aa r X i v : . [ m a t h . N T ] S e p Algebraic Degree Periodicity in Recurrence Sequences
Daqing Wan , Hang Yin October 1, 2020 Department of Mathematics, University of California, Irvine, CA [email protected] Institute of Mathematics, Chinese Academy of Sciences, [email protected]
Abstract . The degree sequence of the algebraic numbers in an algebraic linear recurrencesequence is shown to be virtually periodic. This is proved using the Skolem-Mahler-Lechtheorem. It has applications to the degree sequence and the minimal polynomial sequencefor exponential sums over finite fields. The degree periodicity also holds for some morecomplicated non-linear recurrence sequences. We give one example from the iterations of apolynomial map. This depending on the dynamic Mordell-Lang conjecture which has beenproved in some cases.
This note is motivated by the following number theoretic question on algebraic degrees ofexponential sums over finite fields.Let p be a prime, ζ p be a primitive p -th root of unity. For each positive integer k , let F p k denote the finite field of p k elements. Let f ( x , x , ..., x n ) ∈ F p [ x , x , ..., x n ] be a polynomial.For each positive integer k , we define the k -th exponential sum of f to be S k ( f ) = X ( x ,x ,...,x n ) ∈ ( F pk ) n ζ Tr k ( f ( x ,x ,...,x n )) p ∈ Z [ ζ p ] , k = 1 , , · · · , where Tr k : F p k → F p is the absolute trace map.It is clear that S k ( f ) ( k = 1 , , · · · ) is a sequence of algebraic integers in the p -th cyclo-tomic field. As an algebraic integer, its degree deg S k ( f ) is a divisor of p −
1. A naturalproblem is to understand and compute this degree sequence deg S k ( f ) ( k = 1 , , · · · ). Theproblem is trivial if p = 2. We can assume p >
2. As we shall see, this degree problem isfar from being well understood, even in the simplest classical case when f ( x ) = x d is a onevariable monomial.A consequence of our main result is the following general structural theorem which showsthat the degree sequence is virtually periodic.1 heorem 1. The sequence deg S k ( f ) is virtually periodic in k . That is, there are positiveintegers N and r depending on f and p such that for all k > N , we have deg S k + r ( f ) =deg S k ( f ) . Theoretically, this theorem suggests that the computation of the infinite degree sequencecould be done in a finite amount of time. In reality, this is far from being so. The problemis that the constant N is not effective, although r (a period) can be made effective. Aninteresting open problem is to compute the degree sequence in effective finite amount oftime. Equivalently, one would like to have an effective upper bound for the total degree ofthe rational function ∞ X k =1 (deg S k ( f )) T k ∈ Q ( T ) . A similar non-effective problem occurs in explicit lower bound for the complex absolute valueof the sequence S k ( f ) as k varies, see Bombieri-Katz [BK10].To get some feeling on this degree problem, let us examine the simplest classical onevariable example when f ( x ) = x d . Historically, S k ( x d ) was sometimes called a Gauss sum ora Gauss period in the literature. It has been studied extensively, see the survey in Berndtand Evans [BE81]. A trivial reduction shows that one can assume that ( d, p ) = 1. If p ≡ d , it is known [W20] thatdeg S k ( x d ) = d ( d, k ) , k = 1 , , · · · . This formula is due to Gauss when k = 1 and to Myerson [My81] when ( d, k ) = 1. It followsthat the degree sequence deg S k ( x d ) is periodic (not just virtually periodic) and completelydetermined when p ≡ d . Another known case [W20] is when ( p − , d ) = 1, in whichcase, deg S k ( x d ) = 1 for all k . Note that ( p k − , d ) = 1 in general, even if ( p − , d ) = 1. Theseresults imply that the degree sequence deg S k ( x d ) ( k = 1 , , · · · ) is completely determinedfor all primes p when d is a prime. If d is not a prime, effectively determining the degreesequence deg S k ( x d ) for all primes p is not known yet.Another classical example is the Kloosterman sum. Fix an integer a not divisible by p .For positive integer n , the sequence of n -dimensional Kloosterman sums is defined byKl k ( n, a ) = X ( x ,x ,...,x n ) ∈ ( F ∗ pk ) n ζ Tr k ( x + x + ··· + x n + ax ··· xn ) p ∈ Z [ ζ p ] , k = 1 , , · · · . The degree sequence deg Kl k ( n, a ) ( k = 1 , , · · · ) is again far from being well understood. Astraightforward Galois theoretic argument shows thatdeg Kl k ( n, a ) | p − n + 1 , p − , k = 1 , , · · · . These two numbers are not expected to be equal in general. One simple result in [W95] saysthat if k is not divisible by p , thendeg Kl k ( n, a ) = p − n + 1 , p − . n = 1 and k = 1 goes back to Sali´e [Sa32]. If p > (2( n + 1) k + 1) ,this formula also follows from Fisher’s result [Fi92] on distinctness of Kloosterman sums,proved using ℓ -adic cohomology. Effectively computing the degree sequence deg Kl k ( n, a )( k = 1 , , · · · ) is currently unknown in general when k is divisible by p .Our main result of this note is to prove that the degree sequence of any linear recurrencesequence in any finite extension of a field of characteristic zero is virtually periodic, seeTheorem 3 for a precise statement. The proof combines Galois theory together with thewell known Skolem-Mahler-Lech theorem on zeros of linear recurrence sequences. The non-effective part of our result comes from the non-effective part of the Skolem-Mahler-Lechtheorem. Theorem 1 on exponential sums then follows from this and the fact that thesequence S k ( f ) ( k = 1 , , · · · ) is a linear recurrence sequence of algebraic integers. The latterfact is a consequence of the well known Dwork-Bombieri-Grothendick rationality theoremwhich says that the L-function L ( f, T ) = exp( ∞ X k =1 S k ( f ) k T k )is a rational function. More generally, the virtual periodicity result stated in Theorem 1holds for the degree sequence arising from any motivic L-function over finite fields. It wouldbe interesting to make the computation of the degree sequence effective in various importantnumber theoretic cases, in addition to the above exponential sum examples.The Skolem-Mahler-Lech theorem is equivalent to a similar theorem on the iterationsequence of a linear map on a vector space. From this dynamic point of view, a remark-able generalization is the dynamic Mordell-Lang conjecture for the iteration sequence of apolynomial map, as proposed by Ghioca and Tucker [GT09]. This conjecture is still openin general but has been proved in some cases. As an illustration, we show that the degreesequence arising from the iteration of any one variable polynomial with algebraic coefficientsevaluated at an algebraic point is also virtually periodic. This application depends on Xie’swork [Xi17] on the two variable case of the dynamic Mordell-Lang conjecture. We first recall some basic definitions about linear recurrence sequences.
Definition 1.
Let { a n } n > be a sequence in a field K .1. We say { a n } n > is an LRS (linear recurrence sequence) if there exist some m ∈ N and c i ∈ K (1 i m ) such that a n = c a n − + · · · + c m − a n − m +1 + c m a n − m holds for all n > m .2. The sequence { a n } n > is called periodic if there exists some integer r > such that a n = a n + r holds for all n ≥ . The sequence { a n } n > is called virtually periodic if thereexist some N, r ∈ N such that for all n > N, a n + r = a n . In both cases, such an integer r is called a period of this sequence. . We say { a n } n > is a rational sequence if its generating function f ( x ) = P n a n x n is arational function. Proposition 1.
The following properties are well known and easy to check.1. A virtually periodic sequence is clearly a rational sequence.2. A sequence { a n } n > is a rational sequence if and only if for some m ∈ Z ≥ , the newsequence { a n } n > m is an LRS. Furthermore, an LRS is exactly a rational sequence whosegenerating function vanishes at ∞ i.e. P n a n x n = f ( x ) g ( x ) with deg( f ) < deg( g ) .3. The sub-sequence { a i + nr } ( n ≥ ) of any rational sequence { a n } is again a rationalsequence for all integers i, r ∈ Z ≥ .4. A sequence { a n } n > in an algebraic closure ¯ K of K is a rational sequence if and onlyif there are finite number of polynomials h i ( x ) and elements β i over ¯ K ( ≤ i ≤ m )such that for all sufficiently large n , we have a n = m X i =1 h i ( n ) β ni . The last property gives the following lemma which will be used in our proof.
Lemma 1.
For each integer ≤ j ≤ ℓ , let { a jn } be a rational sequence in K . Let g ( x , · · · , x ℓ ) ∈ K [ x , · · · , x ℓ ] . Then, the new sequence g ( a n , · · · , a ℓn ) is a rational sequencein K . We first remind ourselves of the Skolem-Mahler-Lech theorem.
Theorem 2 (Skolem-Mahler-Lech) . Let K be a field of characteristic 0. Let { a n } n > be arational sequence with coefficients in K , then the set { n ∈ N | a n = 0 } is virtually periodic. We now present our result.
Theorem 3.
Let { a n } n > be a rational sequence with coefficients in a finite extension L ofa field K of characteristic zero. We have the following. • (1). The sequence { deg( a n ) } n > is virtually periodic, where deg( a n ) = [ K ( a n ) : K ] . • (2). Let P n ( T ) denote the minimal polynomial of a n over K . The sequence { P n ( T ) } n > is a rational sequence, that is, its generating function P n P n ( T ) x n is a rational functionin x with poles algebraic over K .Proof. Without loss of generality, we may enlarge L to assume
L/K is finite Galois. Let σ ∈ G = Gal( L/K ). Since { a n } is a rational sequence, applying σ to it we get anotherrational sequence { σ ( a n ) } . Now the difference { σ ( a n ) − a n } is also a rational sequence.By the Skolem-Mahler-Lech theorem, we deduce that { n ∈ N | σ ( a n ) − a n = 0 } is virtuallyperiodic. In another word, the indices of those a n fixed by σ is virtually periodic.4or each σ ∈ G , let N σ and r σ be positive integers such that for all n > N σ , a n is fixedby σ if and only if a n + r σ is fixed by σ . Let N = max σ ∈ G ( N σ ), r = Q σ ∈ G r σ . We see that N, r work for all σ ∈ G uniformly. Now for all n > N and all σ ∈ G , σ fixes a n if and onlyif it fixes a n + r , which implies the fixed subgroup H n of a n satisfies H n = H n + r . By Galoistheory, the degree of a n is just [ G : H n ], hence for n > N, deg( a n ) = deg( a n + r ). This showsthat { deg( a n } is virtually periodic.Now, for each integer 0 ≤ i ≤ r −
1, we have H N + i = H N + i + kr for all integers k ≥
0. Let σ i , · · · , σ id i be a set of representatives for the coset G/H N + i = G/H N + i + kr . The minimalpolynomial of a N + i + kr over K is P N + i + kr ( T ) = Y σ ∈ G/H N + i + kr ( T − σ ( a N + i + kr )) = d i Y j =1 ( T − σ ij ( a N + i + kr )) . For σ ∈ G , the sequence σ ( a N + i + kr ) ( k ≥
0) is an LRS. Each coefficient of the abovepolynomial is an elementary symmetric polynomial of finitely many such linear recurrencesequences. It follows that the generating function of the subsequence { P N + i + kr ( T ) } ( k ≥ x for each i . Hence, the generating function of the total sequence { P n ( T ) } ( n ≥
0) is also a rational function in x . Furthermore, the poles of this rationalfunction are clearly algebraic over K . The proof is complete.The following corollary was first suggested by Shaoshi Chen, based on his computercalculations. Corollary 1.
Let α be an element in a finite extension L of a field K of characteristiczero. For integer n ≥ , let P n ( T ) be the minimal polynomial of α n over K . Then, thesequence P n ( T ) ( n ≥ ) is a rational sequence, that is, its generating function P n P n ( T ) x n is a rational function in x with poles in the Galois closure of K ( α ) over K . Note that this corollary can be proved directly without the Skolem-Mahler-Lech theorem,as the set { n | σ ( α n ) = α n } is obviously an arithmetic progression. This means that thecorollary is actually effective. More generally, Theorem 3 is effective when the order of thesequence { a n } is at most 2. This is because that the sequence { σ ( a n ) − a n } then has orderat most 4 and the Skolem-Mahler-Lech theorem is effective for LRS with order up to 4. The degree periodicity property can hold for sequences of algebraic numbers coming fromcertain non-linear recurrence relations. We summarize the key property that makes it workin the following Lemma, whose proof is the same as our proof in section 2.
Lemma 2.
Let { a n } ( n ≥ ) be a sequence of elements in a finite Galois extension L of a field K . Assume that for each σ in the Galois group of L over K , the set { n ∈ N | σ ( a n ) − a n = 0 } is virtually periodic. Then the degree sequence { deg( a n ) } n > is virtually periodic, where deg( a n ) = [ K ( a n ) : K ] . Furthermore, the sequence of the minimal polynomial of a n over K is a rational sequence.
5n the case that the sequence is an LRS in a field L of characteristic zero, the conditionin the Lemma is satisfied by the Skolem-Mahler-Lech theorem and hence one gets the degreeperiodicity. The condition is also satisfied for some more complicated sequences. As anillustration, we give one such example coming from arithmetic dynamics. Theorem 4.
Let
L/K be a finite extension of number fields. Let f ( x ) ∈ L [ x ] be a polynomialand a ∈ L . Then the sequence { deg K f ( n ) ( a ) } is virtually periodic, where f ( n ) denotes the n -th iterate of the polynomial f . Furthermore, the sequence of the minimal polynomial of f ( n ) ( a ) over K is a rational sequence.Proof. As before, we may assume that L is a finite Galois extension of K with Galois group G . For σ ∈ G , σ ( f ) ∈ L [ x ] is another polynomial over L . By the Lemma, it is enough toprove that the set { n ∈ N | σ ( f ( n ) ( a )) = f ( n ) ( a ) } = { n ∈ N | σ ( f ) ( n ) ( σ ( a )) = f ( n ) ( a ) } is virtually periodic for every σ ∈ G . Define the polynomial map F σ : A L −→ A L , F σ ( x , x ) = ( f ( x ) , σ ( f )( x )) . Let V be the diagonal { ( x, x ) } in A . On checks that { n ∈ N | σ ( f ( n ) ( a )) = f ( n ) ( a ) } = { n ∈ N | F ( n ) σ ( a, σ ( a )) ∈ V } . The last set is virtually periodic by the affine plane A case of the dynamic Mordell-Langconjecture as proved in Xie [Xi17]. The proof is complete.A natural generalization of the above theorem is the following conjecture. Conjecture 1.
Let
L/K be a finite extension of number fields and m ∈ N . Let f = ( f ( x , · · · , x m ) , · · · , f m ( x , · · · , x m )) ∈ L [ x , · · · , x m ] m be a polynomial map from L m to L m . For any element a = ( a , · · · , a m ) ∈ L m , the se-quence { deg K f ( n ) ( a ) } is virtually periodic, where deg K f ( n ) ( a ) denotes the degree of the fieldextension over K obtained by adjoining the coordinates of the point f ( n ) ( a ) ∈ L m . The above theorem shows that this conjecture is true in the case m = 1. For m >
1, it isa consequence of the affine A m case of the dynamic Mordell-Lang conjecture, which is stillopen. Acknowledgements
This note grew out of the Number Theory Online Mini-Course held at Xiamen Universityin August 2020. It is a pleasure to thank the many audience, particularly, Shaoshi Chen andhis students, for their interests, questions and careful note-taking.6 eferences [BE81] B. Berndt and R. Evans, The determination of Gauss sums,
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