On the orbit of a post-critically finite polynomial of the form x d +c
aa r X i v : . [ m a t h . N T ] J u l ON THE ORBIT OF A POST-CRITICALLY FINITE POLYNOMIALOF THE FORM x d + c VEFA GOKSELA
BSTRACT . In this paper, we study the critical orbit of a post-critically finitepolynomial of the form f c,d ( x ) = x d + c ∈ C [ x ] . We discover that in manycases the orbit elements satisfy some strong arithmetic properties. It is wellknown that the c values for which f c,d has tail size m ≥ and period n arethe roots of a polynomial G d ( m, n ) ∈ Z [ x ] , and the irreducibility or not of G d ( m, n ) has been a great mystery. As a consequence of our work, for anyprime d , we establish the irreducibility of these G d ( m, n ) polynomials for in-finitely many pairs ( m, n ) . These appear to be the first known such infinitefamilies of ( m, n ) . We also prove that all the iterates of f c,d are irreducibleover Q ( c ) if d is a prime and f c,d has a fixed point in its post-critical orbit.
1. I
NTRODUCTION
Let f ( x ) ∈ C [ x ] be a polynomial of degree at least . We denote by f n ( x ) the n th iterate of f ( x ) . Given a ∈ C , one fundamental object in dynamics is theorbit O a ( f ) = { f ( a ) , f ( a ) , . . . } of a under f . When this orbit is finite for all critical points of f , we call f post-critically finite (PCF). In this paper, we study a special case, namely the PCFpolynomials of the form f c,d ( x ) = x d + c ∈ C [ x ] for d ≥ . is the unique critical point of f c,d . Suppose f c,d is PCF, i.e., there exist m, n ∈ Z with n = 0 such that f mc,d (0) = f m + nc,d (0) . We say f c,d has exact type ( m, n ) if n is the minimal positive integer such that f mc,d (0) = f m + nc,d (0) and f kc,d (0) = f k + nc,d (0) for any k < m . When m ≥ , a number c for which f c ,d has exact type ( m, n ) is called a Misiurewicz point with period (m,n) . It is knownthat Misiurewicz points with period ( m, n ) are the roots of a monic polynomial G d ( m, n ) ∈ Z [ x ] . In particular, c is always an algebraic integer. ([6], Corollary . ). To explain how the polynomial G d ( m, n ) is defined, we will follow thenotation in ([6]): Let Φ ∗ f,n ( x ) = Y k | n ( f k ( x ) − x ) µ ( n/k ) Mathematics Subject Classification.
Primary 11R09, 37P15.
Key words and phrases.
Misiurewicz point, post-critically finite, rigid divisibility sequence. be the standard dynatomic polynomial, and define the generalized dynatomicpolynomial Φ ∗ f,m,n by Φ ∗ f,m,n ( x ) = Φ ∗ f,n ( f m ( x ))Φ ∗ f,n ( f m − ( x )) . Then, the polynomial G d ( m, n ) is defined by G d ( m, n ) = ( Φ ∗ f,m,n (0)Φ ∗ f, ,n (0) d − if m = 0 and n | ( m − ∗ f,m,n (0) otherwise . Having defined the polynomials G d ( m, n ) , it is natural to ask: Which triples ( d, m, n ) make G d ( m, n ) irreducible? This question is wide open. For a refer-ence, see for instance [6] and [8]. There does not appear to exist any prior workwhich gives an infinite family of ( d, m, n ) for which G d ( m, n ) is irreducible. Inthis direction, the following corollary to our main theorems gives the first knownsuch infinite families of ( d, m, n ) . Corollary 1.1. G d ( m, n ) is irreducible in the following cases:(i) m = 0 , n = 1 , d is any prime.(ii) m = 0 , n = 2 , d = 2 . We also would like to mention that this corollary inspired a subsequent paperof Buff et al. ([2]), where they extended this result by proving that for k ≥ ,both G p k ( m, and G p k ( m, have precisely k different irreducible factors for m ≥ ([2], Theorem and Corollary ). They also proved that G ( m, is irre-ducible for m ≥ ([2], Corollary ).Before giving the next corollary to our main theorems, we recall a definitionfrom the theory of polynomial iteration: Let F be a field, and f ( x ) ∈ F [ x ] be apolynomial over F . We say that f is stable over F if f n ( x ) is irreducible over F for all n ≥ . Corollary 1.2.
Let d be a prime, and suppose f c,d ( x ) = x d + c has exact type ( m, for some m = 0 . Set K = Q ( c ) . Then, f c,d is stable over K . Note that the simplest example of Corollary . is the polynomial f − , ( x ) = x − ∈ Q [ x ] , which is already well-known to be stable. Thus, Corollary . can be thought of as a generalization of this well-known example.We also would like to say a few words about why the stability question isharder when f c,d has exact type ( m, n ) with n > : By a result of Hamblen et al.([5], Theorem ), proving the stability of f c,d comes down to show that there are OST-CRITICALLY FINITE POLYNOMIAL OF THE FORM x d + c no ± d th powers in the critical orbit. However, when n > , one of our main the-orems implies that there always exist some unit elements in the critical orbit, andchecking if these units are ± d th powers or not appear to be a difficult problem.Both Corollary . and Corollary . will follow from the next theorem,which establishes that the critical orbit elements for the PCF polynomials f c,d satisfy suprisingly strong properties when d is a prime. We first fix the followingnotation, which we will also use throughout the paper:Let K be a number field, and O K its ring of integers. Take a ∈ O K . Through-out, we will use ( a ) to denote the ideal a O K . Also, for f c,d with exact type ( m, n ) ,we will use O f c,d = { a , a , . . . , a m + n − } to denote the critical orbit, where weset a i = f ic,d (0) . Whenever we use a i for some i > m + n − , we again obtainit by setting a i = f ic,d (0) and using periodicity of f c,d . Theorem 1.3.
Let f c,d ( x ) = x d + c ∈ ¯ Q [ x ] be a PCF polynomial having exacttype ( m, n ) with m ≥ . Set K = Q ( c ) , and let O f c,d = { a , a , . . . , a m + n − } ⊂O K be the critical orbit of f c,d . Then the following holds:(a) If n i , then a i is a unit.(b) If d is a prime and n | i , then one has ( a i ) A = ( d ) , where A = (cid:26) d m − ( d − if n m − d m − − d − if n | m − . Having stated Theorem . , two remarks are in order here:Firstly, taking i = 1 in Theorem . , it follows that a = c is always a unitunless n = 1 , which is what is proven in ([1], Proposition ). Hence, our theoremgeneralizes this result of Buff.Secondly, our proof of the part n | i of Theorem . only works when d isa prime. In fact, it is easy to come up with counterexamples for this part ofthe statement when d is not a prime. For example, taking ( d, m, n ) = (4 , , ,MAGMA gives that ( a i ) = (4) for all i , although Theorem . would imply ( a i ) = (4) . However, when d is a prime power, based on MAGMA computa-tions, perhaps interestingly, it appears that some power of ( a i ) gives the ideal ( d ) for all i divisible by n . See Appendix for some more details about these compu-tations. The question of whether for all prime powers d such a power exists ornot remains open. V. GOKSEL
In our next theorem, we are able to get rid of the condition that d is a prime.However, it comes with the price that we do not get as much information as inTheorem . . Theorem 1.4.
Let f c,d ( x ) = x d + c ∈ ¯ Q [ x ] be a PCF polynomial with exact type ( m, n ) . Suppose m = 0 . Set K = Q ( c ) , and let O f c,d = { a , a , . . . , a m + n − } ⊂O K be the critical orbit of f c,d . Then, ( a i ) | ( d ) for all ≤ i ≤ m + n − . Theorem . has an application to the Galois theory of polynomial iterates,which we state as our next corollary: Corollary 1.5.
Let f c,d ( x ) = x d + c ∈ ¯ Q [ x ] be a PCF polynomial with exact type ( m, n ) . Suppose m = 0 . Set K = Q ( c ) , let K n be the splitting field of f nc,d ( x ) over K . If a prime p of K ramifies in K n , then p | ( d ) .
2. P
ROOFS OF M AIN R ESULTS
We first start by stating the following lemma, which we will use throughoutthe paper. Although it is simple, it becomes suprisingly useful.
Lemma 2.1.
Let f c,d ( x ) = x d + c ∈ ¯ Q [ x ] be a PCF polynomial with exacttype ( m, n ) . Set K = Q ( c ) , and let O f c,d = { a , a , . . . , a m + n − } ⊂ O K be thecritical orbit of f . Then for any i, j ≥ , there exists a polynomial P i,j ( t ) ∈ Z [ t ] such that f ic,d ( a j ) = a d i j + a jd P i,j ( c ) + a k , where k is the integer satisfying ≤ k ≤ j and k ≡ i ( mod j ) . Moreover, if d is a prime, we have P i,j ( t ) d ∈ Z [ t ] .Proof. Note that the consant term of f ic,d ( t ) ∈ Z [ c ][ t ] is a i , and all the other termsare divisible by t d . Write(2.1) f ic,d ( t ) = t d i + t d F ( t ) + a i for some F ( t ) ∈ Z [ c ][ t ] . Plugging a j into (2 . , get(2.2) f ic,d ( a j ) = a d i j + a dj F ( a j ) + a i . If i ≤ j , by taking P i,j ( c ) = F ( a j ) , (2 . already proves the statement (recallthat a j ∈ Z [ c ] ). Suppose i > j . Let i = jr + k for r ≥ and ≤ k ≤ j . By thedefinitions of a i and a j , we have(2.3) a i = f i − jc,d ( a j ) = a dj G ( c ) + a k for some G ( t ) ∈ Z [ c ][ t ] . Combining this with (2 . , the first part of the statementdirectly follows. To prove the last statement: Note that by the definition of f c,d and using binomial expansion repeatedly, some (cid:0) dl (cid:1) for ≤ l ≤ d − will appearin each coefficient of P i,j ( t ) , which proves the result, since d | (cid:0) dl (cid:1) when d is aprime. (cid:3) OST-CRITICALLY FINITE POLYNOMIAL OF THE FORM x d + c Next, we state another lemma which will be one of the ingredients in theproof of Theorem . . We first need to recall rigid divisibility sequences: Definition 2.2. ([5]) Let A = { a i } i ≥ be a sequence in a field K . We say A is arigid divisibility sequence over K if for each non-archimedean absolute value | . | on K , the following hold:(1) If | a n | < , then | a n | = | a kn | for any k ≥ .(2) If | a n | < and | a j | < , then | a gcd ( n,j ) | < . Remark 2.3.
Let K be a number field, and O K be its ring of integers. Suppose { a i } i ≥ ⊂ O K is a rigid divisibility sequence. Then, the following is a straight-forward consequence of Definition . :(i) ( a i ) | ( a ki ) for all k ≥ .(ii) (( a i ) , ( a j )) = ( a ( i,j ) ) . Lemma 2.4.
Let f c,d ( x ) = x d + c ∈ ¯ Q [ x ] be a PCF polynomial with exacttype ( m, n ) . Set K = Q ( c ) , and let O f c,d = { a , a , . . . , a m + n − } ⊂ O K be thecritical orbit of f c,d . Then, ( a i ) = ( a j ) for all i, j with ( i, n ) = ( j, n ) .Proof. First note that the sequence { a i } is a rigid divisibility sequence (For aproof of this fact, see ([5], Lemma )). We will now prove the lemma by show-ing that ( a i ) = ( a ( i,n ) ) for all i . Since the period is n , we can choose large enoughinteger k such that a i + nik = a i + n ( ik +1) . Using the second part of Remark . , wehave(2.4) (( a i + nik ) , ( a i + n ( ik +1) )) = ( a ( i + nik,i + n ( ik +1)) ) = ( a ( i,n ) ) . Hence, we get(2.5) ( a i + nik ) = ( a i + n ( ik +1) ) = ( a ( i,n ) ) . Using the first part of Remark . , the equalities in (2 . give ( a i + nik ) = ( a ( i,n ) ) | ( a i ) , since ( i, n ) | i . On the other hand, we also have i | i + nik , thus ( a i ) | ( a i + nik ) =( a ( i,n ) ) . Combining these two, we get ( a i ) = ( a ( i,n ) ) , which finishes the proof. (cid:3) We can finally prove Theorem . : Proof of Theorem 1.3.
We will prove (1) and (2) simultaneously. First suppose n i . Using Lemma . , we can find m ≤ j ≤ m + n − such that ( a i ) = ( a j ) .So, it is enough to prove the statement for a j . Since the exact type is ( m, n ) , each a k for m ≤ k ≤ m + n − is a root of the polynomial φ ( x ) = f nc,d ( x ) − x ∈ V. GOKSEL Z [ c ][ x ] . The constant term of φ ( x ) is a n . There exists a polynomial P ( x ) ∈ Z [ c ][ x ] satisfying(2.6) φ ( x ) = ( m + n − Y k = m ( x − a i )) P ( x ) . So, in particular P (0) ∈ Z [ c ] . We also have(2.7) (( − n m + n − Y i = m a i ) P (0) = a n . Note that there exists a unique m ≤ k ≤ m + n − such that n | k , and applyingLemma . to this k we have ( a k ) = ( a n ) . Hence, dividing both sides of the lastequation by a k , right-hand side becomes a unit, which implies a j is a unit (sinceit appears on the left-hand side), thus a i is a unit.Now, suppose n | i . Using Lemma . , there exists an integer k such that m ≤ nk ≤ m + n − and ( a i ) = ( a nk ) . So, it is enough to prove the statement for ( a nk ) . Note that since f has exact type ( m, n ) , we have(2.8) f m + nkc,d (0) = f mc,d (0) . We also have f m + nkc,d (0) = f mc,d ( a nk ) , so we obtain(2.9) f mc,d ( a nk ) = f mc,d (0) . (2.10) ⇐⇒ [ f m − c,d ( a nk )] d + c = [ f m − c,d (0)] d + c. (2.11) ⇐⇒ ( f m − c,d ( a nk ) − f m − c,d (0))( d − X i =0 [ f m − c,d ( a nk )] i [ f m − c,d (0)] d − − i ) = 0 . Because f c,d has exact type ( m, n ) , we get(2.12) d − X i =0 [ f m − c,d ( a nk )] i [ f m − c,d (0)] d − − i = 0 . Using Lemma . , since d is a prime, we can find a polynomial P ( t ) ∈ Z [ t ] satisfying(2.13) f m − c,d ( a nk ) = a d m − nk + da dnk P ( c ) + a m − . Putting this into (2 . , we get(2.14) d − X i =0 ( a d m − nk + da dnk P ( c ) + a m − ) i a d − − im − = 0 . Using (2 . , we can find a polynomial Q ( t ) ∈ Z [ t ] such that(2.15) d − X i =0 ( a d m − nk + a m − ) i a d − − im − + da dnk Q ( c ) = 0 . OST-CRITICALLY FINITE POLYNOMIAL OF THE FORM x d + c (2.16) ⇐⇒ d − X i =0 ( i X j =0 (cid:18) ij (cid:19) a d m − jnk a i − jm − ) a d − − im − + da dnk Q ( c ) = 0 . (2.17) ⇐⇒ d − X j =0 ( d − X i = j (cid:18) ij (cid:19) ) a d m − jnk a d − − jm − + da dnk Q ( c ) = 0 . Using the hockey-stick identity, (2 . becomes(2.18) d − X j =0 (cid:18) dj + 1 (cid:19) a d m − jnk a d − − jm − + da dnk Q ( c ) = 0 . Observe that all terms of (2 . except da d − m − and a d m − ( d − nk are divisible by da dnk because d is a rational prime. Therefore we get(2.19) da d − m − + a d m − ( d − nk ≡ ( mod da dnk ) , and(2.20) a d m − ( d − nk = d ( − a d − m − + αa dnk ) with some α ∈ O K .If n | m − , then a nk = εa m − with a unit ε , hence (2 . takes the form(2.21) ( εa m − ) d m − ( d − = d ( − a d − m − + α ( εa m − ) d ) , and after dividing by a d − m − we get(2.22) ε d m − ( d − a ( d m − − d − m − = d ( − α ( ε d a m − )) and since the ideals ( − αε d a m − ) and ( a m − ) are co-prime we get the asser-tion.If n m − , then a m − is a unit, hence the ideals ( a nk ) and ( − a d − m − + αa dnk ) are co-prime and the assertion follows. (cid:3) We will now prove the Corollary . and Corollary . . We first need to recalla basic fact from algebraic number theory:Let L be a finite extension of a number field K . Let p be a prime ideal in K .Suppose that p factors in L as p O L = P e . . . P g e g . Set f i = |O L / P i | for ≤ i ≤ g , and n = [ L : K ] . Then, we have(2.23) g X i =1 e i f i = n. V. GOKSEL
Proof of Corollary 1.1. (1) Set K = Q ( c ) , and let N = [ K : Q ] . For n = 1 , by ([6], Corollary . ), G d ( m, n ) has degree ( d m − − d − . Thus, we have N ≤ ( d m − − d − .On the other hand, second part of Theorem . gives ( a i ) ( d m − − d − = ( d ) forany i . Factor ( a i ) into prime factors as ( a i ) = P e . . . P g e g . Taking the ( d m − d − th power of each side, we get ( d ) = P ( d m − − d − e . . . P g ( d m − − d − e g . Using (2 . , we get g X i =1 ( d m − − d − e i f i = N, which implies N ≥ ( d m − − d − . Hence, we obtain N = ( d m − − d − ,which shows G d ( m, n ) is irreducible.(2) Let N be as in the first part of the proof. For n = 2 , d = 2 , by ([6], Corollary . ), G d ( m, n ) has degree m − − if | m − , and has degree m − if m − .On the other hand, using the first part of Theorem . , we have ( a i ) m − − = (2) for any | i if m is odd, and ( a i ) m − = (2) for any | i if m is even. Similar tothe first part, for both cases N becomes equal to the degree of G d ( m, n ) , whichproves that G d ( m, n ) is irreducible. (cid:3) Remark 2.5. In (1) , d is totally ramified in K , and for all i , ( a i ) is the uniqueprime ideal of O K that divides ( d ) . In (2) , totally ramified in K , and for all i even, ( a i ) is the unique prime ideal of O K that divides (2) .Proof of Corollary 1.2. By ([5], Theorem ), it suffices to show that there is no ± d th power in the orbit. By the first part of Remark . , a i is a prime element of O K for all i . Hence, a i can never be ± d th power in O K for d ≥ , which finishesthe proof. (cid:3) Proof of Theorem 1.4.
From Theorem . , we already have that a i is a unit when n i , which gives ( a i ) | ( d ) . So, we only need to show ( a i ) | ( d ) for n | i . There existsa unique k such that m ≤ nk ≤ m + n − , i.e., a nk is periodic under f . ByLemma . , it is enough to prove the statement for a nk . This is the same situationas in the second part of the proof of Theorem . . We rewrite the equation (2 . : d − X i =0 [ f m − c,d ( a nk )] i [ f m − c,d (0)] d − − i = 0 . OST-CRITICALLY FINITE POLYNOMIAL OF THE FORM x d + c Recalling that d is not necessarily a prime, and making the obvious modificationsin the equations (2 . − (2 . accordingly, we obtain(2.24) da d − m − + a d m − ( d − nk = αa dnk with some α ∈ O K .If n | m − , then a nk = εa m − with a unit ε , hence (2 . takes the form(2.25) da d − m − + ( εa m − ) d m − ( d − = α ( εa m − ) d and after dividing by a d − m − we get(2.26) d + ε d m − ( d − a ( d m − − d − m − = αε d a m − , which proves the assertion.If n m − , then a m − is a unit, hence (2 . directly proves the assertion. (cid:3) We finish this section by proving Corollary . , and giving a remark about it. Proof of Corollary 1.5.
Recall that if a prime p of K is ramified in K n , then itmust divide Disc ( f nc,d ) . Set △ n = Disc ( f nc,d ) . By ([7], Lemma 2.6), we have therelation(2.27) △ n = ±△ n − d d n a n . Proceeding inductively, (2 . shows that if a prime p of K ramifies in K n , then p divides ( d ) or ( a i ) for some i . By Theorem . , this directly implies p | ( d ) , asdesired. (cid:3) Remark 2.6.
Let f c, ( x ) = x + c have exact type ( m, or ( m, , and set K = Q ( c ) . is totally ramified in K (by Remark . ), let p ⊂ O K be the uniqueprime above it (which is generated by one of the critical orbit elements). Then,it follows from Corollary . that K n is unramified outside of the set { p , ∞} forall n ≥ . Hence, this way we can get infinitely many explicit examples of pro- extensions of various number fields unramified outside of a finite prime andinfinity.
3. P
ERIODIC CASE
In this section, we will give a simple observation about the case m = 0 . Lemma 3.1.
Let f c,d ( x ) = x d + c ∈ ¯ Q [ x ] be a PCF polynomial with exact type (0 , n ) . Set K = Q ( c ) , and let O f c,d = { a , a , . . . , a n = 0 } ⊂ O K be the criticalorbit of f c,d . Then, a i is a unit in O K for all ≤ i ≤ n − . Proof.
In view of a i +1 = a di + c , a j +1 = a dj + c one gets after subtraction andmultiplication the equality Y i = j a di − a dj a i − a j = 1 , hence all elements a di − a dj a i − a j ( i = j ) are units, and putting here j = n we obtain that for i = j a d − i is a unit, and so is a i . (cid:3)
4. A
PPENDIX
We finish the paper by presenting some data about the question stated at thebottom of page . Note that G d ( m, n ) is not irreducible in the cases below, andMAGMA computations show that we get different values of A (with the notationof Theorem . ) depending on the minimal polynomial of c over Q . • ( d, m, n ) = (4 , ,
1) = ⇒ ( a i ) = (4) or ( a i ) = (4) for all i . • ( d, m, n ) = (4 , ,
2) = ⇒ ( a i ) = (4) or ( a i ) = (4) for all | i . • ( d, m, n ) = (4 , ,
3) = ⇒ ( a i ) = (4) or ( a i ) = (4) for all | i . • ( d, m, n ) = (4 , ,
1) = ⇒ ( a i ) = (4) or ( a i ) = (4) for all i . • ( d, m, n ) = (4 , ,
2) = ⇒ ( a i ) = (4) or ( a i ) = (4) for all | i . • ( d, m, n ) = (4 , ,
1) = ⇒ ( a i ) = (4) or ( a i ) = (4) for all i . • ( d, m, n ) = (8 , ,
1) = ⇒ ( a i ) = (8) or ( a i ) = (8) or ( a i ) = (8) forall i . • ( d, m, n ) = (9 , ,
1) = ⇒ ( a i ) = (9) or ( a i ) = (9) for all i .We also would like to note that this phenomenon does not necessarily holdwhen d is not a prime power. For instance, taking ( d, m, n ) = (6 , , , one seesthat (6) = p p for some prime ideals p , p in Q ( c ) , which shows that (6) cannot be a perfect power. OST-CRITICALLY FINITE POLYNOMIAL OF THE FORM x d + c Acknowledgments.
The author owes Nigel Boston and Sarah Koch a debt ofgratitude for their very helpful comments on this work. The author also wouldlike to thank the anonymous referee for their careful reading of the article andhelpful comments. R
EFERENCES [1] X.Buff. On postcritically finite unicritical polynomials, Preprint. Available at .[2] X.Buff; A.Epstein; S.Koch. Irreducibility and postcriti-cally finite unicritical polynomials, Preprint. Available at .[3] L.Danielson and B.Fein. On the irreducibility of the iterates of x n − b . Proc. Amer.Math. Soc. 130(6), 1589-1596, 2002.[4] A.Epstein. Integrality and rigidity for postcritically finite polynomials. Bull. Lond.Math. Soc., 44(1):39-46, 2012. With an appendix by Epstein and Bjorn Poonen.[5] S.Hamblen; R.Jones; K.Madhu. The density of primes inorbits of z d + c . Int. Math. Res.Not. IMRN, no. 7, 1924-1958, 2015.[6] B.Hutz; A.Towsley. Misiurewicz points for polynomial maps and transversality. NewYork J. Math. 21, - , 2015.[7] R.Jones. The density of prime divisors in the arithmetic dynamics of quadratic polyno-mials. J. Lond. Math. Soc. (2), 78(2): - , 2008.[8] J.Milnor. Arithmetic of unicritical polynomial maps, Frontiers in Complex Dynamics:In Celebration of John Milnor’s 80th Birthday 15-23, 2012.[9] G.Pastor; M.Romera; F.Montoya. Misiurewicz points in one-dimensional quadraticmaps. Phys. A, 232(1-2): - , 1996.M ATHEMATICS D EPARTMENT , U
NIVERSITY OF W ISCONSIN , M
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