ON THE QUANTITATIVE DYNAMICAL MORDELL-LANG CONJECTURE 5 1 0 ALINA OSTAFE AND MIN SHA 2 c Abstract. ThedynamicalMordell-Langconjecture concernsthe e structure of the intersection of an orbit in an algebraic dynamical D system and an algebraic variety. In this paper, we bound the size 7 of this intersection for various cases when it is finite. 2 ] T N 1. Introduction . h 1.1. Motivation. Let be an algebraic variety defined over the com- at plex numbers C, and leXt Φ : be a morphism. For any integer m n 0, we denote by Φ(n) the nX-t→h itXeration of Φ with Φ(0) denoting the ≥ [ identity map. 2 Throughout the paper, a single integer is viewed as an arithmetic v progression with common difference 0. 3 The following is the well-known dynamical Mordell-Lang conjecture 4 5 for self-morphisms of algebraic varieties in the dynamical setting; see 2 [11, 16, 17]. 0 . 1 Conjecture 1.1 (Dynamical Mordell-Lang Conjecture). Let and 0 X Φ be given as the above, let V be a closed subvariety, and let 5 ⊆ X 1 P (C). Then, the following subset of integers : ∈ X v n 0 : Φ(n)(P) V(C) i { ≥ ∈ } X is a finite union of arithmetic progressions. r a Conjecture1.1hasbeenstudiedextensivelyinrecentyears. However, sofarthereareonlyafewrelatedresults. Theseincluderesultsonmaps of various special types [4, 5, 7, 14, 16, 17, 23, 24] (especially diagonal maps), and analogues for Noetherian spaces [6] and Drinfeld modules [15]. Recently, Silverman and Viray [23, Corollary 1.4] have given results regardingtheuniformboundedness(onlyintermsofm)ofintersections 2010 Mathematics Subject Classification. Primary 37P55; Secondary 11B37, 11D61, 11D72. Key words and phrases. Dynamical Mordell-Lang conjecture, linear recurrence sequence, exponential polynomial, linear equation. 1 2 ALINAOSTAFEAND MIN SHA of orbits of the power map (with the same exponent) at a point of the projective m-space Pm(C) with non-zero multiplicatively independent coordinates, with any linear subspace of Pm(C). However, they have not provided quantitative results. In fact, such a result follows, even in a more general case, directly from the uniform bound on the number of zeros of simple and non-degenerate linear recurrence sequences. We also note that the uniform boundedness condition has recently beenconsideredin[10], whereseveral resultsaregivenforthefrequency of the points in an orbit of an algebraic dynamical system that belong to a given algebraic variety under the reduction modulo a prime p. 1.2. Our Results. In this paper, we study the quantitative version of Conjecture 1.1 for polynomial morphisms of several special types when is the affine m-space Am(C) and V is a hypersurface; see X Section 3. Our main objective is to find as many classes of polynomial morphismsaspossible havinguniformbounds(orascloseaspossible to uniformity), andnot toinvestigate detailedly thequality ofthebounds. To the best of our knowledge, this is the first work on the quantitative dynamical Mordell-Lang conjecture. Here, we extend the results of Silverman and Viray [23] in two direc- tions. First, we consider monomial systems with different exponents. Second, we estimate the size of the intersection of an orbit with a hy- persurface rather than with a hyperplane. For example, we illustrate a typical case of our results; see Theorem 3.1 for more details. Let Φ = Xd,...,Xd with integer d 2 be a (cid:0) 1 m(cid:1) ≥ diagonal endomorphism of Am(C). Fix a hypersurface V, defined by a non-zero polynomial G = a Xi1 Xim C[X ,...,X ]. X i1,...,im 1 ··· m ∈ 1 m i1,...,im Then, for any w (C∗)m with multiplicatively independent coordi- ∈ nates, the size of the intersection of V and the orbit of Φ at the point w is at most (8n(G))4n(G)5, where n(G) is the number of monomials of G. Our methods rely on estimates (when finite) for integer solutions of certain polynomial-exponential equations. For the case of the power map studied by Silverman and Viray [23] we employ results on the number of zeros in linear recurrence sequences due to [1, 2] and [19]. For more general monomial systems we use results on the number of solutions in a finitely generated subgroup of (C∗)k of linear equations of the form a x +...+a x = 0, a ,...,a C∗, as well as solutions 1 1 k k 1 k ∈ QUANTITATIVE DYNAMICAL MORDELL-LANG CONJECTURE 3 to more general polynomial-exponential equations due to Schlickewei and Schmidt [20]. In fact, by [16, Theorem 1.8] the Dynamical Mordell-Lang Conjec- ture is known to hold in the cases we consider, because the morphisms can essentially restrict to endomorphisms of (C∗)m. Besides, the meth- ods we use might be not applicable on other kinds of morphisms, see Section 4 for more details. 1.3. Convention and notation. For integer m 2, let ≥ Φ = (F ,...,F ) : Am(C) Am(C), F ,...,F C[X ,...,X ], 1 m 1 m 1 m → ∈ be a morphism defined by a system of m polynomials in m variables over C. For each i = 1,...,m, we define the n-th iteration of the polynomials F by the recurrence relation i F(0) = X , F(n) = F F(n−1),...,F(n−1) , n = 1,2,..., i i i i(cid:16) 1 m (cid:17) so that Φ(n) = F(n),...,F(n) . (cid:16) 1 m (cid:17) See [3, 21, 22] for a background on dynamical systems associated with such iterations. For a vector w = (w ,...,w ) Cm, we denote by 1 m ∈ Orbw(Φ) = Φ(n)(w) : n = 0,1,2,... (cid:8) (cid:9) theorbitofΦatw. ForanalgebraicvarietyV = Z(G ,...,G )defined 1 s by the equations G = = G = 0, G C[X ,...,X ], i = 1,...,s, 1 s i 1 m ··· ∈ we consider the elements of the orbit Orbw(Φ) which fall into V and denote (1.1) w(Φ,V) = n 0 : Φ(n)(w) V . S (cid:8) ≥ ∈ (cid:9) We say that the complex numbers α ,...,α are multiplicatively in- 1 n dependent if all of them are non-zero and there is no non-zero integer vector (i ,...,i ) such that αi1 αin = 1. 1 n 1 ··· n In the sequel, we denote by S the cardinality of a finite set S. Our | | objective in this paper is to bound the size of w(Φ,V) for various |S | cases when it is finite. Throughout the paper, let Q be the algebraic closure of the rational numbers Q. For any field K, we write K∗ for the multiplicative group of all the non-zero elements of K. For any multiplicative group Λ and any integer k 1, let Λk be the direct product consisting of k-tuples ≥ x = (x ,...,x ) with x Λ,1 i k. As usual, the multiplication 1 k i ∈ ≤ ≤ of the group Λk is defined by xy = (x y ,...,x y ) for any x,y Λk. 1 1 k k ∈ 4 ALINAOSTAFEAND MIN SHA 2. Preliminaries In this section, we gather some results which are used afterwards. Recall that a linear recurrence sequence (LRS) of order m 1 is a ≥ sequence u ,u ,u ,... with elements in C satisfying a linear relation 0 1 2 { } (2.1) u = a u + +a u (n = 0,1,2,...), n+m 1 n+m−1 m n ··· where a ,...,a C, a = 0 and u = 0 for at least one j in the 1 m m j ∈ 6 6 range 0 j m 1. We assume that relation (2.1) is minimal, that ≤ ≤ − is the sequence u does not satisfy a relation of type (2.1) of smaller n { } length. The characteristic polynomial of this LRS u is n { } k f(X) = Xm a Xm−1 a = (X α )ei C[X] − 1 −···− m Y − i ∈ i=1 with distinct α ,α ,...,α and e > 0 for 1 i k. Then, u can be 1 2 k i n ≤ ≤ expressed as k u = f (n)αn, n X i i i=1 where f is some polynomial of degree e 1 (i = 1,2,...,k). We i i − call the sequence u simple if k = m (that is e = = e = 1) n 1 m { } ··· and non-degenerate if α /α is not a root of unity for any i = j with i j 6 1 i,j k. ≤ ≤ One fundamental problem of the LRS (2.1) is to describe the struc- ture or bound the size of the following set n 0 : u = 0 , n { ≥ } which iscalled thezero set of thesequence (2.1). Equivalently, wewant to study the integer roots of the exponential polynomial k f (z)αz. Pi=1 i i The well-known Skolem-Mahler-Lech theorem says that the zero set ofany LRSis afinite union ofarithmetic progressions, and furthermore it is a finite set if the sequence is non-degenerate; for example see [9, Theorem 2.1]. There are rich results on the quantitative version of this theorem. Here we restate some results in the setting of exponential polynomials, which are used later on. In the rest of this section, we fix an exponential polynomial over C k (2.2) F(z) = f (z)αz X i i i=1 QUANTITATIVE DYNAMICAL MORDELL-LANG CONJECTURE 5 with distinct α ,...,α C∗, and non-zero f C[z] for 1 i k. 1 k i ∈ ∈ ≤ ≤ We also define m = degf + +degf +k 1 k ··· and denote (F) = n Z : F(n) = 0,n 0 . Z { ∈ ≥ } Note that F(z) corresponds to an LRS of order m, and the set (F) Z is exactly the zero set of the corresponding sequence. Especially, when f ,...,f are constants, F(z) corresponds to a simple LRS. 1 k The following result comes from [1, Corollary 6.3] and [2, Theorem 1.1]. Lemma 2.1. Let F(z) be given by (2.2). Then the set (F) is the Z union of at most exp(exp(70m)) arithmetic progressions. Moreover, if f ,...,f are non-zero constants, then the set (F) is the union of at 1 k Z most (8m)4m5 arithmetic progressions. Lemma2.1canyieldsomequantitativeresultsconcerningConjecture 1.1. However, here we are more interested with the case when the subset of integers in Conjecture 1.1 is a finite set. As mentioned above, if F(z) corresponds to a non-degenerate LRS, the set (F) is in fact a finite set, and furthermore we can bound the Z cardinality (F) . The following result follows from [1, Corollary 6.3] |Z | and [2, Theorem 1.2]. Lemma 2.2. Let F(z) be given by (2.2). Suppose that F(z) corre- sponds to a non-degenerate LRS, and degf + 1 a for 1 i k. i ≤ ≤ ≤ Then we have (F) (8ka)8k6a; |Z | ≤ furthermore if f ,...,f are non-zero constants, then we have 1 k (F) (8m)4m5. |Z | ≤ In fact, if there exists some index i such that the ratio α /α is not i j a root of unity for any j = i, then the set (F) is still a finite set; see 6 Z [9, Theorem 2.1 (iii)]. Here we want to bound (F) in this case by |Z | using Lemma 2.2 and following the arguments in [9]. Lemma 2.3. Let F(z) be given by (2.2). Let D be the order of the group of roots of unity generated by all those roots of unity which are of the form α /α for some 1 i,j k. Suppose that there exists some i j ≤ ≤ index i such that the ratio α /α is not a root of unity for any j = i , 0 i0 j 6 0 and degf +1 a for 1 i k. Then we have i ≤ ≤ ≤ (F) D(8ka)8k6a . |Z | ≤ 6 ALINAOSTAFEAND MIN SHA Proof. We partition α ,...,α into equivalence classes according to 1 k the equivalence relation where b c if and only if the ratio b/c is ∼ a root of unity. By renumbering, we can assume that α ,...,α are 1 s representatives of these equivalence classes. Then, fixing an integer b with 0 b < D, we consider the equation ≤ F(b+nD) = 0 with integer unknown n 0. By the choice of D, we can express ≥ F(b+nD) as s F(b+nD) = g (n) αD n X i (cid:0) i (cid:1) i=1 for some polynomials g C[z] with degg +1 a. Under the assump- i i ∈ ≤ tion on α , there indeed exists some index j such that g = 0. So, i0 j 6 using Lemma 2.2, we deduce that the cardinality of the set n 0 : { ≥ F(b+nD) = 0 is at most (8ka)8k6a. The final result follows from the fact that there}are D choices of the integer b. (cid:3) Note that if F(z) corresponds to a non-degenerate sequence, then D = 1 and Lemma 2.3 is exactly the first upper bound in Lemma 2.2. Moreover, ifα ,...,α arerootsofapolynomialf(X)over anumber 1 k field K, then the quantity D can be bounded by (2.3) D < exp 1.05314+√6d mlog(dm) , (cid:16)(cid:16) (cid:17)p (cid:17) where d = [K : Q] and m = degf 2; see [12, Theorem 1]. ≥ Except for studying the set (F), we also need to estimate the num- Z berofintegersn 0suchthatF(n)isequaltoafixednon-zerocomplex ≥ number. Corollary 2.4. Let F(z) be given by (2.2). Define α = 1. Suppose k+1 that there exists some index i such that the ratio α /α is not a root 0 i0 j of unity for any j = i with 1 j k +1. Let D be the order of the 0 6 ≤ ≤ group of roots of unity generated by all those roots of unity which are of the form α /α for some 1 i,j k +1. Assume that degf +1 a i j i ≤ ≤ ≤ for 1 i k. Then for any µ C with µ = 0, we have ≤ ≤ ∈ 6 n Z : F(n) = µ,n 0 D(8(k+1)a)8(k+1)6a ; |{ ∈ ≥ }| ≤ furthermore if F(z) corresponds to a non-degenerate LRS, no α (1 i ≤ i k) is a root of unity, and f ,...,f are non-zero constants, we have 1 r ≤ n Z : F(n) = µ,n 0 (8(m+1))4(m+1)5. |{ ∈ ≥ }| ≤ QUANTITATIVE DYNAMICAL MORDELL-LANG CONJECTURE 7 Proof. Under the assumptions, we can get the desired results by ap- plying Lemmas 2.2 and 2.3 to the following equation k f (n)αn +( µ) 1 = 0, n 0, X i i − · ≥ i=1 with coefficients f (n),...,f (n), µ. (cid:3) 1 k − Whenα ,...,α arealgebraicnumbers, theresultsinLemma2.3and 1 k Corollary 2.4 can be improved in some sense. The following lemma is derived from [19, Theorem 1] with a slight refinement. Although the arguments in [19] were presented only for non-degenerate sequences, they are also valid for more general cases under minor changes. Lemma 2.5. Let F(z) be given by (2.2). Suppose that α ,...,α are 1 k algebraic numbers, and let D be the order of the group of roots of unity generated by all those roots of unity which are of the form α /α for i j some 1 i,j k. Denote K = Q(α ,...,α ), and assume that 1 k ≤ ≤ f ,...,f K[z]. Let d = [K : Q], and let ω be the number of prime 1 k ∈ ideals occurring in the decomposition of the fractional ideals α in K. i h i Assume that there exists some index i such that the ratio α /α is not 0 i0 j a root of unity for any j = i . Then, we have 0 6 (F) < D(4(d+ω))2(d+1)(m 1); |Z | − furthermore if K/Q is a Galois extension but not a cyclic extension, we have (F) < D(4(d+ω))d+2(m 1). |Z | − Proof. Here, we sketch the proof for the convenience of the reader. We first choose a rational prime p such that none of the prime ideals p ,...,p from the decomposition in K of the ideals (α ) (1 i k) 1 ω i ≤ ≤ divides the ideal p . Let p be a prime ideal of K lying above p, and h i let f denote the residue class degree of K over Q , where Q is the p p p p-adic completion of Q and K is the completion of K with respect to p p. Let C be the completion of the algebraic closure of Q . We denote p p the valuation of C by , normalised such that p = p−1. Note that p p p | | | | Q K C . p p p ⊆ ⊆ By the choice of p, we have α = 1, i = 1,...,k. i p | | Furthermore, by [19, Equation (3.4)] we know αpf−1 1 < p−1/(p(p−1))−1/(p−1), i = 1,...,k. (cid:12)(cid:12) i − (cid:12)(cid:12)p (cid:12) (cid:12) 8 ALINAOSTAFEAND MIN SHA Then, we have αD(pf−1) 1 αpf−1 1 < p−1/(p(p−1))−1/(p−1), i = 1,...,k. (cid:12)(cid:12) i − (cid:12)(cid:12)p ≤ (cid:12)(cid:12) i − (cid:12)(cid:12)p (cid:12) (cid:12) (cid:12) (cid:12) Fix an integer a with 0 a < D(pf 1), we consider the equation ≤ − F a+nD pf 1 = 0 (cid:0) (cid:0) − (cid:1)(cid:1) with integer unknown n 0. ≥ As in the proof of Lemma 2.3, we partition α ,...,α into equiv- 1 k alence classes, and we assume that α ,...,α are representatives of 1 s these equivalence classes. Then, by the choice of D, we can express F a+nD pf 1 as (cid:0) (cid:0) − (cid:1)(cid:1) s F a+nD pf 1 = g (n) αD(pf−1) n (cid:0) (cid:0) − (cid:1)(cid:1) X i (cid:16) i (cid:17) i=1 for some polynomials g K[z]. Under the assumption of α , there i ∈ i0 indeed exists some index j such that g = 0. j 6 As solving the equation (3.6) of [19], we immediately see that the cardinality of the set n Z : F a+nD pf 1 = 0,n 0 is at (cid:8) ∈ (cid:0) (cid:0) − (cid:1)(cid:1) ≥ (cid:9) most (m 1)(p+1). Thus, we obtain − (F) D pf 1 (m 1)(p+1). |Z | ≤ (cid:0) − (cid:1) − From [19, Section 4], the prime p can be chosen such that p < (4(d+ω))2. Then, the first desired upper bound follows from the fact that f d. ≤ Now, we assume that K/Q is a Galois extension but not a cyclic extension. Inordertoprove thesecondclaimed upper bound, itsuffices to show that p does not remain inert in K. Because if this is true, then f d/2, which can conclude the proof. ≤ Let D denote the decomposition group of p in K/Q. Suppose that p p remains inert in K. Then, f = d, and D is a cyclic group of order p d. Since [K : Q] = d, D is exactly the Galois group Gal(K/Q). So, p K/Q is a cyclic extension, this leads to a contradiction. (cid:3) Applying the same arguments as in the proof of Corollary 2.4, we can obtain similar results as Lemma 2.5 for the cardinality n Z : |{ ∈ F(n) = µ,n 0 , where µ is a non-zero algebraic number. ≥ }| We also need a result on solutions of linear equations in several vari- ables. The following result is given in [2, Lemma 2.1] and is derived from [1, Theorem 6.2]. QUANTITATIVE DYNAMICAL MORDELL-LANG CONJECTURE 9 Lemma 2.6. Let Γ be finitely generated subgroup of (C∗)k of rank r, and let a ,...,a C∗. Then, up to proportionality, the equation 1 k ∈ (2.4) a x + +a x = 0 1 1 k k ··· has less than (8k)4(k−1)4(k+r) non-degenerate solutions in Γ. Here, “up to proportionality” means that two solutions (x ,...,x ) 1 k and (y ,...,y ) of (2.4) are equivalent if there is some non-zero c such 1 k that (y ,...,y ) = (cx ,...,cx ). 1 k 1 k Besides, we call a solution of (2.4) non-degenerate if no subsum of the left hand side of the equation vanishes. LetK beanumber field, letS beafinitesetofplacesofK containing all the Archimedean places and write ∗ for the group of S-units of OS K. If the coefficients a ,...,a K 0 , then the number of such 1 k ∈ \ { } solutions of (2.4) in Γ ( ∗)k can be bounded better than Lemma ⊆ OS 2.6; for example see [13, Theorem 3]. So, some results in this paper can be improved in this case. Let be a partition of the set I = 1,...,k . The subsets λ I P { } ⊆ occurring in the partition are considered as elements of . Then, P P the system of equations (2.4 ) a x = 0 (λ ) P X i i ∈ P i∈λ is a refinement of (2.4). Given a partition of the set I, we say that P two solutions (x ,...,x ) and (y ,...,y ) of (2.4) are equivalent up to 1 k 1 k proportionality with respect to ifbothofthemarealsosolutionsofthe P system (2.4 ), and for each λ the two solutions (x ) and(y ) i i∈λ i i∈λ P ∈ P of the corresponding equation are equivalent up to proportionality. Finally, two solutions (x ,...,x ) and (y ,...,y ) of (2.4) are called 1 k 1 k equivalent up to weak proportionality if there exists a partition of the P set I such that they are equivalent up to proportionality with respect to . P Now, we want to count all the solutions of (2.4) up to weak propor- tionality. Corollary 2.7. Let Γ be finitely generated subgroup of (C∗)k of rank r, and let a ,...,a C∗. Then, up to weak proportionality, the equation 1 k ∈ (2.4) has less than (0.5k)k(8k)4(k−1)4(k+r) solutions in Γ. 10 ALINAOSTAFEAND MIN SHA Proof. Let be a partition of the set I = 1,...,k . Note that in P { } order to ensure that the system (2.4 ) has solutions in Γ we must P have that λ 2 for any λ . So, we can assume that k/2. | | ≥ ∈ P |P| ≤ If is another partition of I such that is a refinement of , then Q Q P the system (2.4 ) implies (2.4 ). Let ( ) consist of solutions of Q P T P (2.4 ) in Γ up to proportionality with respect to , which are not P P solutions of any (2.4 ) where is a proper refinement of . Q Q P According to the partition , we can treat Γ as a direct product P Γ = Γ(λ), Y λ∈P where Γ(λ) is the projection of Γ corresponding to λ. For each λ ∈ Γ, Γ(λ) is also a finitely generated group and let r(λ) be its rank. Obviously, we have r(λ) = r. X λ∈P For each equation in (2.4 ) P a x = 0, X i i i∈λ by Lemma 2.6 it has less than (8 λ )4(|λ|−1)4(|λ|+r(λ)) non-degenerate so- | | lutions in Γ(λ) up to proportionality. Thus, we have ( ) < (8 λ )4(|λ|−1)4(|λ|+r(λ)) |T P | Y | | λ∈P < (8k)4(k−1)4(k+r). Recall that the Bell numbers count the number of partitions of a set. By [8, Theorem 2.1], the number of partitions of I is less than (0.792k/log(k +1))k. However, not all such partitions are suitable for our settings. We have indicated that a suitable partition should satisfy that λ 2 for any P | | ≥ λ . So, the number of these suitable partitions is not greater than ∈ P (0.5k)k. Note that every solution of the equation (2.4) is a solution of the system (2.4 ) for some partition , and we can assume that k 2. P P ≥ So, up to weak proportionality, the number of solutions of (2.4) in Γ is at most ( ) < (0.5k)k(8k)4(k−1)4(k+r), X|T P | P