The orbit intersection problem in positive characteristic
aa r X i v : . [ m a t h . N T ] F e b THE ORBIT INTERSECTION PROBLEM IN POSITIVECHARACTERISTIC
S. S. ROUT
Abstract.
In this paper, we study the orbit intersection prob-lem for the linear space and the algebraic group in positive char-acteristic. Let K be an algebraically closed field of positive char-acteristic and let Φ , Φ : K d −→ K d be affine maps, Φ i ( x ) = A i ( x ) + x i (where each A i is a d × d matrix and x ∈ K d ). Ifeach a i ∈ K d is not Φ i -preperiodic, then we prove that the set (cid:8) ( n , n ) ∈ Z | Φ n ( a ) = Φ n ( a ) (cid:9) is p -normal in Z of order atmost d . Further, for the regular self-maps Φ , Φ : G dm −→ G dm ,we show that the set { ( n , n ) ∈ N | Φ n ( a ) = Φ n ( a ) } (where a , a ∈ G dm ( K )) lie in a finite number of linear and exponentialone-parameter families. To do so, we use results on linear equa-tions over multiplicative groups in positive characteristic and someresults on systems of polynomial-exponential equations. Introduction
Let N denote the set of positive integers and N := N ∪ { } . For a set X endowed with self map Φ and for any m ∈ N , we denote by Φ m the m -th iteration Φ ◦ · · · ◦ Φ of Φ with Φ denoting the identity map on X .If x ∈ X , we define the forward orbit Orb +Φ ( x ) := { Φ m ( x ) | m ∈ N } .Similarly, if the self map Φ is invertible, then the backward orbit isdefined as the collection of inverse images Orb − Φ ( x ) := ∪ n ≥ Φ − n ( x ). Inthis paper, we are dealing with some problems of dynamical Mordell-Lang conjecture in positive characteristics.1.1. Overview in characteristic . Let C be a curve of genus g ≥ K of characteristic 0. The classical Mordell con-jecture states that the curve C has finitely many K - rational points.Further, Lang conjectured that if V is a subvariety of a semiabelian va-riety G defined over C and Γ is a finitely generated subgroup of G ( C ),then V ( C ) ∩ Γ is a union of at most finitely many translates of subgroups
Mathematics Subject Classification.
Primary 37P55, Secondary 11B37,11D61.
Key words and phrases.
Dynamical Mordell-Lang Conjecture, algebraic variety,linear recurrence sequences, orbit intersections. of Γ. This conjecture was proved by Faltings [9] for abelian varietiesand Vojta [23] for semiabelian varieties. Motivated by the classicalMordell-Lang conjecture, various authors [5, 1, 12] have proposed thefollowing dynamical Mordell-Lang conjecture.
Conjecture 1.1.
Let X be a quasiprojective variety defined over a field K of characteristic , let Y be any subvariety of X , let α ∈ X ( K ) andlet Φ be an endomorphism on X . Then the set (1) { m ∈ N | Φ m ( α ) ∈ Y ( K ) } is a finite union of arithmetic progressions. An arithmetic progression is a set of the form { mk + ℓ : k ∈ N } forsome m, ℓ ∈ N and when m = 0, this set is a singleton. Conjecture 1.1implies the cyclic case in the classical Mordell-Lang conjecture when X is a semiabelian variety and Φ the translation by a point x ∈ X ( K ).It is natural to ask for a generalization of dynamical Mordell-Langconjecture to a statement which would contain as a special case the fullstatement of classical Mordell-Lang conjecture. The general dynamicalMordell-Lang conjecture is stated as follows: Suppose Φ , . . . , Φ r arecommuting K -morphisms from X to itself and x ∈ X ( K ). Then theset { ( n , . . . , n r ) ∈ N r | Φ n ◦ · · · ◦ Φ n r r ( x ) ∈ V ( K ) } is a finite union of translates of subsemigroups of N r .There is an extensive literature proving various special cases of Con-jecture 1.1 (see [3, 12, 24]) and for a survey of recent works (see [2]).Ghioca et. al., [14] have obtained various results for the general dynam-ical Mordell-Lang problem when X is a semiabelian variety and the selfmaps are endomorphisms satisfying certain technical conditions. Fur-ther, not much is known even when we restrict to the following specialcase called the orbit intersection problem : Question 1.1.
Let X be a variety over algebraically closed field K of characteristic , let r ≥ be a positive integer. For ≤ i ≤ r ,let Φ i be a K -morphism from X to itself, and let α i ∈ X ( K ) is not Φ i -preperiodic. When can we conclude that the set S := { ( n , . . . , n r ) ∈ N r | Φ n ( α ) = · · · = Φ n r r ( α r ) } is a finite union of sets of the form { ( n k + ℓ , . . . , n r k + ℓ r ) : k ∈ N } for some n , . . . , n r , ℓ , . . . , ℓ r ∈ N ? Question 1.1 is known for the case when X = P K and each Φ i is apolynomial of degree larger than 1 (see [13, 15]). Later, Question 1.1is answered when X is a semiabelian variety and when X = A nK and HE ORBIT INTERSECTION PROBLEM IN POSITIVE CHARACTERISTIC 3 the self maps are affine transformations [11]. Further in [20], variousupper bounds are derived for the orbit intersection problem when X is an affine n -space and self maps are polynomial morphisms of specialtypes.1.2. Overview in positive characteristic.
The picture in positivecharacteristic is very much different. From the following example, onecan observe that the Mordell-Lang conjecture is not true in positivecharacteristic. From now onwards, we let p be a prime number and q = p e for some positive integer e .Let K = F p ( t ) be the field of rational functions over the field of size p . Let G := G m be a semiabelian variety defined over K and V be thesubvariety defined by the equation x + y = 1. Let Γ be the subgroupof G ( K ) generated by ( t, − t ). Then V ( K ) ∩ Γ = { ( t p e , (1 − t ) p e ) | e ∈ N } and this set cannot be expressed as a finite union of translates ofsubgroups of Γ.Using model theoretic ideals, Hrushovski [16] proved the classicalMordell-Lang conjecture in positive characteristic. Moosa-Scanlon [18]proved a form of the classical Mordell-Lang conjecture for semiabelianvarieties defined over finite fields. Motivated by the work of Moosa-Scanlon [18], the following conjecture was proposed (see [4, 10]). Conjecture 1.2 (Dynamical Mordell-Lang Conjecture in postive char-acteristic) . Let X be a quasiprojective variety defined over a field K of characteristic p , let Y be any subvariety of X defined over K , let α ∈ X ( K ) and let Φ : X −→ X be an endomorphism defined over K .Then the set { m ∈ N | Φ ( m ) ( α ) ∈ Y ( K ) } is a union of finitely manyarithmetic progressions along with finitely many sets of the form (2) ( m X j =1 c j p k j n j | n j ∈ N , j = 0 , , . . . , m ) , for some m ∈ N , for some c j ∈ Q , and some k j ∈ N . As of now, Conjecture 1.2 is known in the following cases. If X is asemiabelian variety defined over finite field and Φ is an algebraic groupendomorphism whose action on the tangent space at the identity isgiven through a diagonalization matrix (see [2, Proposition 13.3.0.2]).Further, if X = G Nm and Y ⊂ X is a curve (see [10, Theorem 1.3]) andif X = G Nm , Y ⊂ X be a subvariety of dimension at most equal to 2and Φ is any regular self map (see [4, Theorem 1.2]). Also, Conjecture1.2 holds for any subvariety Y ⊂ G Nm assuming Φ is an algebraic groupendomorphism with the property that no iterate of it restricts to beinga power of the Frobenius on the proper algebraic subgroup G Nm (see S. S. ROUT [4, Theorem 1.3]). Conjecture 1.2 seems to be very difficult. To solvein the case of arbitrary subvarieties of X = G Nm , it leads to difficultquestions involving polynomial-exponential equations.In this paper, we study Question 1.1 in positive characteristic. Fromthe following example (see [11]), one can see that Question 1.1 failsfor affine maps defined over F p ( t ). Let Φ i : A −→ A be affine mapsdefined by Φ ( x ) = t ( x −
1) + 1 and Φ ( x ) = ( t + 1) x . Now it is easyto see Φ n (2) = t n + 1 and Φ n (1) = ( t + 1) n . Then the set (cid:8) ( n , n ) ∈ N | Φ n (2) = Φ n (1) (cid:9) = { ( p n , p n ) | n ∈ N } which is not a finite union of arithmetic progressions. This exam-ple motivate us to consider the orbit intersection problem in positivecharacteristic. In order to state our theorems, we need the followingdefinitions (see [8]):Suppose that k ≥ , c , c , . . . , c k ∈ Q m with ( q − c i ∈ Z m for all i , c + · · · + c k ∈ Z m . Then we define S q ( c ; c , . . . , c k ) = (cid:8) c + c q f + · · · + c k q f k | f , . . . , f k ∈ N (cid:9) . The conditions on c , c , . . . , c k imply that S q ( c , c , . . . , c k ) ⊂ Z m .We call it an elementary p -nested set in Z m of oder at most k . Wedefine a p -normal set in Z m of order at most k as a finite union ofsingletons and sums R + S , where R is a subgroup of Z m and S is eithera singleton or an elementary p -nested set in Z m of order at most k . For m = 1, this definition was first introduced by Derksen [6] to study thezero set of linear recurrence sequences in positive characteristic. Nowwe are ready to state our first result. Theorem 1.1.
Let K be an algebraically closed field of characteristic p and let d ∈ N . Let Φ , Φ : K d −→ K d be affine maps (that is thereexist a d × d matrix A i ∈ GL d ( K ) and x i ∈ K d such that Φ i ( x ) = A i ( x ) + x i for i = 1 , . Let a i ∈ K d be not Φ i -preperiodic for i = 1 , .Then the set (3) (cid:8) ( n , n ) ∈ Z | Φ n ( a ) = Φ n ( a ) (cid:9) is p -normal in Z of order at most d . An one-parameter linear family in Z is of the form ( x ( t ) , y ( t )) =( at + c, bt + d ) , t ∈ Z for some integers a, b, c and d . An one-parameterexponential family in Z is of the form ( x ( s ) , y ( s )) = ( acξ s + γ, bcξ s + ηs + µ ) , s ∈ N where ξ ∈ Z > and a, b, η ∈ Z \ { } and c = 0 , µ, γ are in Q with x ( s ) , y ( s ) ∈ Z for each s ∈ N . We get the one-parameter linearand exponential family in N by intersecting N with one-parameterlinear and exponential family in Z and removing some singletons. HE ORBIT INTERSECTION PROBLEM IN POSITIVE CHARACTERISTIC 5
Theorem 1.2.
Let K be an algebraically closed field of characteristic p and let d ∈ N . Let Φ , Φ : G dm −→ G dm be regular self-maps and let a , a ∈ G dm ( K ) . Then the set (4) (cid:8) ( n , n ) ∈ N | Φ n ( a ) = Φ n ( a ) (cid:9) is a finite union of one-parameter linear and exponential families. Proof of Theorem 1.1
Auxiliary results.
Let K be a field of positive characteristic p and let G be a finitely generated subgroup of the multiplicative group K ∗ . For any subgroup G of K ∗ and a positive integer n it makes senseto write P n ( G ) for the set of points in projective space defined over G .For n ≥
2, let V be a linear variety in P n defined over K . We write V ( G ) = V ∩ P n ( G ) for the set of points defined over G .We will need the radical G = √ G . For us this remains in K ; thus itis the group of γ in K for which there exists a positive integer s suchthat γ s lies in G . We denote by ϕ = ϕ q the Frobenius with ϕ ( x ) = x q ,where q = p e for some positive integer e . For points π , π , · · · , π h , wedefine the set(5)( π , π , · · · , π h ) := ( π , π , · · · , π h ) q = π ∞ [ e =0 · · · ∞ [ e h =0 ( ϕ e π ) · · · ( ϕ e h π h )with the interpretation π itself if h = 0. Proposition 2.1 (Derksen-Masser [7]) . Let K be a field of positivecharacteristic p , let V be an arbitrary linear variety defined over K ,and suppose that √ G in K is finitely generated. Then there is apower q of p such that V ( G ) is an effectively computable union of sets ( π , π , · · · , π h ) q H ( G ) with points π , π , · · · , π h (0 ≤ h ≤ n − definedover √ G and subgroups H . We also need nested sets in abelian groups. Let A be a finitelygenerated abelian group. For k ≥
1, let C , C , . . . , C k in A , we define U q ( C ; C , . . . , C k ) = { C + C q f + · · · C k q f k | f , . . . , f k ∈ N } in A . We call it an elementary integral p -nested set of order at most t . Proposition 2.2 (Derksen-Masser [8]) . Let B , . . . , B d in A , let B bea subgroup of A and denote by R the subgroup of all ( k , . . . , k d ) ∈ Z d such that k B + · · · + k d B d lies in B . Let U be an elementary integral p -nested set of order at most k in A . Then the set of all ( k , . . . , k d ) in Z d such that k B + · · · + k d B d lies in B + U is either empty or R + S , S. S. ROUT where S is a finite union of singletons and elementary p -nested sets oforder at most k in Z d . By an application of Laurent’s theorem [17], we obtain the followingresult.
Proposition 2.3.
The intersection of two elementary p -nested sets isa finite union of elementary p -nested sets. Proof of Theorem 1.1.
Let ˜ a be a fixed point of Φ , that is, A ˜ a + a = ˜ a . This is permissible since A − I d is invertible. Define ψ ( a ) = a + ˜ a , so that ψ − ◦ Φ ◦ ψ ( a ) = A ( a ). From this one candeduce, Φ n ( a + ˜ a ) = A n ( a ) + ˜ a . Similarly, let ˜ a be a fixed pointof Φ , then we have Φ n ( a + ˜ a ) = A n ( a ) + ˜ a . Hence, we reduce theproblem of studying the set of pairs ( n , n ) ∈ Z satisfying(6) A n b = A n b + b where b := ˜ a − ˜ a and b , b are given vectors such that b (re-spectively b ) is not preperiodic under the map x A x (respectively x A x ).Suppose that J and J are Jordan normal form of matrix A and A respectively. Hence, we can write A = C − J C and A = D − J D ,where C, D ∈ GL d ( K ). With these expressions of A and A , equation(6) becomes(7) J n C b = CD − J n D b + C b . Let λ ∈ K, ℓ ∈ N and n ∈ N . Let J λ,ℓ be the Jordan matrix of size ℓ and eigenvalue λ and we have the formula(8) J nλ,ℓ = λ n (cid:0) n (cid:1) λ n − (cid:0) n (cid:1) λ n − · · · (cid:0) nℓ − (cid:1) λ n − ℓ +1 λ n (cid:0) n (cid:1) λ n − · · · (cid:0) nℓ − (cid:1) λ n − ℓ +2 ... ... ... ... ...... ... ... ... ...0 0 0 · · · λ n . Since (cid:0) nk (cid:1) is a polynomial in n of degree k , there is a power Q of p such that all (cid:0) nk (cid:1) in (8) depend only on the values of n modulo Q . Thusfor all n in each fixed residue class n + Q Z in Z , we may write (8) as(9) J nλ,ℓ = λ n γ , λ n γ , λ n · · · γ ,ℓ − λ n λ n γ , λ n · · · γ ,ℓ − λ n ... ... ... ... ...... ... ... ... ...0 0 0 · · · λ n . HE ORBIT INTERSECTION PROBLEM IN POSITIVE CHARACTERISTIC 7 If b = ( b , . . . , b ℓ ) T , is a fixed column vector in K n ( T denote thetranspose) and n ≥ ℓ , we have(10) J nλ,ℓ b = ( γ λ n , . . . , γ ℓ λ n ) T . Let r be number of Jordan blocks in J , let J λ i ,g i for 1 ≤ i ≤ r, λ i ∈ K, g i ∈ N and P i g i = d be the Jordan blocks of J . Let s be numberof Jordan blocks in J , let J δ i ,h i for 1 ≤ i ≤ s, δ i ∈ K, h i ∈ N and P i h i = d be the Jordan blocks of J .Assume that n (resp. n ) is in fixed residue class n ′ + Q Z (resp. n ′ + Q Z ) in Z . This is possible as for any ( n ′ , n ′ ) ∈ Z and p -normalset R + S , the set ( n ′ , n ′ )+ Q ( R + S ) = ˜ R + ˜ S for the subgroup ˜ R = QR and the elementary p -nested set ˜ S = ( n ′ , n ′ ) + QS . Thus, by (10),(11) J n C b = ( b (1)1 , λ n , . . . b (1)1 ,g λ n , b (1)2 , λ n , . . . b (1)2 ,g λ n , · · · , b (1) r, λ n r , . . . b (1) r,g r λ n r ) T , and(12) J n D b = ( b (2)1 , δ n , . . . b (2)1 ,h δ n , b (2)2 , δ n , . . . b (2)2 ,h δ n , · · · , b (2) s, δ n s , . . . b (2) s,h s δ n s ) T . Write C b = ( b (3)1 , . . . , b (3) d ) and let k -th row of the matrix CD − is( c k , , . . . , c k ,h , c k , , . . . , c k ,h , . . . , c ks, , . . . , c ks,h s ) T . For fix i ∈ { , . . . , r } and ℓ ∈ { , . . . , g i } , our claim is that the setof ( n , n ) ∈ Z satisfying (7) is R + S , where R is the subgroup of Z and S is the finite union of singletons and elementary p -nested sets oforder at most d in Z . If we assume this claim is true, then applicationof Proposition 2.3 finishes the proof of of the theorem.Now fix an i ∈ { , . . . , r } and ℓ ∈ { , . . . , g i } . From (7), (11) and(12), we have(13) b (1) i,ℓ λ n i = s X j =1 h j X k =1 c g ′ i − + ℓj,k b (2) j,k δ n j + b (3) g ′ i − + ℓ , with g ′ i = P ij =1 g j . We rewrite (13) as(14) b (1) i,ℓ λ n i = s X j =1 d i,j,ℓ δ n j + b (3) g ′ i − + ℓ where d i,j,ℓ := P h j k =1 c g ′ i − + ℓj,k b (2) j,k . S. S. ROUT
We want to apply Proposition 2.1 to the linear variety V defined bythe corresponding equation(15) s X j =1 d i,j,ℓ X j + ( − b (1) i,ℓ ) X s +1 + b (3) g ′ i − + ℓ X s +2 = 0 . Here we are in projective space P s +1 and also we work inside the fieldgenerated by d i,j,ℓ , b (1) i,ℓ , b (3) g ′ i − + ℓ , λ i , δ j over F p and G as the radical of thegroup generated by λ i , δ j . This G is also finitely generated. Equation(14) gives a point η on V ( G ). Let A be a finitely generated additiveabelian group. We may identify the group G s +1 with P s +1 ( G ) anddefine an isomorphism log from these to A . Then we have(16) log η = n B + n B where B is the log of the point with X = δ , . . . , X s = δ s , X s +1 = X s +2 = 1 and B is the log of the point with X = · · · = X s = X s +2 =1 , X s +1 = λ i .By Proposition 2.1, V ( G ) is a finite union of sets Z := ( π , π , . . . , π h ) q H ( G )with points π , π , . . . , π h defined over G ,( π , π , · · · , π h ) q = π ∞ [ e =0 · · · ∞ [ e h =0 ( ϕ e π ) · · · ( ϕ e h π h )and linear subgroups H defined by the equation X i = X j . Now a point π Z ∈ Z has(17) log π Z = C + C q f + · · · + C h q f h + B with C k = log π k for k = 0 , , . . . , h and B = log σ in H ( G ). Let B = log H ( G ). Thus, the set of all such π m such that log π m forms asum B + U , where U is an elementary integral p -nested set of order atmost h ≤ s ≤ d in A . Now from (16) and (17) and Proposition 2.2,one can observe that for each Z , the set ( n , n ) ∈ Z is R Z + S Z , with R Z is group of all ( n , n ) in Z with n B + n B in B and S Z is afinite union of singletons and elementary p -nested sets in Z of orderat most d . This completes the proof of the theorem. (cid:3) Proof of Theorem 1.2
Linear recurrence sequences.
We define linear recurrence se-quences which are essential in our proof.
HE ORBIT INTERSECTION PROBLEM IN POSITIVE CHARACTERISTIC 9
A linear recurrence sequence of order k is a sequence ( U n ) n ≥ satis-fying a relation(18) U n = b U n − + · · · + b k U n − k where b , . . . , b k ∈ C with b k = 0 and U , . . . , U k − are integers not allzero. The characteristic polynomial of U n is defined by(19) f ( x ) := x k − b x k − − · · · − b k = t Y i =1 ( x − α i ) m i ∈ C [ X ]where α , . . . , α t are distinct and m , . . . , m t are positive integers. Thenas it is well-known (see e.g. Theorem C1 in part C of [22]) we have arepresentation of the form(20) U n = t X i =1 P i ( n ) α ni for all n ≥ . Here P i ( x ) is a polynomial of degree m i − i = 1 , . . . , t ). We callthe sequence { U n } simple if t = k . The sequence ( U n ) n ≥ is called degenerate if there are integers i, j with 1 ≤ i < j ≤ t such that α i /α j is a root of unity; otherwise it is called non-degenerate . For more detailson linear recurrence sequences we refer the reader to ([21, 22]).To unify the notation in the sequel, we will consider instead of (20)the function(21) P ( z ) = t X i =0 P i ( z ) α zi of polynomial-exponential type, where α is a root of unity, α i /α j for i = j is not a root of unity and the P i are polynomials of degree m i − ≤ i ≤ t and P i = 0 for 1 ≤ i ≤ t .Similarly consider V m = Q ( m ) with(22) Q ( w ) = t ′ X i =0 Q i ( w ) β wi where β is a root of unity, β i /β j for i = j is not a root of unity andthe Q i are polynomials of degree m i − ≤ i ≤ t ′ and Q i = 0 for1 ≤ i ≤ t ′ . Now P and Q are said to related if t = t ′ and after suitablereordering of β , . . . , β t we have(23) α ai = β bi , for every i ∈ { , . . . , t } and for some non-zero integers a and b . Suppose that P and Q are related with (23) and that t is even. Now P and Q are said to doubly related if after reordering, we have both(23) and(24) α a ′ i = β b ′ i +1 , α a ′ i +1 = β b ′ i for i ∈ { , . . . , t } , i oddand for some non-zero integers a ′ and b ′ . A pair P and Q that isrelated but not doubly related is called simply related . Now considerthe equation(25) P ( x ) = Q ( y )in intergers x and y .Schlickewei and Schmidt [19] proved that when P and Q are simplyrelated, then all but finitely many solutions of (25) satisfy the systemof equations(26) P i ( x ) α xi = Q i ( y ) β yi , for i ∈ { , . . . , t } . We also need the following definition from [19]. We call the orderedpair F, G exceptional if(a) P and Q are simply related,(b) there is a natural number N > α i and each β i with 1 ≤ i ≤ t .(c) either | α i | > ≤ i ≤ t , or | α i | < ≤ i ≤ t ,(d) P and Q are constant,(e) each Q i is constant and for 1 ≤ i ≤ t, P i ( x ) = a i ( x − A ) l i where A is rational and l i > a and b are non-zero integers satis-fying (23). Proposition 3.1 (Schlickewei-Schmidt [19]) . Suppose that P and Q are related, but neither P, Q nor
Q, P is exceptional. Then (26) ei-ther has only finitely many solutions or it has finitely many solutionstogether with a one-parameter linear family of solutions (27) x ( t ) = at + a ′ , y ( t ) = bt + b ′ , t ∈ Z , with certain a ′ , b ′ ∈ Z and a and b are non-zero integers satisfying (23) . Proposition 3.2 (Schlickewei-Schmidt [19]) . Suppose that P and Q are exceptional. Then in addition to finitely many solutions, the so-lution to (26) (and therefore to (25) ) will comprise a finite number(possibly zero) of exponential families D , . . . , D g of the type (28) x ( s ) = ac j ξ s + γ, y ( s ) = bc j ξ s + ηs + b j , ( s ∈ N ) . HE ORBIT INTERSECTION PROBLEM IN POSITIVE CHARACTERISTIC 11
Here ξ ∈ Z , ξ > and ξ is a rational power of each α i and each β i with ≤ i ≤ t and c j = 0 , b j ∈ Q with x ( s ) , y ( s ) ∈ Z for each s ∈ N (1 ≤ j ≤ g ) . Proposition 3.3. (a)
The intersection of two one parameter linearfamilies is again a family of linear type. (b)
The intersection of two one parameter exponential families isagain a family of exponential type. (c)
The intersection of an one parameter linear family with an oneparameter exponential family is a finite union of exponential fam-ilies.Proof.
Let ( a t + c , b t + d ) and ( a t + c , b t + d ) with t ∈ Z be twolinear families. Here ( a i , b i ) for i = 1 , α a i i = β b i i as in (23). Let ( a, b ) be the least common multiple of ( a i , b i ) with α ai = β bi . If the intersection of the two linear families in non-empty andif ( n, m ) lies in this intersection, then the intersection consists of thepairs ( at + n, bt + m ) , t ∈ Z . This completes the proof of part (a).For part (b), let( x ( s ) , y ( s )) = ( ac ξ s + γ , bc ξ s + η s + b )and ( x ( s ) , y ( s )) = ( ac ξ s + γ , bc ξ s + η s + b )be two one parameter exponential families. Since we are interestedto study the intersection of these two families, hence we consider thefollowing equations(29) ac ξ s + γ = ac ξ s + γ , and bc ξ s + η s + b = bc ξ s + η s + b . Now (29) imply(30) γ = γ ; η s + b = η s + b ; and c ξ s = c ξ s with finitely many exceptions. If there are infinitely many solutions s , s of (30), then we have a family s = w s + u , s = w s + u with s ∈ N and then x ( s ) = x ( s ) = x ( w s + u ) = ac ξ u ξ w s + γ ,y ( s ) = y ( s ) = bc ξ u ξ w s + η w s + ηu + b is again a family of exponential type. This completes the proof of part(b). Let x ( t ) = at + a ′ , y ( t ) = bt + b ′ with t ∈ Z be a linear family and x ( s ) = ac j ξ s + γ, y ( s ) = bc j ξ s + ηs + b j with s ∈ N be an exponentialfamily. From the linear family we have bx − ay is a constant (= ba ′ − ab ′ )and whereas in exponential family, | bx − ay | tends to infinity. Thus,their intersection is finite. This completes the proof of Proposition3.3. (cid:3) Proof of Theorem 1.2.
Since Φ is a regular self map on G dm ,then there exists a group endomorphism Φ : G dm −→ G dm and thereexists y ∈ G dm ( K ) such that for any x ∈ G dm ( K ), we have Φ ( x ) =Φ ( x ) + y . Further, Φ acts on G dm as follows:Φ ( x , . . . , x d ) = d Y k =1 x a ,k k , . . . , d Y k =1 x a d,k k ! for some integer a i,j , and therefore Φ is integral over Z , i.e., there existintegers c , . . . , c d − such that(31) (Φ ) d ( x ) + c d − (Φ ) d − ( x ) + · · · + c Φ ( x ) + c x = 0 , for each x ∈ G dm ( K ). Then as shown in [10, Claim 4.2], there ex-ist linear recurrence sequences { U i,n } n ∈ N ⊂ Z for 1 ≤ i ≤ d and { V i,n } n ∈ N ⊂ Z for 0 ≤ i ≤ d − n ∈ N , we have(32) Φ n ( a ) = d X i =1 U i,n i − X j =0 (Φ ) j ( y ) ! + d − X i =0 V i,n (Φ ) i ( a ) . Similarly, for the regular self map Φ on G dm and n ∈ N , we have(33) Φ n ( b ) = d X i =1 U ′ i,n i − X j =0 (Φ ) j ( z ) ! + d − X i =0 V ′ i,n (Φ ) i ( b ) , where Φ is a group endomorphism on G dm , { U ′ i,n } n ∈ N ⊂ Z for 1 ≤ i ≤ d and { V ′ i,n } n ∈ N ⊂ Z for 0 ≤ i ≤ d − z ∈ G dm ( K ). For each i = 1 , . . . , d , we use the following notation(34) P i := i − X j =0 (Φ ) j ( y ) and P ′ i := i − X j =0 (Φ ) j ( z ) . Now using (32), (33) and (34), we rewrite the equation Φ n ( a ) = Φ n ( b )as(35) d X i =1 U i,n P i + d − X i =0 V i,n (Φ ) i ( a ) = d X i =1 U ′ i,n P ′ i + d − X i =0 V ′ i,n (Φ ) i ( b ) . Let Γ be a finitely generated abelian group containing(Φ ) j ( y ) , (Φ ) j ( z ) , (Φ ) j ( a ) and (Φ ) j ( b )for each j = 0 , . . . , d −
1. Since Γ is finitely generated abelian group,we know that Γ is isomorphic to a direct sum of a finite subgroup Γ HE ORBIT INTERSECTION PROBLEM IN POSITIVE CHARACTERISTIC 13 with a subgroup Γ which is isomorphic to Z r for some r ∈ N . Let { Q , . . . , Q r } be a Z -basis for Γ . Then we write each P i = P i, + r X k =1 e i,k Q k , P i, ∈ Γ , and each e i,k ∈ Z for i = 1 , . . . , d and also write each(Φ ) i ( a ) = T i, + r X k =1 f i,k Q k , T i, ∈ Γ and each f i,k ∈ Z . Similarly, we write each P ′ i = P ′ i, + P rk =1 e ′ i,k Q k for i = 1 , . . . , d ,where P ′ i, ∈ Γ and each e ′ i,k ∈ Z and also write each (Φ ) i ( b ) for i = 1 , . . . , d as T ′ i, + P rk =1 f ′ i,k Q k with T ′ i, ∈ Γ and each f ′ i,k ∈ Z .Comparing the torsion and free part of (35), we have(36) d X i =1 U i,n P i + d − X i =0 V i,n T i, = d X i =1 U ′ i,n P ′ i + d − X i =0 V ′ i,n T ′ i, and d X i =1 U i,n r X k =1 e i,k Q k + d − X i =0 V i,n r X k =1 f i,k Q k = d X i =1 U ′ i,n r X k =1 e ′ i,k Q k + d − X i =0 V ′ i,n r X k =1 f ′ i,k Q k . (37)Since each of the points P i, , T i, , P ′ i, and T ′ i, belongs to a finitegroup Γ and so they all have finite order bounded by | Γ | . Thus { U i,n P i, } n is preperiodic for each i . (Here we recall that a sequence { W n } n is preperiodic if there exists a positive integer ℓ such that thesubsequence { W n } n ≥ ℓ is periodic). Therefore, the set of ( n , n ) ∈ N which satisfy (36) is a finite union of sets of the form { u m + v , u m + v : m , m ∈ N } for some u , u , v , v ∈ N .Using Proposition 3 .
3, it suffices to prove that the set of all ( n , n )satisfying (37) is a finite union of exponential families along with finiteunion of linear families. Now comparing coefficient of each Q k in bothsides of (37) for k = 1 , . . . , r , we have(38) d X i =1 e i,k U i,n + d − X i =0 f i,k V i,n = d X i =1 e ′ i,k U ′ i,n + d − X i =0 f ′ i,k V ′ i,n . Since linear combination of linear recurrence sequence is a linear recur-rence sequence, both sides of (38) are again linear recurrence sequences.Now (37) is equivalent to the simultaneous solutions of the equations(39) W k,n = W ′ k,n for each k = 1 , . . . , r, where W k,n := P di =1 e i,k U i,n + P d − i =0 f i,k V i,n and W ′ k,n := P di =1 e ′ i,k U ′ i,n + P d − i =0 f ′ i,k V ′ i,n . Further, the characteristic roots of the linear recurrencesequence W k,n (resp. W ′ k,n ) are positive integer powers of the roots ofminimal polynomial of the group endomorphism Φ (resp. Φ ). Nowwe will discuss about the set Z := { ( n , n ) ∈ N | W k,n = W ′ k,n for each k = 1 , . . . , r } . If W k,n and W ′ k,n are not related (in the sense of Section 2.1), thenby Laurent’s theorem Z is finite. Suppose that W k,n and W ′ k,n arerelated. If neither W k,n , W ′ k,n nor W ′ k,n , W k,n is exceptional, thenby Proposition 3.1, there is a unique linear one parameter family ofsolutions ( n , n ) = ( ut + u ′ , vt + v ′ ) ∩ N , t ∈ Z , with certain u, v ∈ Z \ { } and u ′ , v ′ ∈ Z .Finally, if W k,n and W ′ k,n are exceptional, then by Proposition 3.2,there is a finite number of one parameter exponential families of solu-tions of the type(40) ( n , n ) = ( u ER ℓ + A, v ER ℓ + F ℓ + G ) ∩ N , ℓ ∈ N , where R ∈ Z > , u , v , F ∈ Z \ { } and E = 0 , A, G ∈ Q . Thus, foreach fixed k ∈ { , . . . , r } , Z is a finite union of one parameter linearand exponential families. Now applying Proposition 3.3 finishes theproof of Theorem 1.2. References [1] J. P. Bell,
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