Reduction of polynomial dynamical systems modulo primes
aa r X i v : . [ m a t h . N T ] F e b REDUCTION OF POLYNOMIAL DYNAMICAL SYSTEMSMODULO PRIMES
S. S. ROUT
Abstract.
We study the algebraic dynamical systems generated by triangu-lar systems of rational functions and estimate the height growth of iterationsgenerated by such systems. Further, using a result on the reduction moduloprimes of systems of multivariate polynomials over the integers, we study theperiodic points and the intersection of orbits of such dynamical systems overfinite fields. Introduction
Let V ⊂ P N be a quasi-projective variety defined over a field K and letΦ : V −→ V be an endomorphism. For any m ∈ N = N ∪ { } , we denote by Φ ( m ) = Φ ◦ · · · ◦ Φthe m -th iteration of Φ with Φ (0) denoting the identity map. For a given point P ∈ V ( K ), the (forward) orbit of P is the setOrb Φ ( P ) = { P, Φ (1) ( P ) , Φ (2) ( P ) , . . . } . The point P is called a periodic point for Φ if Φ ( n ) ( P ) = P for some n ≥ n is called the period of P . The point P is called preperiodic ifsome iterate Φ ( m ) ( P ) is periodic.The area of algebraic dynamics was introduced by Northcott [9] and later,Silverman [11] greatly developed all aspects of the theory of algebraic dynamics.For a background of the dynamical systems associated with iterations, one canrefer to [10, 11]. In [12], Silverman studied the orbit length for the reductionmodulo a prime p for any self morphism of a quasi-projective variety defined overa number field. Later, this result has been improved in [1]. Since then there havebeen many advances in the study of periodic points and period lengths in thereductions of orbits of dynamical systems modulo distinct primes p .Motivated by the work of Towsley [13] on Hasse principle for periodic points,D’Andrea et. al., [3], using several tools from arithmetic geometry, have provednew results about the orbits of the reductions modulo a prime p of algebraic Mathematics Subject Classification.
Primary 37P05, Secondary 37P25, 11G25, 13P15.
Key words and phrases.
Algebraic dynamical system, arithmetic Nullstellensatz, reductionof systems of polynomials, orbit intersection, periodic point. dynamical systems over Q . Later in [2], Changa et. al., gives a lower boundfor the orbit length of the reduction modulo primes of parametric polynomialdynamical systems defined over integers. As a by-product, their result recoversa result in [1] and slightly improves a result in [12].The results in [3] depends on the growth of the degree and the height of the it-erates. When this growth is slower than generic, one can expect stronger bounds.Although for a typical system an exponential degree growth is expected, thereare rich families of multivariate polynomial systems with a much slower degreegrowth (see [4, 5, 6, 7]). For example, for triangular system of polynomials, it hasbeen shown in [6] that degrees of the iterations of the polynomials in triangularsystem grow very slowly.In this paper, we consider the following class of rational dynamical systemswith slow degree growth.Let(1) F = ( F , . . . , F n ) , F , . . . , F n ∈ Q ( X )be a system of n rational functions in n variables ( X , . . . , X n ) over Q where F ( X , . . . , X n ) = X e G ( X , . . . , X n ) + H ( X , . . . , X n ) · · · F n − ( X , . . . , X n ) = X e n − n − G n − ( X n ) + H n − ( X n ) F n ( X , . . . , X n ) = g n X e n n + h n , (2)with e , . . . , e n ∈ {− , } , G i , H i ∈ Z [ X i +1 , . . . , X n ] , i = 1 , . . . , n − g n , h n ∈ Z , g n = 0. We define the iterations of the rational function F i as follows. G ( ℓ ) i ( X i +1 , . . . , X n ) = G i ( F ( ℓ − i +1 , . . . , F ( ℓ − n ) ,H ( ℓ ) i ( X i +1 , . . . , X n ) = H i ( F ( ℓ − i +1 , . . . , F ( ℓ − n ) . Our first main result in this paper gives a bound for the number of pointsof a given period in the reduction modulo p of the algebraic dynamical systemdefined in (2). Also, we give a bound for the frequency of the points in an orbitof the reduction modulo p of the algebraic dynamical systems defined in (2) lyingin a given algebraic variety. To prove these results, we use a deep result fromarithmetic geometry [3, Theorem 2.1].2. Notation and Main Results
Let X denotes the group of variables ( X , . . . , X n ) so that Z [ X ] denotes thering of polynomials Z [ X , . . . , X n ] and Q ( X ) denotes the field of rational functions Q ( X , . . . , X n ). Let K denote an algebraic closure of a field K of characteristiczero. EDUCTION OF POLYNOMIAL DYNAMICAL SYSTEMS MODULO PRIMES 3
For a polynomial L ∈ Z [ X ], we define its height as the logarithm of the maxi-mum of the absolute values of its coefficients and denote it by h ( L ). For a rationalfunction F ∈ Q ( X ), we write F = L/K with coprime
L, K ∈ Z [ X ] and we definethe degree and the height of F , respectively, as the maximum of the degrees andof the heights of L and K , that is,deg F = max { deg L, deg K } and h ( F ) = max { h ( L ) , h ( K ) } . To give explicit formula for degree growth of the iterates of the system in (2), weimpose the following conditions on the degrees of the polynomials G i and H i for i = 1 , . . . , n − e i = 1, we assume that the polynomial G i has a unique leading monomial X s i,i +1 i +1 · · · X s i,n n , that is G i = g i X s i,i +1 i +1 · · · X s i,n n + ˜ G i , where g i ∈ Z \ { } and ˜ G i ∈ Z [ X i +1 , . . . , X n ] with(3) deg X j ˜ G i < s i,j , deg X j H i < s i,j , j = i + 1 , . . . , n. If e i = −
1, we assume that the polynomial H i has a unique leading monomial X s i,i +1 i +1 · · · X s i,n n , that is H i = h i X s i,i +1 i +1 · · · X s i,n n + ˜ H i , where h i ∈ Z \ { } and ˜ H i ∈ Z [ X i +1 , . . . , X n ] with(4) deg X j ˜ H i < s i,j , deg X j G i < s i,j , j = i + 1 , . . . , n. We define the orbit of a given point w ∈ Q n with respect to the system ofrational functions in (1) as the set(5) Orb F ( w ) = { w k | w = w and w k = F ( w k − ) , k = 1 , , . . . } . If w k is a pole of F , then the orbit terminates and in this case Orb F ( w ) is a finiteset. Further, given k ≥
1, we say that w ∈ Q n is k -periodic if the element w k exists in the orbit (5) and we have w k = w . Further, we put(6) S ( w ) = F ( w ) ∈ N ∪ {∞} . Let p ∈ Z be a prime. For each prime p , set F ( m ) i,p = F ( m ) i (mod p ). We definethe reduction modulo p of the iteration F ( m ) and denote this by F ( m ) p = ( F ( m )1 ,p , . . . , F ( m ) n,p ) ∈ F p [ X ] n . Let L = ( L , . . . , L s ) ∈ Z [ X ] be a system of polynomials of degree at most D and height at most H . We denote by V the subvariety of the affine space A n Q defined by this system of polynomials. For a prime p , we denote by L i,p ∈ F p [ X ]the reduction modulo p of L i and by V p the subvariety of A n F p defined by thesystem L i,p , i = 1 , · · · , s . S. S. ROUT
Given the functions f, g : N −→ R the symbols f = O ( g ) and f ≪ g both mean that there is a constant c ≥ | f ( k ) | ≤ cg ( k ) for all k ∈ N . To emphasize the dependence of theimplied constant c on a list of parameters, say n, d, h , we write f = O n,d,h ( g ) and f ≪ n,d,h g .The following result is concerned with the number of points of a given periodin the reduction modulo p of triangular systems of polynomials as in (1) with e i = 1. Theorem 2.1.
Let F , . . . , F n ∈ Z [ X ] be as in (2) with e i = 1 , satisfying thecondition (3) such that s i,i +1 = 0 , i = 1 , . . . , n − . Set d = max j =1 ,...,n deg F j and h = max j =1 ,...,n h ( F j ) . Suppose that F = ( F , . . . , F n ) has finitely many periodic points of order k over C . Then there exists an integer B ∈ N satisfying log B ≪ d,h,n k n (3 n − and such that if p is a prime number not dividing B , then the reduction of F modulo p has O d,h,n ( k n ( n − / ) periodic points of order k . In the following theorem, we study the same result as in Theorem 2.1 for systemin (1) with e i = − Theorem 2.2.
Let n ∈ N with n ≥ . For i = 1 , . . . , n , let F i be rationalfunctions defined by (2) satisfying the condition (4) such that s i,i +1 = 0 , i =1 , . . . , n − and e i = − for i = 1 , . . . , n . Set d = max j =1 ,...,n deg F j and h = max j =1 ,...,n h ( F j ) . Suppose that F = ( F , . . . , F n ) has finitely many periodic points of order k over C . Then there exists an integer B ∈ N satisfying log B ≪ d,h,n k n (3 n +8 n +9) / and such that if p is a prime number not dividing B , then the reduction of F modulo p has O d,h,n ( k n ) periodic points of order k . Next we obtain an upper bound for the frequency of the orbit intersections ofa rational function system. More generally, we bound the number of points insuch an orbit that belong to a given algebraic variety.For ℓ ∈ N , let p be a prime such that the iterations F ( j ) with j = 0 , . . . , ℓ − p . Given a point w ∈ F np , we define I w ( F , V ; p, ℓ ) = (cid:8) m ∈ { , , . . . , ℓ − } | F ( m ) p ( w ) ∈ V p ( F p ) (cid:9) . EDUCTION OF POLYNOMIAL DYNAMICAL SYSTEMS MODULO PRIMES 5
We say that the iterations of F generically escape V if for every integer k ≥ k -th iteration of F is well defined and the set (cid:8) w ∈ C n | (cid:0) w , F ( k ) ( w ) (cid:1) ∈ V ( C ) × V ( C ) (cid:9) . is finite. Theorem 2.3.
Let n ∈ N with n ≥ . For i = 1 , . . . , n , let F i be ratio-nal functions defined by (2) satisfying the conditions (3) and (4) and such that s i,i +1 = 0 , i = 1 , . . . , n − . Set d = max j =1 ,...,n deg F j and h = max j =1 ,...,n h ( F j ) . Let V be the subvariety of A n Q defined by the system of polynomials ( L , . . . , L s ) ∈ Z [ X ] of degree at most D and height at most H . Assume that the iterations of F generically escape V . Then, there is a constant C > (depending on D, H, d, h,n, s) such that for any real ǫ > and ℓ ∈ N with ℓ ≥ n s D s ǫ ( n − s +2 there exists D ∈ N with log D ≤ C/ǫ n (3 n − such that if p is a prime number not dividing D , then for any w ∈ F np with S ( w ) ≥ ℓ , I w ( F , V ; p, ℓ ) ≤ ǫℓ. Next we obtain a better result for the problem of bounding the frequency ofthe points in an orbit lying in a given variety under a restrictive condition.Let F ∈ Q [ X ] n be a system of rational functions over K and let V ⊆ A n Q be anaffine variety. The intersection of the orbit F with V is L -uniformly bounded ifthere is a constant L depending only on F and V such that for all initial values w ∈ Q n , { m ∈ N | w m ∈ V ( Q ) } ≤ L, where w m is defined in (5). Theorem 2.4.
Let n ∈ N with n ≥ . For i = 1 , . . . , n , let F i be ratio-nal functions defined by (2) satisfying the conditions (3) and (4) and such that s i,i +1 = 0 , i = 1 , . . . , n − . Set d = max j =1 ,...,n deg F j and h = max j =1 ,...,n h ( F j ) . Let V be the subvariety of A n Q defined by the system of polynomials = ( L , . . . , L s ) ∈ Z [ X ] of degree at most D and height at most H . Assume that the intersection of S. S. ROUT orbits of F with V is L -uniformly bounded. There is a constant C > (dependingon D, H, d, h, n, L, s) such that for any real ǫ > there exists C ∈ N with log C ≤ Cǫ ( n − n +2)+( n + L +2) such that if p is a prime number not dividing C , then for any integer ℓ ≥ L/ǫ + 1 and for any initial point w ∈ F np with S ( w ) ≥ ℓ , we have I w ( F , V ; p, ℓ ) ≤ ǫℓ. Preliminaries
In this section, we gather some bounds on the heights and the degrees oftriangular polynomial systems. We start with bounds for the heights of sumsand products of polynomials, which follows from [8, Lemma 1.2].
Lemma 3.1.
Let K , . . . , K t ∈ Z [ X ] . Then(1) h (cid:16) P ti =1 K i (cid:17) ≤ max ≤ i ≤ t h ( K i ) + log t ;(2) − n +1) P ti =1 deg K i ≤ h Q ti =1 K i ! − P ti =1 h ( K i ) ≤ log( n +1) P ti =1 deg K i . The following is the standard bound for the degree and height of the composi-tion of polynomials with integer coefficients (see [8, Lemma 1.2(1.c)]).
Lemma 3.2.
Let L ∈ Z [ Y , . . . , Y t ] , K , . . . , K t ∈ Z [ X ] . Set d = max i =1 ,...,t deg K i and h = max i =1 ,...,t h ( K i ) . Then, deg( L ( K , . . . , K t )) ≤ d deg Lh ( L ( K , . . . , K t )) ≤ h ( L ) + deg L ( h + log( t + 1) + d log( n + 1)) . The following is an extension of Lemma 3.2 to the composition of rationalfunctions (see [3]).
Lemma 3.3.
Let
L, K , . . . , K n ∈ Q [ X ] such that the composition L ( K , . . . , K n ) is well defined.. Set d = max i =1 ,...,n deg K i and h = max i =1 ,...,n h ( K i ) . Then, deg( L ( K , . . . , K n )) ≤ dn deg Lh ( L ( K , . . . , K n )) ≤ h ( L ) + h deg L + (3 dn + 1) deg L log( n + 1) . EDUCTION OF POLYNOMIAL DYNAMICAL SYSTEMS MODULO PRIMES 7
The following lemma gives the degree growth of the iterations of function de-fined by (2)(see [7, Theorem 2]).
Lemma 3.4.
Let F , . . . , F n be rational functions defined by (2) satisfying theconditions (3) and (4) and such that s i,i +1 = 0 , i = 1 , . . . , n − . Then degrees ofthe iterations of F , . . . , F n grow as follows deg F ( k ) i = 1( n − i )! k n − i s i,i +1 · · · s n − ,n + ψ i ( k ) , i = 1 , . . . , n − , deg F ( k ) n = 1 where ψ i ( T ) ∈ Q [ T ] is a polynomial of degree deg ψ i < n − i . Lemma 3.5.
For i = 1 , . . . , n , let G i ∈ Z [ X i , X i +1 , . . . , X n ] be a triangularsystem of polynomials with a unique leading monomial of the form X s i,i +1 i +1 · · · X s i,n n and F i as in (2) . Set d = max i =1 ,...,n deg F i , and h = max i =1 ,...,n h ( F i ) . The height of the iterations of G , . . . , G n for k ≥ grow as follow: (7) h ( G ( k ) i ) ≤ k − X j =1 deg G ( j ) i ! ( h + log( n − i )( n + 1) d ) + h. Moreover, for any positive integer k ≥ and ≤ i ≤ n , h ( G ( k ) i ) ≪ h,d,n k n − i +1 . (8) Proof.
The inequality (7) for the height follows by induction on the number ofiterates k . We set for any k ≥ ≤ i ≤ nd i,k = deg G ( k ) i , h i,k = h (cid:16) G ( k ) i (cid:17) . For k = 2, we have h ( G (2) i ) = G i ( F (1) i +1 , . . . , F (1) n ). Now applying Lemma 3.2 tothis, h ( G (2) i ) = h ( G i ) + deg( G i )( h + log( n − i ) + d log( n + 1))= h i, + d i, ( h + log( n − i ) + d log( n + 1)) ≤ ( d i, + 1) h + d i, log( n − i ) + dd i, log( n + 1) . Thus, the inequality (7) is true for k = 2. Now assume that inequality (7) is truefor the first k − G ( k ) i = G ( k − i ( F i +1 , . . . , F n ) , S. S. ROUT h ( G ( k ) i ) = h ( G ( k − i ) + deg( G ( k − i )( h + log( n − i ) + d log( n + 1)) ≤ k − X j =1 deg G ( j ) i ! ( h + log( n − i )( n + 1) d ) + h + d i,k − ( h + log( n − i ) + d log( n + 1)) . This proves inequality (7). Now from (7) and Lemma 3.4, we have h ( G ( k ) i ) ≤ k − X j =1 deg G ( j ) i ! ( h + log( n − i )( n + 1) d ) + h = k − X j =1 deg G i (cid:16) F ( j − i +1 , . . . , F ( j − n (cid:17)! ( h + log( n − i )( n + 1) d ) + h = k − X j =1 deg (cid:16) ( F ( j − i +1 ) s i,i +1 · · · ( F ( j − n ) s i,n (cid:17)! ( h + log( n − i )( n + 1) d ) + h = k − X j =1 (cid:18) n − i − j − n − i − s i,i +1 · · · s n − ,n + · · · + ( j − s i,n s n − ,n + 1 (cid:19)! × ( h + log( n − i )( n + 1) d ) + h ≪ h,d,n k n − i +1 . This completes the proof. (cid:3)
Let us define the sets I + = { ≤ i ≤ n | e i = 1 } , I − = { ≤ i ≤ n | e i = − } . Lemma 3.6.
Let F , . . . , F n be rational functions defined by (2) satisfying theconditions (3) and (4) and such that s i,i +1 = 0 , i = 1 , . . . , n − . Then height ofthe iterations of F , . . . , F n grow as follows: h ( F ( k ) i ) ≤ ( k + 1) deg (cid:16) F ( k ) i (cid:17) log( n + 1) + k X j =1 h (cid:16) G ( j ) i (cid:17) + log 2 for every i ∈ I + and for every i ∈ I − h ( F ki ) ≤ ( k + 1) deg (cid:16) F ( k ) i (cid:17) log( n + 1) + k X j =1 h (cid:16) H ( j ) i (cid:17) + ( k + 1) log 2 . Moreover, h (cid:16) F ( k ) i (cid:17) ≪ d,h,n k n − i +2 . EDUCTION OF POLYNOMIAL DYNAMICAL SYSTEMS MODULO PRIMES 9
Proof.
First we prove the case when i ∈ I + . The explicit structure of the iterationsof the rational functions F i are given in [7]. By [7, Lemma 2], we have(9) F ki = ( X i G i,k + H i,k , for i < ng kn X n + ( g k − n + · · · + g n + 1) h n for i = n, where G i,k = G i G (2) i · · · G ( k ) i ,H i,k = H i G (2) i · · · G ( k ) i + H (2) i G (3) i · · · G ( k ) i + · · · + H ( k − i G ( k ) i + H ( k ) i . Applying Lemma 3.1 in equation (9) for i < n , h ( F ki ) ≤ h ( X i G i,k ) + log 2 = h (cid:16) X i G i G (2) i · · · G ( k ) i (cid:17) + log 2 ≤ deg (cid:16) X i G i G (2) i · · · G ( k ) i (cid:17) log( n + 1) + k X j =1 h (cid:16) G ( j ) i (cid:17) + h ( X i ) + log 2 ≤ ( k + 1) deg (cid:16) F ( k ) i (cid:17) log( n + 1) + k X j =1 h (cid:16) G ( j ) i (cid:17) + log 2 . Again using Lemma 3.1 in (9) for i = n , h ( F ( k ) n ) = h (cid:0) g kn X n + ( g k − n + · · · + g n + 1) h n (cid:1) ≤ h ( g kn X n ) + log( k + 1) ≤ log (cid:0) g kn ( n + 1)( k + 1) (cid:1) . Now consider the case i ∈ I − and i < n . In this case, by [7, Lemma 2], we have(10) F ( k ) i = X i R i,k + S i,k X i R i,k − + S i,k − , where R i,k , S i,k are defined by the recurrence relations(11) R i,k = G ( k ) i R i,k − + H ( k ) i R i,k − , S i,k = G ( k ) i S i,k − + H ( k ) i S i,k − for k ≥ R i, = 1 , S i, = 0 , R i, = H i , S i, = G i . From Lemma 3.1 and (10), h ( F ki ) ≤ max { h ( X i R i,k + S i,k ) , h ( X i R i,k − + S i,k − ) }≤ h ( X i R i,k ) + log 2 ≤ deg ( X i R i,k ) log( n + 1) + h ( R i,k ) + h ( X i ) + log 2 . (12) Applying Lemma 3.1 in (11), one can inductively show that(13) h ( R i,k ) ≤ (deg( R i, · · · R i,k )) log( n + 1) + k X j =1 h ( H ( j ) i ) + k log 2Thus, for i ∈ I − and i < n , from (12) and (13) we conclude h ( F ki ) ≤ ( k + 1) deg (cid:16) F ( k ) i (cid:17) log( n + 1) + k X j =1 h (cid:16) H ( j ) i (cid:17) + h ( X i ) + ( k + 1) log 2 . For the case e n = −
1, we have F ( k ) n = ( A k ) , X n + ( A k ) , ( A k ) , X n + ( A k ) , , where A k = (cid:18) h n g n (cid:19) k = (cid:18) ( A k ) , ( A k ) , ( A k ) , ( A k ) , (cid:19) . One can observe that the entries of the matrix A k are polynomials in the integer h n and g n . Hence h ( F ki ) ≤ max (cid:8) h (cid:0) ( A k ) , X n + ( A k ) , (cid:1) , h (cid:0) ( A k ) , X n + ( A k ) , (cid:1)(cid:9) ≤ h (cid:0) ( A k ) , X n (cid:1) + h (( A k ) , ) + log 2 ≤ h (cid:0) ( A k ) , (cid:1) + h (cid:0) ( A k ) , (cid:1) + log(2( n + 1)) ≤ log( h kn ( k + 1)( n + 1)) . This completes the estimates of h ( F ki ) for i ∈ I − and i ≤ n . Also, k X j =0 j n − i +1 = 1 n − i + 2 ( B n − i +2 ( k + 1) − B n − i +2 (0)) , where B n − i +2 is the Bernoulli polynomial of degree n − i +2 with leading coefficientequal to 1. Thus, from Lemma 3.5, the height of the k -th iteration of F i is atmost h (cid:16) F ( k ) i (cid:17) ≪ d,h,n k n − i +2 . (cid:3) The following result is on the reduction modulo primes of systems of multivari-ate polynomials over the integers, whose proof relies on the arithmetic Nullsten-llensatz (see [3, Theorem 2.1]).
Lemma 3.7.
Let H , . . . , H s ∈ Z [ X ] be polynomials of degree at most d ≥ andheight at most h , whose zero set in C n has a finite number T distinct points.Then there is an integer A ≥ with log A ≤ (11 n + 4) d n +1 h + (55 n + 99) log((2 n + 5) s ) d n +2 such that if p is a prime number not dividing A , then the zero set in F np of thesystem of polynomials H i ( mod p ) , i = 1 , . . . , s consists of exactly T distinct points. We also need the following combinatorial result [3].
EDUCTION OF POLYNOMIAL DYNAMICAL SYSTEMS MODULO PRIMES 11
Lemma 3.8.
Let ≤ M ≤ N/ . For any sequence ≤ n < · · · < n M ≤ N, there exists r ≤ N/ ( M − such that n i +1 − n i = r for at least ( M − / N values of i ∈ { , . . . , M − } . Now we are ready to proof our results. The proof is motivated by the ideas ofD’Andrea et. al., [3]. 4.
Proof of Main Results
Proof of Theorem 2.1.
Consider the system of equations F ( k ) i − X i = 0 , i = 1 , . . . , n. The set of k -periodic points of F coincides with the zero set V k = Z (cid:16) F ( k )1 − X , . . . , F ( k ) n − X n (cid:17) . For i = 1 , . . . , n ,(14) deg (cid:16) F ( k ) i − X i (cid:17) = 1( n − i )! k n − i ( s i,i +1 · · · s n − ,n ) + 1 , and h (cid:16) F ( k ) i − X i (cid:17) ≤ max n h (cid:16) F ( k ) i (cid:17) , h ( X i ) o + log 2 ≪ d,h,n k n − i +2 . (15)Now apply Lemma 3.7 and derivelog B ≤ C ( n, d, h )( k n − ) n +1 k n +1 + C ( n, d, h )( k n − ) n +2 ≪ d,h,n k n (3 n − . Suppose T k is the number of points of V k over C and this equal to the number ofperiodic points of order k of F over C . By Bezout’s theorem, T k ≤ n Y i =1 k n − i ≪ n,d,h k n ( n − / . This completes the proof. (cid:3)
Proof of Theorem 2.2.
By equation (10), the iterates of the system ofrational functions F is given by F ( k ) i = X i R i,k + S i,k X i R i,k − + S i,k − =: Γ i,k Ψ i,k with Ψ i,k = 0 and consider the system of equationsΓ i,k − X i Ψ i,k = 0 , i = 1 , . . . , n. To extract the poles of F ( j ) i , j ≤ k from the solutions of the system, we introducea new variable X . Now the set of k -periodic points of F coincides with the zeroset V k = Z Γ ,k − X Ψ ,k , . . . , Γ n,k − X n Ψ n,k , − X n Y i =1 k Y j =1 Ψ i,j ! . For i = 1 , . . . , n deg(Γ i,k − X i Ψ i,k ) ≤ k n − i + 1 ≤ C ( n, d ) k n − i and h (Γ i,k − X i Ψ i,k ) ≤ h ( F ( k ) i ) + log 2 ≤ C ( n, d, h ) k n − i +2 . Now deg X n Y i =1 k Y j =1 Ψ i,j ! ≤ n X i =1 k X j =1 j n − i ≤ C ( n, d, h ) k n ( n +1) / . By Lemma 3.3 h X n Y i =1 k Y j =1 Ψ i,j ! = h n Y i =1 k Y j =1 Ψ i,j ! ≤ n X i =1 k X j =1 h (Ψ i,j ) + log( n + 1) n X i =1 k X j =1 deg Ψ i,j ! ≤ n X i =1 k X j =1 j n − i +2 + C log( n + 1) nk n ( n +1) / ≤ C ( n, d, h ) k n ( n +5) / . We apply Lemma 3.7 with n + 1 polynomials and n + 1 variables,log B ≪ n,d,h ( k n ( n +1) / ) n +1)+1 k n ( n +5) / + ( k n ( n +1) / ) n +1)+2 ≪ d,h,n k n (3 n +8 n +9) / . Again by Bezout’s theorem, T k ≤ k n ( n +1) / n Y i =1 k n − i ≪ n,d,h k n . This completes the proof of theorem. (cid:3)
EDUCTION OF POLYNOMIAL DYNAMICAL SYSTEMS MODULO PRIMES 13
Proof of Theorem 2.3.
Let p ∈ Z be a prime and let L = ( L , . . . , L s ) ∈ Z [ X ] be a system of polynomials of degree at most D and height at most H .We denote by V the subvariety of the affine space A n Q defined by this system ofpolynomials. We also denote the reduction modulo p of the iteration F ( m ) and Vby F ( m ) p and V p , respectively. Here we fix an initial point w ∈ F np and let A = (cid:8) m ∈ { , , . . . , ℓ − } | F ( m ) p ( w ) ∈ V p ( F p ) (cid:9) . Suppose that(16)
A > ǫℓ ≥ . Take γ ≤ ℓ/ ( A −
1) and let B be number of m ∈ { , , . . . , ℓ − } with(17) F ( m ) p ( w ) ∈ V p and F ( m + γ ) p ( w ) = F ( γ ) p (cid:0) F ( m ) p ( w ) (cid:1) ∈ V p . By Lemma 3.8,(18) B ≥ ( A − ℓ ≫ ǫ ℓ and hence we have γ ≪ /ǫ. Since the iterations generically escape V , the set { u ∈ V | F ( γ ) ( u ) ∈ V } isfinite and this set is defined by the following 2 s equations(19) L j ( X ) = L j (cid:0) F ( γ ) (cid:1) ( X ) = 0 , j = 1 , . . . , s. By Lemma 3.3 and 3.4, we havedeg L j (cid:0) F ( γ ) (cid:1) ≤ Dnγ n − and from B´ezout’s theorem { u ∈ V | F ( γ ) ( u ) ∈ V } ≤ D s ( Dnγ n − ) s ≪ nD s ǫ ( n − s . (20)From Lemma 3.6, we have h (cid:16) F ( γ ) i (cid:17) ≪ d,h,n γ n − i +2 and hence by Lemma 3.3, h ( L j (cid:0) F ( γ ) (cid:1) ) ≤ H + Dγ n +1 + (3 Dnγ n − + 1) D log( n + 1) ≪ d,h,H,n Dnγ n +1 . Here the degree and height of 2 s polynomials in (19) are bounded by Dnγ n − and Dnγ n +1 respectively. By Lemma 3.7, there is a positive integer D withlog D ≤ (11 n + 4)( Dnγ n − ) n +1 ( Dnγ n +1 )+ (55 n + 99) log((2 n + 5) s )( Dnγ n − ) n +2 ≤ C ǫ n (3 n − such that if p ∤ D , then { u ∈ V | F ( γ ) ( u ) ∈ V } = { u ∈ V p | F ( γ ) p ( u ) ∈ V p } . Since S ( w ) ≥ ℓ , the points F ( m ) p ( w ) , m = 0 , . . . , ℓ − B ≤ { u ∈ V p | F ( γ ) p ( u ) ∈ V p } . From (18) and (20), we have ǫ ℓ ≤ n s D s ǫ ( n − s . This is a contradiction as ℓ > n s D s ǫ ( n − s +2 . Thus, A ≤ ǫℓ and this completes theproof of theorem (cid:3) .4.4. Proof of Theorem 2.4.
Set β = (cid:22) Lǫ (cid:23) + 1;thus ℓ ≥ β. For each set B ⊆ { , . . . , β − } of cardinality B = L + 1, weconsider the system of equations(21) L j (cid:0) F ( k ) (cid:1) = 0 , k ∈ B, j = 1 , . . . , s.
Since k ∈ B , we have k ≤ β −
1. By Lemma 3.3 and 3.4,deg L j (cid:0) F ( k ) (cid:1) ≤ k n − Dn ≤ ( β − n − Dn.
Again, by Lemma 3.3 and Lemma 3.6, we have h (cid:0) L j (cid:0) F ( k ) (cid:1)(cid:1) ≤ H + Dk n +1 + (3 nk n − + 1) D log( n + 1) ≪ H,n D ( β − n +1 . Since the intersection of orbits of F with V is L -uniformly bounded and k ∈ B ,the system of equations in (21) has no common solution w ∈ Q n . By Lemma3.7, there exists C B ∈ N withlog C B ≤ (11 n + 4)( Dn ( β − n − ) n +1 D ( β − n +1 + (55 n + 99) log((2 n + 5) s )( Dn ( β − n − ) n +2 ≤ C ( Dn ( β − n − ) n +2 ( β − n +1 . such that if p is a prime and p ∤ C B , then the reduction modulo p of the systemof equations (21) has no solutions in F np .Now set C = Y B ⊆{ ,...,β − } B = L +1 C B and hence log C ≪ D,d,h,H,n,L (cid:18) βL + 1 (cid:19) ( Dn ( β − n − ) n +2 ( β − n +1 ≤ C ( D, d, h, H, n, L, s ) ǫ ( n − n +2)+( n + L +2) . (22) EDUCTION OF POLYNOMIAL DYNAMICAL SYSTEMS MODULO PRIMES 15
Let p be a prime with p ∤ C . Suppose that for some u ∈ F np there are at least ǫℓ values of n ∈ { , . . . , ℓ − } with F ( n ) p ( u ) ∈ V p . Since ℓ ≥ β , there is a nonnegative integer i ≤ ⌊ ℓ/β ⌋ such that there are at least ǫℓ ⌊ ℓ/β ⌋ + 1 ≥ ǫβ > L values of n ∈ { iβ, . . . , ( i + 1) β − } with F ( n ) p ( u ) ∈ V p . Now consider L + 1 values iβ < iβ + δ < · · · < iβ + δ L +1 < ( i + 1) β. Then for j = 1 , . . . , s and t = 1 , . . . , L + 1, L j (cid:16) F p ( δ t ) (cid:16) F p ( iβ ) (cid:17)(cid:17) = 0 . Setting w = F p ( iβ ) ∈ F np , then for all j, tL j (cid:16) F p ( δ t ) ( w ) (cid:17) = 0 . This implies that p ∤ C B with B = { δ , . . . , δ L +1 } which is a contradiction. Thiscompletes the proof of theorem (cid:3) . References [1] A. Akbary and D. Ghioca,
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