Periods of Iterated Rational Functions over a Finite Field

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1 Periods of Iterated Rational Functions over a Finite Field Charles Burnette Department of Mathematics Drexel University Philadelphia, PA cdb72@drexel.edu Eric Schmutz Department of Mathematics Drexel University Philadelphia, PA Eric.Jonathan.Schmutz@drexel.edu August 2, 205 Abstract Choose a random polynomial f uniformly from among the q d (q polynomials of degree d in F q [x]. Let c k be the number of cycles of length k in the directed graph on F q with edges {(v, f(v} v Fq. If d = d(q as q, then the numbers c, c 2,..., c b are asymptotically independent Poisson(/k, just as in the classical theory of random mappings. However the random polynomials differ from random mappings in important ways. For example, the restriction of f to its periodic points is not, in general, a uniform random permutation of those points. Such differences complicate research on the distribution of T(f = the ultimate period of f under ( compositional iteration. We prove that if ω = ω(q slowly and d = d(q > exp log q, then for ω /3 all sufficiently large prime powers q, E(T > q ω. A similar bound holds when random polynomials are replaced by random rational functions. No non-trivial upper bound for E(T is known. Introduction The classical theory of random mappings concerns functions chosen uniform randomly from among the q q functions whose domain and codomain is a given q-element set V. (See, for example, [6], [9], and chapter 9 of [4]. Basic facts about random mappings have motivated conjectures and theorems in arithmetic dynamics [], [2], [], [3]. [6], [7]. This paper begins with F q [x]-analogs of known facts about random mappings cycle lengths and the lengths of their (ultimate periods under function composition.

2 For each f, let cyclic(f be the set of periodic points, i.e the set of vertices that lie on cycles in the functional digraph G f that is formed by putting an edge from v to f(v for each v. Let Z(f = cyclic(f and let σ f be the restriction of f to cyclic(f. Finally, let T(f be the least common multiple of the cycle lengths, i.e. the order of σ f as an element of the group Sym(cyclic(f. Equivalently, T(f is the ultimate period of f, defined as the smallest positive integer t such that, for every n q, f (n+t = f (n. By the main theorem in Harris [8], the random variable T is asymptotically lognormal. An immediate consequence of his theorem is that, if ɛ > 0, then for all but o(q q functions f, e ( 8 ɛ log2 q < T(f < e ( 8 +ɛ log2 q. (. Igor Shparlinski and Alina Ostafe proposed the challenging problem of estimating the expected value of T for random polynomials and for random rational functions. Let Ω(q, d be the set of all q d (q polynomials of degree d with coefficients in the finite field F q. Let M = M q,d be the uniform probability measure on Ω(q, d, and let E M ( be the corresponding expectation. Then T is a random variable on each Ω(q, d, and it makes sense to ask for the asymptotic distribution of the random variables as q through the increasing sequence of prime powers 2 < 3 < 4 < 5 < 7 < 8 < This paper provides a lower bound for these expected values: if ω = ω(q is a non-decreasing function such that ω(q, and if d = d(q > exp ( log q ω, then /3 ( log q E(T > exp ω for all sufficiently large prime powers q. The proofs include almost sure bounds that are analogous to those in (., but presumably much cruder. In both [5] and Harris proof of lognormality, essential use was made of the following observation about classical random mappings: Observation.. (Folklore Suppose M is the uniform probaility measure on the V V functions f : V V. Then, for all non-empty A V and all σ Sym(A, M (σ f = σ cyclic(f = A = A! In other words, σ f is a uniform random permutation of the cyclic vertices. However a simple counting argument shows that the analogue of Observation. cannot be true in general for Ω(d, q. Consider, for example, the case where A is all of F q. If it were true that M (σ f = σ cyclic(f = A = > 0 for all σ in Sym(A, then, for each σ Sym(A, A! Ω(q, d would contain at least one element f for which σ f = σ. This implies that Ω(q, d A!. But if A is all of F q, and if d = o(q and, then Ω(q, d < A! for all sufficiently large prime powers q. Although it is not uniformly random, σ f is a permutation of cyclic(f. By an old theorem of Landau, [0], [2], the maximum order that a permutation of [m] can have is e m log m(+ɛ(m, where ɛ(m 0. However we do not know enough about the distribtuon of Z to draw any conclusions about E M (T. To date, we have only the trivial upper bound E M (T e q log q(+o( that follows from Z q +. 2

3 2 Joint Distribution of Small Cycle Lengths Flynn and Garton [7] used Lagrange interpolation to calculate the expected number of cycles of length k. In this section, Flynn and Garton s methods are combined with the method of factorial moments to determine the asymptotic joint distribution of the cycle length multiplicities c, c 2,..., c b for fixed b. Lemma 2.. Let A F q, and let σ be a permutation of A. If d A, then there are exactly q d A (q polynomials in F q [x] of degree d that extend σ to F q. Proof. Let E d consist of those polynomials f of degree d that extend σ, i.e. such that f(x = σ(x for all x A. Also let K d be the set of polynomials of degree d such that f(x = 0 for all x A. By the Lagrange interpolation theorem, f E d if and only if f(x = L(x + k(x, where L is the minimum degree interpolating polynomial (whose degree is less than d, and k(x K d. Since F q [x] is a Euclidean domain, a polynomial of degree d is in K d if and only if it is divisible by g(x = (x a. Thus k K d if and only if there is a polynomial h of a A degree d A such that k(x = g(xh(x. The number of ways to choose the polynomial h is q d A (q. Therefore E d = K d = q d A (q. Theorem 2.2. If d = d(q, then for any non negative integers m, m 2,..., m b, lim M(c k = m k, k b = q e k. m k!k m k Proof. Let Z q,k be the set of all ( q k (k! possible k-cycles. Let IC (f = if C is a cycle for f and zero otherwise. Thus c k = I C. Also let (c k r = r (c k j. To prove C Z q,k j=0 the theorem, it suffices to show that the joint factorial moments E M ((c r (c 2 r2 (c b rb converge to those of the corresponding independent Poisson distributions. In other words, it suffices to check that, for any choice of r, r 2,..., r b, ( lim E M (c k rk =. q k r k It is well known (and easy to prove by induction on r that (c k r = I C I C2 I Cr, where the sum is over all sequences C, C 2,..., C r of r distinct k cycles in Z q,k. Hence E M ( (c k rk = E M ( r k j= I Ck,j, (2. 3

4 where the sum is over all ordered choices of b r k disjoint cycles for which there are exactly r k cycles of length k for k b. Consider a single term in the sum corresponding to a particular permutation σ of a set A such that there are r k cycles of length k (k =, 2,..., b. By the lemma, E M ( r k j= I Ck,j = qd A (q q d (q = q A. For a given value of a = b kr k, the number of terms in the sum is q! (q a! k! r k (k! r k. Therefore E M ( (c k rk = q! (q a! Since a and b are fixed, it follows that ( lim E (c k rk = q k r k k r k. q a. 3 Lower Bound for the Expected Order The goal of this section is to prove a lower bound for T that is analagous to that in ( (.. log q Assume ω = ω(q is a non-decreasing function such that and ω(q. Let ξ = exp L(q ( 3 log q and ξ 2 = exp, where L = L(q is the positive solution to L3 L(q log L = ω. Note that L(q. For all sufficiently large q, L is non-decreasing and ξ < ξ 2. Theorem 3.. If d = d(q > 2ξ 2 for all sufficently large q, then M(T > q ω = o(. Proof. Let δ k (f = if c k > 0 and k is prime, δ k = 0 otherwise. For any choice of ξ < ξ 2, the period is at least as large as the product (without multiplicity of prime cycle lengths in the interval [ξ, ξ 2 ]: T(f ξ 2 p=ξ p δp 4

5 If we replace each prime p in the product by ξ, then we get a lower bound: where D(f = ξ 2 p= ξ T ξ D, (3. δ p is the number, without multiplicity, of primes in the interval [ξ, ξ 2 ] that occur as cycle lengths. The remainder of the proof is a second moment argument: we calculate the mean and variance of D, and use Chebyshev s inequality to deduce a lower bound for D that holds with probability o(. ξ 2 E M (D = E M (δ p, and E M (δ p will be calculated using the Principle of Inclusion- p= ξ Exclusion (PIE. Let Z p be the set of all ( q p (q! possible p-cycles. If C is a cycle, let AC be the event that f has C as a cycle. Finally define ( r S r = M A i, {C,C 2,...,C r} where the sum is over all r element subsets of Z p. If p is prime, then by PIE, E M (δ p = ( j+ S j, and by the alternating inequalities j i= S S 2 E M (δ p S. (3.2 Because we assumed d = d(q > 2ξ 2, we can again use Lagrange Interpolation to estimate S and S 2. For each of the q(q (q 2p+ sets of two p-cycles, we can apply lemma 2. to get 2p 2 S 2 = q(q (q 2p+ <. Since S 2p 2 q 2p p 2 = ( q (p!, we have ( p p q p p q p S. For p [ξ p, ξ 2 ], we have p = o( 3 q. Therefore S ( p p = O(. Putting these estimates for S p p 3 p p 2 and S 2 into (3.2, we get E M (δ p = ( p + O. (3.3 p 2 Let π(x = the number of primes less than or equal to x so that π(k π(k = when k is a prime number and 0 otherwise and ξ 2 ξ 2 π(k π(k E M (D = + O. k k 2 Summing by parts, we get E M (D = ξ 2 k= ξ k= ξ k= ξ π(k k(k + + π( ξ 2 π( ξ ( + O. ξ 2 ξ ξ Using the Prime Number Theorem, in the weak form π(x = 5 x + O( x log(+x log 2 x, we get

6 ξ 2 ( E M (D = (k + log(k + + O. log ξ k= ξ Approximating the sum with an integral, we get E M (D = ξ 2 ξ ( ( dt t log t + O = 2 log L + O. log ξ log ξ For the variance, we first use first the identity E M (D(D = the identity By lemma lemma 2., E M (δ p δ p2 p,p 2 E M (δ p δ p2, and then V ar M (D = E M (D(D + E M (D E M (D 2. (3.4 C Z p,c 2 Z p2 M(A C A C2 < q! (p!(p 2! p!p 2!(q p p 2! q p +p 2 <. p p 2 Therefore E M (D(D ( ξ2 π(k π(k k k=ξ 2 ( ( 2 = E M (D + O. ξ Combining this with the identity (3.4, we get V ar M (D = E M (D( + O(/ξ. By Chebyshev s inequality, M(D log L = O( V ar M(D = O(. Therefore, with asymptotic probability O(, log (log L 2 log L L T ξ D ξ log L = exp ( log q ω. If d > exp( log q, and if ω = o(log q, then then hypothesis d 2ξ ω /3 2 is satisfied. therefore have the following corollary. We Corollary 3.2. If ω = ω(q is a nondecreasing function such that ω(q but ω = o(log q, and if If d = d(q > exp( log q ω /3, then for all sufficiently large q, E M (T > e log q ω. 6

7 4 Rational Functions If f and g are polynomials in F q [x], let δ(g = the degree of g and let mgcd(f, g be the greatest monic common divisor of f and g. Let Ω d = {(f, g : δ(f = δ(g = d and mgcd(f, g = and g is monic } It is known [3], [4] that = q 2d+ ( q 2. For each pair (f, g Ω d, the rational function R(x = f(x can be regarded as a function g(x from F q { } to F q { } with the convention that R(x = whenever g(x = 0 and R( = the leading coefficient of f. Note that, because of the conditions on f and g, the polynomials f and g are uniquely determined by the rational function R. Let P d be the uniform probability measure on Ω d, and let E d denote the corresponding expectation. Since V = F q { } is finite, the ultimate period T is a well-defined random variable. Our goal in this section is to prove a lower bound for E d (T. To avoid dealing with the exceptional point, define B(R = { 0 if cyclic(r T(R else Then T(R B(R for all R Ω d, and E d (T E d (B. Hence it suffices to prove a lower bound for E d (B. Next we prove a rational function analogue of lemma 2.. Lemma 4.. Let A F q, and let σ be a permutation of A. If d > 2 A, then ( A P d (R extends σ = q ( A + O. q Proof. P d (R extends σ = g {f : f g Ω d and f(x g(x = σ(x for all x A}, where the sum is over monic polynomials g, of degree d, that have no roots in A. As an auxiliary device, consider a larger ambient space in which the coprimality restriction is removed: Ω d = {(f, g : δ(f = δ(g = d and g is monic}. Then we can conveniently write P d (R extends σ = T T 2 T 3, (4. where T is an overestimate obtained by removing the mgcd condition, T 2 is the excess contribution from those pairs in Ω d Ω d with small gcd, and T 3 is the remaining contribution from pairs with larger mgcd. More precisely: T = {f : δ(f = d and f extends σ}, g g 7

8 where the sum is over monic, degree d polynomials g for which g(x 0 for all x A. T 2 = T 3 = A k=2 {h: δ(h=k, h monic} d k= A + {h: δ(h=k, h monic} {(f, g : gcd(f, g = h and f/h g/h {(f, g : gcd(f, g = h and f/h g/h extends σ} extends σ} By inclusion-exclusion, the number of degree d monic polynomials with no root in A is q d ( q A = q d ( + O( A. Because d > A, we can use Lagrange interpolation again: for q a given g, {f : δ(f = d and f extends σ} = g qd A (q. Thus ( A T = q ( A + O. (4.2 q If the degree of h = mgcd(f, g is k, then f = f/h and g = g/h have degree d k. Given k, there are q k choices of a monic polynomial h = mgcd(f, g. We again remove the mgcd condition to get an upper bound. Hence {(f, g : mgcd(f, g = h and f/h g/h extends σ} {(f, g : δ(f = δ(g = d k and f extends σ}. g Again by Lagrange interpolation, this is less than q d k A q d k. (We have d k > A because k A and d > 2 A. Therefore T 2 A k=2 q k q d k A q d k = q( q 2 A k=2 q k A = O ( T q 3. (4.3 Thus Finally T 3 is also negligible: T 3 q 2d+ ( q 2 d k= A + {h: δ(h=k, h monic} d k= A + q k q d k q d k T 3 = O {(f, g : mgcd(f, g = h } q( q 2 ( T d k= A + Putting (4.2, (4.3, and (4.4 into (4., we get lemma (4.. q 2 ( q k = O. q A +2. (4.4 8

9 With lemma 4. in hand, the proof of a lower bound for E d (B is essentially the same as in corollary 3.2. Taking σ to be a p-cycle, we can estimate the sum S = ( ( ( q p P d (C is a cycle of R = (p!q p + O = p ( p p q + O. (4.5 2 C Z p Taking σ to be an unordered pair of p-cycles, we get S 2 = P d (C, C 2 are cycles of R {C,C 2 } q(q (q 2p + q 2p 2p 2 ( + O ( p 2 < q p. 2 The remainder of the lower bound proof is verbatim the same as that for polynomials, so we have the following theorem. Theorem 4.2. If d = d(q > 2ξ 2, then for all sufficently large q, E d (T q /ω. Acknowledgement: We thank Igor Shparlinski and BIRS, for providing reference [7] and stimulating our interest in this subject. We also thank Derek Garton for helpful comments. References [] Bach, Eric, Toward a theory of Pollard s rho method, Inform. and Comput., Information and Computation, 90, (99, no.2, [2] Benedetto, Robert L. and Ghioca, Dragos and Hutz, Benjamin and Kurlberg, Pär and Scanlon, Thomas and Tucker, Thomas J., Periods of rational maps modulo primes, Math. Ann., 355, (203, no.2, [3] Benjamin, Arthur T. and Bennett, Curtis D., The probability of relatively prime polynomials, Math. Mag., 80, 2007, no.3, [4] Corteel, Sylvie and Savage, Carla D. and Wilf, Herbert S. and Zeilberger, Doron, A pentagonal number sieve, J. Combin. Theory Ser. A, 82 (998, no. 2, [5] Erdős, P. and Turán, P., On some problems of a statistical group-theory. III, Acta Math. Acad. Sci. Hungar., 8, 967, [6] Flajolet, Philippe and Odlyzko, Andrew M., Random mapping statistics, in Advances in cryptology EUROCRYPT 89 (Houthalen, 989, Lecture Notes in Comput. Sci., 434, , Springer, Berlin,

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