {1, . . . , r}, are periodic for f c0 with period n and belong to pairwise distinct cycles, it follows that the points ζ j , with j ∈ {1, . . . , r}, are periodic points for f : S → S with period n, which belong to pairwise distinct cycles. Consequently, there exists a unique monic polynomial

Ψ n ∈ S[z] such that Φ f n (z) = Ψ n (z) r j=1 n-1 k=0 z -f •k (ζ j ) ,
and we have

M f n (λ) n = res z Ψ n (z), λ -(f •n ) (z) r j=1 (λ -λ j ) n ,
where

λ j = (f •n ) (ζ j
) is the multiplier of f : S → S at ζ j for j ∈ {1, . . . , r}.

Therefore, there exists a unique monic polynomial N n ∈ S[λ] that satisfies

M f n (λ) = N n (λ) r j=1 (λ -λ j ) . Now, define κ n (c 0 ) = 1≤j<k≤r ord c0 (λ j -λ k ) ∈ Z ≥0 ,
which is positive if and only if the map f c0 has two distinct cycles with period n and the same multiplier since λ j (c 0 ) = f •n c0 (z j ) for all j ∈ {1, . . . , r}. Let us consider three different cases.

Assume that f c0 has neither a cycle with period a proper divisor k of n and multiplier a primitive n k th root of unity nor a cycle with period n and multiplier 1. Then, by Proposition 43, we have

M fc 0 n (λ) = r j=1 λ -f •n c0 (z j ) = r j=1 (λ -λ j (c 0 )) ,
and hence r = ν(n) n and N n = 1. Therefore, we have

∆ n = disc M f n = 1≤j<k≤ ν(n) n (λ j -λ k ) 2 ,
and hence ord c0 ∆ n = 2κ n (c 0 ). Assume now that f c0 has a cycle with period a proper divisor k of n and multiplier a primitive n k th root of unity. Then, by Proposition 43, we have

M fc 0 n (λ) = (λ -1) r j=1 (λ -λ j (c 0 )) ,
and hence r = ν(n) n -1 and there exists a holomorphic map ρ : U → C such that ρ (c 0 ) = 1 and N n (λ) = λ -ρ. Therefore, we have

∆ n = ν(n) n -1 j=1 (ρ -λ j ) 2 1≤j<k≤ ν(n) n -1 (λ j -λ k ) 2 ,
and hence ord c0 ∆ n = 2κ n (c 0 ) since

ν(n) n -1 j=1 (ρ -λ j ) (c 0 ) 2 = ν(n) n -1 j=1 1 -f •n c0 (z j ) 2 = 0 .
Finally, assume that f c0 has a cycle with period n and multiplier 1. Then, by Proposition 43, we have

M fc 0 n (λ) = (λ -1) 2 r j=1 (λ -λ j (c 0 )) ,
and hence r = ν(n) n -2 and there exist holomorphic maps σ 1 , σ 2 :

U → C such that σ 1 (c 0 ) = 2 , σ 2 (c 0 ) = 1 and N n (λ) = λ 2 -σ 1 λ + σ 2 .
Therefore, we have

∆ n = σ 2 1 -4σ 2 ν(n) n -2 j=1 λ 2 j -σ 1 λ j + σ 2 2 1≤j<k≤ ν(n) n -2 (λ j -λ k ) 2 ,
and hence ord c0 ∆ n = ord c0 σ 2 1 -4σ 2 + 2κ n (c 0 ) since we have

ν(n) n -2 j=1 λ 2 j -σ 1 λ j + σ 2 (c 0 ) 2 = ν(n) n -2 j=1 f •n c0 (z j ) -1 4 = 0 .
We have σ 2 1 -4σ 2 (c 0 ) = 0. Since the polynomial M n (c, 1) ∈ Z[c] is separable by Proposition 47, we have .

∂M n ∂c (c 0 , 1) = (-σ 1 (c 0 ) + σ 2 (c 0 )) ν(n) n -2 j=1 (1 -λ j (c 0 )) = 0 , and hence σ 2 1 -4σ 2 (c 0 ) = -4 (-σ 1 (c 0 ) + σ 2 (c 0 )) = 0 . Therefore, we have ord c0 σ 2 1 -4σ 2 = 1,
Note that the polynomials Q 1 and R 1 have no common roots, which shows that there is no parameter c 0 ∈ C for which the map f c0 has both a fixed point with multiplier 1 and two distinct fixed points with the same multiplier. Using the software SageMath, we observe that the same is true of the polynomials Q n and R n for small values of d and n. Thus, it seems likely that the following question has a negative answer.

Question 54. Does there exist an integer n ≥ 1 such that the polynomials Q n and R n have a common root? Equivalently, does there exist an integer n ≥ 1 and a unicritical polynomial map f : C → C of degree d that has both a cycle with period n and multiplier 1 and two distinct cycles with period n and the same multiplier? Finally, note that, if n ≥ 1 and c 0 ∈ C is a parameter such that the map f c0 has a rational multiplier at each cycle with period 1 or n, then c d-1 0 is rational and ∆ n (c 0 ) is the square of a rational number, and hence R n (c 0 ) = 0 or a n Q n (c 0 ) is the square of a rational number. Thus, it would be interesting to determine the integers a n , with n ≥ 1. We proved in Example 53 that a 1 = ±1. Using the software SageMath, we also obtain that a n = ±1 for small values of d and n, which suggests the following: Question 55. Do we have a n = ±1 for all n ≥ 1?

The periodic points for f -t d : z → z d -t d have Laurent expansions in t -(d-1) with coefficients in Q(ω), where ω ∈ C is a primitive dth root of unity, for t around ∞ (compare [Mor96, Lemma 2]), and hence the same is true of their multipliers. Using this fact, it is possible to prove that the question above has a positive answer when d = 2. Claim 58. The ring A/λ 3 A contains exactly 3 cubes, namely the residue classes of -1, 0 and 1 (see Figure 2). More precisely, for every a ∈ A,

• either a ≡ -1 (mod λ) and a 3 ≡ -1 (mod λ 3 ),

• or a ≡ 0 (mod λ) and a 3 ≡ 0 (mod λ 3 ),

• or a ≡ 1 (mod λ) and a 3 ≡ 1 (mod λ 3 ).

Proof of Claim 58. If λ divides a, then λ 3 divides a 3 . If a ≡ 1 (mod λ), then there exists b ∈ A such that a = 1 + λb, and we have a 3 -1 = (a -1)(a -j) a -j 2 = λ 3 b(b + 1)(b + 1 + j) .

If a ≡ -1 (mod λ), then a 3 ≡ -1 (mod λ 3 ) since a 3 = -(-a) 3 .

Finally, Lemma 56 follows immediately from the following more general result:

Lemma 59. Suppose that x, y, z ∈ A and u, v ∈ A × are such that x 3 + uy 3 = 4vz 3 . Then z = 0.

Proof of Lemma 59. To obtain a contradiction, suppose that there exist x, y, z ∈ A, with z = 0, that are relatively prime and u, v ∈ A × such that x 3 + uy 3 = 4vz 3 and the valuation ord λ (z) is minimal. Then x, y and z are pairwise relatively prime. Note that, by Claim 58, x 3 + uy 3 is at distance at most 2 from λ 3 A, while the distance between 4A × and λ 3 A equals √ 7 > 2. Therefore, we have z ≡ 0 (mod λ), u = ±1 and x ≡ -uy (mod λ). It follows that

x + uy ≡ jx + j 2 uy ≡ j 2 x + juy ≡ 0 (mod λ) , and hence there exist a, b, c, d ∈ A such that x + uy = λa , jx + j 2 uy = λb , j 2 x + juy = λc and z = λd . We have

X 3 + U Y 3 = 4V Z 3 ,
where U = u -1 X u Y ∈ A × and V = -u -1 X u Z ∈ A × , and 3 ord λ (Z) = ord λ (σ(c)) = ord λ (abc) = ord λ d 3 = 3 ord λ (z) -3 . This contradicts the minimality of ord λ (z), and thus the lemma is proved.

  and Tan (see [BL14, Theorem 1.2]), Morton (see [Mor96, Corollary 1]) and Schleicher (see [Sch17, Theorem 7.1]).

  is a monic polynomial of degree d and n ≥ 1. Then the multipliers of f at its cycles with period n all lie in R if and only if the polynomial M f n splits into linear factors of R[λ].

  then we have |f c (z 0 )| ≤ 1 + |c| 1 d by Claim 46, and hence

  n ≥ 1. By Proposition 43, Proposition 44 and Proposition 47, the polynomialM n (c, 1) ∈ Z[c] lies in Z d d c d-1 , has leading coefficient ±1 in d d c d-1and its roots are simple and are precisely the roots of the polynomials P k, n k , with k a proper divisor of n, and the parameters c 0 ∈ C for which the map f c0 has a cycle with period n and multiplier 1. Furthermore, the polynomialk|n, k =n P k, n k ∈ Z[c] lies in Z d d c d-1 , has leading coefficient ±1 in d d c d-1and is separable by the discussion above. Therefore, there exists a unique polynomialQ n ∈ Z[c] such that M n (c, 1) = Q n (c) k|n, k =n P k, n k (c)and the polynomial Q n lies in Z d d c d-1 , has leading coefficient ±1 in d d c d-1 and its roots are simple and are precisely the parameters c 0 ∈ C for which the map f c0 has a cycle with period n and multiplier 1 (see [MV95, Theorem A]).

  fc 0 n = 1 by Proposition 36, and hence ∂Φn ∂z (c 0 , z j ) = 0. By the implicit function theorem, it follows that there exist a complex domain U containing c 0 and holomorphic maps ζ 1 , . . . , ζ r : U → C such that, for every j ∈ {1, . . . , r}, we have ζ j (c 0 ) = z j and Φ n (c, ζ j (c)) = 0 for all c ∈ U . Now, embed R = Z[c] into the ring S = H(U ) of holomorphic maps on U .

  and hence ord c0 ∆ n = 1 + 2κ n (c 0 ) . This completes the proof of the proposition.Example 53. By Example 24 and Example 49, we haveQ 1 (c) = M 1 (c, 1) = (-1) d d d c d-1 -(d -1) d-1and ∆ 1 (c) = (-1)d(d-1) 2 d d(d-1) c (d-1)(d-2) d d c d-1 -(d -1) d-1 .

Figure 1 .

 1 Figure1. The ring of Eisenstein integers, its units and its ideal λA.

(Figure 2 .

 2 Figure2. The ring of Eisenstein integers, its cubes and its ideal λ 3 A. The elements of the form x 3 +uy 3 , with x, y ∈ A and u ∈ A × , lie in blue disks. The set 4A × + λ 3 A is indicated by purple dots.

Appendix A. A Fermat-like Diophantine equation

We shall prove here the following statement, which is a crucial argument in our proof of Proposition 17: Lemma 56. Assume that x, y, z ∈ Z satisfy x 3 + y 3 = 4z 3 . Then z = 0.

Our proof of Lemma 56 is adapted from the proof of Fermat's last theorem for exponent 3 presented in [START_REF] Hindry | Arithmetics, Universitext[END_REF] and uses the principle of infinite descent.

Define A = Z[j] to be the ring of Eisenstein integers, where j = exp 2πi 3 ∈ C. The ring A is a Euclidean domain and its group of units A × = ±1, ±j, ±j 2 consists of the 6th roots of unity.

First, observe that λ = 1 -j ∈ A is irreducible.

Claim 57. The quotient ring A/λA consists of the residue classes of -1, 0 and 1 (see Figure 1).

Proof of Claim 57. Every element of A is congruent to an element of Z modulo λ since j ≡ 1 (mod λ) and every element of Z is congruent to either -1, 0 or 1 modulo 3 = -j 2 λ 2 .


Élodie Tailleur

Mathieu Marchivie

Philippe Négrier

Dominique Denux

Stéphane Massip

Denise Mondieig

Guillaume Chastanet

Philippe Guionneau

Elodie Tailleur

Philippe Negrier

Using polymorphism to master the spin crossover mechanism in

Introduction

In the investigation of the Spin crossover (SCO) phenomenon, polymorphism represents an exciting opportunity to understand the complex relationships between the structural properties and the switching features in the solid-state. [1][2][3] Note that despite solvatomorphism is also an interesting route 4 , polymorphism seems more suitable for orienting towards a choice of the SCO properties of a given compound, since for the first one, the possibility of a solvent departure greatly complicates the process. Hereafter, we are thus discussing about solvent-free polymorphism. The SCO originates on the metallic site, here iron(II). However, the SCO features are affected by all the physical scales of the solid, from the atom up to the crystal. 5 The structure-property relationships in SCO materials start to be well documented and the main learning of the literature on this subject is the complexity of corresponding close and subtle links. In this context, having different crystal packing for identical molecular complexes, i. e. polymorphs, makes it possible, for example, to track the structural origin of some physical behavior, such as magnetic properties, by meticulously comparing polymorphs' features. [1][2][3][6][7][8][9] This paper provides a cornerstone to this building by probing two wellcrystallized polymorphs with radically different magnetic properties -one with hysteresis at room temperature and the other one with a continuous conversion occurring at low temperature -using the crystal structures determined both at high temperature when the iron is High-Spin (HS) and at low temperature when the iron is Low-Spin (LS). Crystals of the molecular complex [Fe(PM-PeA)2(NCSe)2] [with PM PeA = N-(2'-pyridylmethylene)-4-(phenylethynyl) aniline)] have been reported as being among the very rare -almost the unique so far -to undergo a thermal-induced SCO centered at room temperature (286 K) with a large hysteresis width (41 K). 10 In that case, the SCO is accompanied by a peculiar structural transition from mP21/c (HS) to oPccn (LS) that was also recently observed with high-pressure acting as a stimulus. 11 The already reported polymorph will be hereafter named polymorph-I since another polymorph, therefore named polymorph-II, has been crystallized and constitutes the main purpose of this paper. The objective, here, is to highlight the structural origin of the SCO features differences shown by the two polymorphs. First, the structural, magnetic and thermodynamics properties of polymorph-II are reported then the comparison of selected features of the two known polymorphs allows some progresses on the knowledge of structure-property relationships in SCO crystals. On the other hand, it was possible to deliberately obtain powder batches of polymorph-II. The powder synthesis is very similar to that of polymorph-I. 10 The difference in the synthesis protocol lies in the addition of the Fe(NCSe)2 solution, drop by drop, to the ligand solution, instead of the reverse operation (addition of the ligand solution to the Fe(NCSe)2 solution). Verywell crystallized powders are obtained by this mean, allowing good-quality Rietveld refinements that, together with elementary analysis, confirmed that these powders are exclusively composed by polymorph-II of [Fe(PM-PeA)2(NCSe)2] (see supplementary material). These pure powders were therefore used for the physical measurements. It is worth noting that among the different attempts to get polymorph-II, it was also obtained powder batches containing a mixture of the two polymorphs, X-ray diffraction allowing easily to identify their presence due to specific Bragg peaks (see supplementary material).

Results and Discussions

To sum-up, though single-crystals of polymorph-II were obtained accidentally, pure batches of polymorph-II powders can be rationally and reproductively obtained. Some trials lead to a mixture of polymorphs which confirms that in some circumstances, these polymorphs can be concomitants. All attempts to transform one polymorph into another by heat treatments in the range [80 -500 K] were not successful. It is also worth to mention that the application of pressure does not allow to transform one polymorph into another. Indeed, the previously reported detailed high-pressure study of polymorph-I as a function of temperature has shown no trace of polymorph-II 11 (the studied range was [0-10 kbar] at [293-353 K]) and, second, the application of pressure on the powder of polymorph-II does not lead to a transformation towards polymorph-I, at least up to 12 kbar. Since powders and single-crystals of polymorph-II were systematically checked to be strictly isostructural and since powders containing only polymorph-II can be easily obtained (see above), magnetic measurements were performed on powder batches. The magnetic properties were studied using a SQUID magnetometer between 50 K and 390 K, under a magnetic field of 5 000 0e, on a 43.1 mg powder sample placed in a polypropylene bag. The measurement was made in temperature scanning mode with a constant maintained rate at cooling and heating. One full spin crossover cycle was measured at 5 K.min -1 . After subtracting the diamagnetic contributions from the polypropylene bag and the compound, the product of molar magnetic susceptibility with temperature, χMT, was plotted as a function of temperature (Figure 1). At 390K, the χMT product around 4.0 cm 3 K mol -1 is slightly higher than the expected value for a iron(II) cation in the HS state. It remains constant on cooling to ca. 300K where it starts to decrease down to 150 K. Upon further cooling, a constant and weak decrease of the χMT product is observed. Together with the slightly high value at 390 K, this could be an indication of the presence of Temperature-Independent Paramagnetism even though a small magnetic impurity cannot be totally excluded. [START_REF] Kahn | Molecular Magnetism[END_REF] The

Magnetism and Heat Capacity of Polymorph-II

Single-Crystal investigations of Polymorph-II in the High-Spin and Low -Spin states

A good-diffracting tiny single-crystal of approximate dimensions 50x10x10 m 3 was mounted on a loop and positioned on a microfocus Rigaku FRX equipped with a rotating anode as X-ray source (CuK, = 1.54184 Å) and a 200K Dectris Pilatus detector. Full data collections were performed at 293 K (HS) and 100 K (LS) (strategy details in supplementary material) and lead to high-quality crystal structures (Table 1). At 293 K, Polymorph-II crystallizes in mP21/c space group in HS with unit-cell parameters a and  notably different from those of polymorph-I though the resulting unit-cell volume appears similar in both polymorphs, which may be related to the possible occurrence of both polymorphs at room temperature. At low temperature, in LS, the space group is unchanged for polymorph-II that exhibits a significant decrease of the unit-cell volume as usually observed in conjunction with the SCO (about 0.8 Å 3 /K here). Note that a quasi-isotropic evolution of the unit cell due to both thermal contraction and SCO is observed upon cooling in contrast with the polymorph-I and other SCO compounds of this family even with those undergoing a gradual conversion (see below and supplementary material for details). 14

Molecular geometry in [Fe(PM-PeA)2(NCSe)2] polymorphs

It is now well recognized that the geometric properties of the coordination sphere not only reflect the spin state -and thus allow visualizing the SCO -but also influence the transition characteristics. 15,16 For example, from one compound to another, if the volume of the coordination sphere always varies by the same order of magnitude (from 13.0(2) Å to 9.9(2)Å) from HS to LS in FeN6), the bond lengths may be very different depending on the concerned ligand while the average variation is always similar (0.2 Å). On the other hand, the distortion of the coordination sphere at the SCO appears very different from one compound to another and probably plays a major role in certain characteristics still to faithfully identify. 17 In this context, the comparison of the geometry of polymorphs' coordination spheres therefore remains an important point, as done in Table 2 for the two polymorphs. The bond lengths and the polyhedron volume reflect the spin state which allows to confirm the occurrence of the SCO. These values for polymorph-II at 293 K show that, while mainly HS the crystal contains a significant LS part (estimated at roughly 30% from the polyhedron volume) and values exhibited at 100 K clearly indicate a 100 % LS, the whole being consistent with magnetic measurements. Keeping in mind such results, the geometry of the coordination sphere appears rather similar in both polymorphs for a given spin state.

The structural difference between both polymorphs is then visible at the upper physical scales, starting by the entire molecular scale, often addressed using the average deviation parameter (RMSD). 5 In polymorph-II, the superposition of the molecular complexes in HS and LS shows a RMSD of 0.13 Å. The latter is slightly underestimated due to the LS residue but this value is much lower than the 0. Please do not adjust margins Please do not adjust margins molecular structural differences between HS and LS quantified by the RMSD follows the nature of the SCO since abrupt transition (L/X = PeA/S 18 , PeA/Se 10 , BiA/S-I 7 ) correspond to RMSD greater than 0.4 Å and gradual transitions (L/X = BiA/S-II 7 , AsA/S 14 , PeA/Se) correspond to RMSD lower than 0.2 Å.

Here, the main difference between the molecular complexes within the two polymorphs comes from the NCSe branches in the HS state, showing a larger distortion in polymorph I (Figure 3). This has certainly repercussions at the crystal-packing scale, which is examined hereafter.

Table 2 : Coordination sphere geometric parameters for the two polymorphs of [Fe(PM-PeA)2(NCSe)2] from single-crystal X-ray diffraction. Selected parameters are the Fe-N bond lengths (Å) with the distinction of the ligand parts (NCSe, Py or PM) and the overall average value, the polyhedron volume (Å 3 ) as well as the angle distortion Σ (°), the trigonal distortion Θ (°) and the length distortion Å) known to be appropriate in this description. LS. The individual atoms involved in the contacts are not the same within polymorphs, however, highlighting the differences in the crystal-packings (The labelling of the atoms is given in supplementary material). The main difference between polymorphs concern the length of the shortest contact, i.e. the low-left tip of the fingerprints (Figure 4), that corresponds to an H...H contact. From HS to LS, the latter decreases from 2.421 Å to 2.311 Å in polymorph-II while it increases from 2.354 Å to 2.483 Å in polymorph-I. This unusual elongation of the H...H contact from HS to LS is probably to be related to the unusual structural behavior of polymorph-I, as already suspected and now emphasized here by the comparison with polymorph-II, though it is difficult to detail it more. Understanding H...H contacts in crystalline cohesion is indeed a current and growing research field. 20,21 The present result suggests to integrate SCO compounds into this field of investigation, both parties would probably benefit greatly from this knowledge crossing.

Unit-cell of Polymorph-II as a function of temperature

The temperature dependence of the unit-cell parameters was investigated using the same single-crystal as for full crystal structure determination from nineteen data collection performed by cooling from 293 to 115 K (Figure 5). In perfect agreement with the magnetic curve, the unit-cell parameters smoothly decrease which contrasts with the discontinuous behavior of the polymorph-I unit-cell at the SCO 11 . The volume decreases by 4.4% from 293 K to 115 K, which includes both SCO and pure thermal effects. In a very similar compound of this molecular complex family that is also showing a continuous SCO, namely [Fe(PM-AzA)2(NCS)2], the same observed volume variation (4.5%) was attributed to about 3% to pure thermal effects and thus 1.5% to the SCO itself. [START_REF] Lakhloufi | [END_REF] This deconvolution was obtained from a meticulous comparison between iron and zinc

Please do not adjust margins Please do not adjust margins isomorphs. In the present case this study was not possible since the zinc isomorph of polymorph-II was not successfully crystallized but the ratio of the thermal/SCO contribution is probably of the same order of magnitude for polymoprh-II of [Fe(PM-PeA)2(NCSe)2] than in the previous study of [Fe(PM-AzA)2(NCS)2] since the unit-cell features, the SCO mechanism and the molecular complexes are very close in both cases. Additionally, the zinc and sulfurocyanide isostructural analogue [Zn(PM-PeA)2(NCS)2] 22 , which does not undergo a SCO, can be used to evaluate the pure thermal decrease of volume at 2.4% leading to a contraction due to the SCO for polymorph-II of 2% approximatively.

The thermal contraction affects the unit-cell parameters concomitantly to the occurrence of the SCO while, on the contrary, the  angle seems only affected by the SCO since this angle does not changed outside the SCO temperature range (Figure 5). Consequently, it can be efficiently used to witness the SCO. The sudden increase of this angle corresponds to the SCO and gives T1/2 at about 253 K. This value is in relatively good agreement with the T1/2 obtained from magnetic and thermodynamic measurements (245 K) especially when taking in mind that the latter are performed on powders and not on single-crystals, which is known to potentially generate transition-temperature shifts.

Conclusions

Following the previous reports on polymorph-I of [Fe(PM-PeA)2(NCSe)2], 10,11 one of the rare SCO compounds exhibiting a large hysteresis centered on room temperature, a second polymorph has been obtained here. This new polymorph undergoes a continuous SCO at low temperature, as attested by magnetism, calorimetry and X-ray diffraction and can be obtained in form of single-crystals or powders; the latter being reproducibly synthetized in a controlled way allowing to master the SCO mechanism for this compound. The drastic change between the SCO features for the two polymorphs -from hysteretic to continuous -is reflected by the structural properties that are appropriately impacted from the coordination-sphere geometry to the unit-cell. Conversely, the structural difference that could be closely related to the distinct SCO mechanisms for the two polymorphs is probably due to differences in the H...H network within the crystal packing, as already observed in other SCO polymorphs. 9 Here, the contrast between the subtlety of the structural differences between polymorphs of [Fe(PM-PeA)2(NCSe)2] and its strong consequences on the spin transition features illustrates the high sensitivity of the SCO phenomenon in the solid-state.

Figure 1 :

 1 Figure 1: χMT as a function of temperature determined on polymorph-II (red) and polymorph-I (light blue) of [Fe(PM-PeA)2(NCSe)2] with in insert, the scheme of the molecular complex..

  polymorph-II of [Fe(PM-PeA)2(NCSe)2] thus shows an almost complete and reversible gradual spin conversion with T1/2=245K. This behavior fully contrasts with the large hysteresis shown by polymorph-I around room temperature but recalls the SCO feature of a few other compounds of the same [Fe(PM-L)2(NCX)2] family such as [Fe(PM-BiA)2(NCS)2]-II 7 or [Fe(PM-AzA)2(NCS)2] 14 for example.Differential scanning calorimetry measurements were carried out with 34.853 mg of polymorph-II powder in the range [125 K -350 K]. Ten successive full thermal cycles were performed using a temperature scanning rate of 10 K.min -1 (Figure2). The Please do not adjust margins Please do not adjust margins very large and extended calorific capacity dependence with temperature is typical of a second-order transition associated with a gradual spin transition, revealing here the full coherence of magnetic and thermodynamic measurements. The SCO temperature, estimated by taking the top of the signal, is very close to the T1/2 temperature determined by the magnetic measurements (around 245 K). First and tenth cycles overlap fine which characterizes a perfect reproducibility of the SCO.

Figure 2 :

 2 Figure 2 : Calorific capacity as a function of temperature for polymorph-II of [Fe(PM-PeA)2(NCSe)2] showing the first (black) and the tenth (light grey) warming.

Table 1 :

 1 Experimental and crystallographic data from singlecrystal X-ray diffraction investigation of the polymorph-II of [Fe(PM-PeA)2(NCSe)2]. ‡ TUnit-cell (Å, °) a = 16.9943 (7) b = 14.9296 (5) c = 17.2155 (6) β = 117.370 (5) a = 16.8161 (4) b = 14.6779 (2) c = 17.1039 (3) β = 117.901 (3) Unit-cell volume (Å

  46 Å determined for polymorph-I, which indicates a much smaller molecular rearrangement between the HS and LS states for polymorph-II than for polymorph-I. This confirms the spectacular nature of the molecular-scale reorganization in polymorph I. The present study highlights that, within the [Fe(PM-L)2(NCX)2] family, the This journal is © The Royal Society of Chemistry 20xx

Figure 3 .

 3 Figure 3. Superposition of the molecular complexes of polymorph -I (dark grey) and polymorph-II (light grey) for [Fe(PM-PeA)2(NCSe)2] at room-temperature showing the main difference in the NCSe ligands.

Figure 4 .

 4 Figure 4. Fingerprints of the intermolecular interactions in [Fe(PM-PEA)2(NCSe)2] polymorph-II (high) and polymorph-I (low) in HS (left) and LS (right). The red circles highlight the relevant H…H intermolecular interaction in both polymorphs.

Figure 5 .

 5 Figure 5. Unit-cell temperature dependence for the polymorph-II of [Fe(PM-PeA)2(NCSe)2] showing the a parameter (top left), the  angle (top right) and the unit-cell volume (down).

  5,15

		Polymorph-I	Polymorph-II
	T(K)	293	230	293	100
	Spin state	HS	LS	HS/LS	LS
	d(Fe-NCSe)	2.063(5)	1.947(7)	2.042(3)	1.942(3)
	d(Fe-N-Py)	2.164 (4)	1.976(6)	2.085(4)	1.968(3)
	d(Fe-N-PM)	2.231(4)	1.977(5)	2.137(5)	1.980(3)
	<dFe-N>	2.152(5)	1.967(6)	2.088(5)	1.963(3)
	Vp	12.88(2)	9.99(2)	11.86(2)	9.96(2)
	Σ	80(2)	60(4)	68(4)	53(2)
	Θ	213(4)	168(8)	187(8)	151(4)
		0.360(5)	0.078(7)	0.184(4)	0.086(3)

This journal is © The Royal Society of Chemistry 20xx J. Name., 2013, 00, 1-3 | 3

Acknowledgements

The authors gratefully acknowledge the Agence Nationale de la Recherche for the financial support under the grant ANR FemtoMat 13-BS046002 and the Region Nouvelle Aquitaine. The University of Bordeaux has provided the PhD grant for ET. This work has benefited from the facilities and expertises of the Biophysical and Structural Chemistry plateform (BPCS) at IECB, CNRS UMS3033, Inserm US001, Bordeaux University. http://www.iecb.ubordeaux.fr/index.php/fr/ plateformestechnologiques.

Conflicts of interest

There are no conflicts to declare