On irreducible divisors of iterated polynomials

D. Gómez-Pérez, A. Ostafe, A. P. Nicolás and D. Sadornil have recently shown that for almost all polynomials f ∈ Fq[X] over the finite field of q elements, where q is an odd prime power, their iterates eventually become reducible polynomials over Fq. Here we combine their method with some new ideas to derive finer results about the arithmetic structure of iterates of f . In particular, we prove that the nth iterate of f has a square-free divisor of degree of order at least n as n → ∞ (uniformly in q).

The polynomial f (n) is called the nth iterate of the polynomial f .
Gómez-Pérez and Nicolás [7], developing some ideas from [16], prove that there are O(q 5/2 (log q) 1/2 ) stable quadratic polynomials over a finite field F q of q elements for an odd prime power q, where the implied constant is absolute. We also note that in [8] an upper bound is given on the number of stable polynomials of degree d 2 over F q .
Here, we continue to study the arithmetic properties of iterated polynomials and obtain several new results about their multiplicative structure.
First, we combine the method of Gómez-Pérez and Nicolás [7] with some new ideas to show that, if q is odd, then for almost all quadratic polynomials f ∈ F q [X] the number r n (f ) of irreducible divisors of the nth iterate f (n) grows at least linearly with n if n is of order at most log q. Our tools to prove this are resultants of iterated polynomials, the Stickelberger's theorem [19] and estimates of certain character sums.
For the values of n beyond this threshold, we use a different technique, related to Mason's proof of the ABC-conjecture in its polynomial version, see [13], [18], to give a lower bound on the largest degree D n (f ) of the irreducible divisors of f (n) . It is interesting to recall that Faber and Granville [4] have used (in a different way) the classical version of the ABC-conjecture for the integers to study the arithmetic of elements in the orbits of polynomial dynamical systems over Z.
Note that our lower bound on D n (f ) is reminiscent of lower bounds on the largest prime divisor of nonlinear recursive sequences over the integers, see [4], [10], and [17].
Our approach and some results used to derive lower bounds on r n (f ) and D n (f ) are readily combined to obtain the lower bound n 1+o(1) as n → ∞ (uniformly in q) on the largest degree of square-free divisors of f (n) .
The outline of the paper is the following. In Section 2 we give the notation used throughout the paper as well as collect some basic properties needed in the proofs of the main results. In Section 3, we collect all results about discriminants and then, in Section 4, we provide bounds on character sums related with discriminants of iterated polynomials. In Section 5 we recall the result of Mason [13]. These preliminary results are used in the following sections. More precisely, Section 6 contains an estimate of the number of distinct irreducible factors of a polynomial iterate. In Section 7 we show that, if f = f d X d , then there is always an irreducible factor of large degree for high order iterates of the polynomial f . Finally, in Section 8 we combine both approaches and also use some of the previous results to derive some nontrivial information about the arithmetic structure of f (n) that applies to any n.

Notation
Let p be an odd prime number and let q = p s for some positive integer s. We denote by F q the finite field of q elements and by χ the quadratic character of F q .
We use F q [X] to denote the ring of polynomials with coefficients in F q . Polynomials in this ring are denoted by the letters f , g and h. We usually use f 0 , . . . , f d to represent the coefficients of a polynomial f ∈ F q [X], that is, Throughout the paper the implied constants in symbols 'O' and ' ' may occasionally, where obvious, depend on a small positive parameter ε but are absolute otherwise (we recall that A = O(B) and B A is equivalent to |A| cB for some positive constant c). Also, we write F (n) = o(G(n)) as n → ∞, which means that

Discriminants and iterates of polynomials
We use the following well-known properties of the discriminant Disc (f ) and the resultant Res (f, g) of polynomials f, g ∈ K[X], see [6], [20], that hold over any field K.
From the definition of the resultant, it is clear that two polynomials f and g are coprime if and only if Res f, g = 0.
To study the discriminants of iterates of polynomials, it is necessary to have a close-form formula for the resultant of polynomials under compositions. In [14], the following chain rule for resultants is proved.

Lemma 2. Let f and g be as in Lemma 1 and let h ∈ K[X] with deg h = and leading coefficient h . Then
It is clear from Lemma 2 that f and g are coprime if and only if for any nonconstant polynomial h we have Res f (h), g(h) = 0 (note that this is also a consequence of the Euclidean algorithm).
Also, Lemma 2 implies the following formula for the discriminant of polynomial iterates.
are the roots of the derivative f . Then, for n 1, we have and we also have Indeed, one can prove this by induction over n and we show it only for deg(f (n) ) as the formula (3.1) for the leading coefficient of f (n) can be obtained using the same idea. As deg f = k, for n = 1 the formula (3.2) is true. We assume that (3.2) is true also for the first n − 1 iterates. We have Thus, applying Lemma 1 (i) we derive Taking into account that f (n) = f · f (n−1) (f ) and applying Lemma 1 (iii) and Lemma 2, we derive while by Lemma 1 (ii) we obtain Substituting (3.5) and (3.6) in (3.4) and using (3.3), we finish the proof. 2 We also note that a similar computation has been given by Jones and Manes (see [12], Lemma 3.1 and Theorem 3.2) for iterated rational functions. For defined as in Lemma 3, it is convenient to introduce the following notation where γ i , i = 1, . . . , k, are the roots of f , which is clearly a polynomial in f d , . . . , f 0 and having the degree O(d n ) in the variable f 0 . We need the following result, which has been proved in [8], Lemma 5.2: K 1 and k 1 , . . . , k μ such that 1 k 1 < · · ·< k μ K, the polynomial

Lemma 4. For fixed integers
is a square polynomial in the variable f 0 up to a multiplicative constant only for

Bounds of some character sums
For an integer n we consider the sums with the quadratic character χ of F q , where k is as in Lemma 3.
For any integer n 1, we have Proof. Squaring and changing the order of summation, we obtain . . , f 0 )) .
We consider the following three cases: Combining the preceding observations, we obtain T 1 (n) = O n 2 d n q d+1/2 + n 2 d 2n q d + n q d+1 , and the first part of the result follows. By the same argument (with some natural simplifications due to a simpler shape of the sum T 2 (n)), we obtain the same estimate for T 2 (n). 2

Polynomial ABC theorem and divisors of iterated polynomials
Some of our results are also based on the Mason theorem [13] that gives a polynomial version of the ABC conjecture, see also [18].
For a polynomial f ∈ F q [X], we denote the product of all monic irreducible divisors of f by rad(f ). Recall that we denote the largest degree of irreducible factors of f (n) by D n (f ). In order to apply Lemma 6 we need the following simple statement.

Lemma 7. For a nonconstant polynomial
Proof. Now assume that D n−1 (f ) = D for some positive integer D. Let g ∈ F q [X] be an irreducible divisor of f (n−1) with deg g = D. Then we obviously have g(f ) | f (n) . Now, if g(f ) has a root α ∈ F q m then g has a root f (α) in F q m too. Because g is irreducible, we have m deg g. Thus g(f ) has an irreducible factor of degree at least D. 2 We denote by Δ n (f ) the largest degree of a square-free divisor of f (n) . That is, Δ n (f ) = deg rad(f (n) ).

Lemma 8. For a nonconstant polynomial
Proof. Assume that where A is the leading coefficient of f (n−1) (see (3.1) for an explicit formula) and g 1 , . . . , g s are the distinct monic irreducible divisors of f (n−1) of multiplicities α 1 , . . . , α s , respectively, with As g 1 , . . . , g s are relatively prime, we see from Lemma 2 that the polynomials g 1 (f ), . . . , g s (f ) are also relatively prime. Thus As in the proof of Lemma 7 we see that deg rad(g i (f )) deg g i , i = 1, . . . , s, which concludes the proof. 2

Growth of the number of irreducible factors under iteration for small n
. We recall that r n (f ) denotes the number of monic irreducible divisors of f (n) . Using the remark after Lemma 2, we have that if g 1 and g 2 are two different irreducible prime factors of f (n) , then g 1 (f ) and g 2 (f ) are coprime. Clearly, this means that r n (f ) is a nondecreasing function. Now, we show that r n (f ) grows at least linearly for n of order at most log q. Proof. Clearly we can discard q d polynomials f with f (0) = 0. We consider first the case when d is even. In this case, . . , f 0 )) = χ(Disc f ( ) ).
We also discard the polynomials f = f d X d + · · · + f 1 X + f 0 ∈ F q [X] corresponding to tuples (f d , . . . , f 0 ) for which . . , f 0 ) ∈ F d+1 q satisfying (6.1). Thus, we see that there are O(n d n q d ) = o(q d+1 ) such polynomials (note that since a zero polynomial is a square polynomial this also follows from Lemma 4).
For odd d we note that r (f ) and r +1 (f ) are of different parity when . . , f 0 ) = −1 and proceed in exactly the same way using Lemma 5 for the sum T 2 . 2

Lower bound on the degrees of irreducible factors of iterates for large n
Recall that for a polynomial f ∈ F q [X] we denote by D n (f ) the largest degree of irreducible factors of f (n) . We are now ready to prove the main result of this section. Then Proof. First, we note that by Lemma 3, Disc f (n) = 0 is possible only if Thus, as in the proof of Theorem 9 (where we count the number of solutions to (6.1)), we see that for any fixed ε > 0, for all but o(q d+1 ) polynomials f ∈ F q [X] of degree d, for every n L with we have Disc f (n) = 0 and thus Δ n (f ) = d n . Therefore, for every n q 1/2 , since by Lemma 8 we know that Δ n (f ) is monotonic, for all but o(q d+1 ) polynomials f ∈ F q [X] of degree d we have For n > q 1/2 , by Theorem 10, for all but Δ n (f ) D n (f ) 1 log q n n log n n 1−ε .

Comments and open questions
We note that an analogue of Theorems 9 and 11 can also be obtained for almost all monic polynomials. Probably the most interesting question is to extend the bound of Theorem 9 to any n (beyond the current threshold n = O(log q)).
Although we do not know how to obtain such a result, we can construct some examples of polynomials for which r n grows linearly (which, as we have mentioned, appears to the expected rate of growth). Indeed, take any quadratic polynomial f (X) = X 2 + 2aX + a 2 − a ∈ F q [X] with a ∈ F q and set γ = −a. Clearly f (γ) = γ.
So, if −1 is a nonsquare in F q (for example, for a prime q = p ≡ 3 (mod 4)), then Disc f (n) is a square or a nonsquare depending only on the parity of n. Therefore, for this polynomial we have r n (f ) n for any n 1. A concrete example is given by f (X) = X 2 + X + 2 ∈ F 3 [X] (we take a = 2 in the above construction).
In [11] the critical orbit of a quadratic polynomial f is defined as the set {f (n) (γ) | n 2} ∪ {−f (γ)}, where γ is the root of the derivative. This coincides with the set It is certainly interesting to investigate various properties of the sequence u n = G n (f 0 , . . . , f d ) for fixed f 0 , . . . , f d ∈ F q . Currently, most of the known results concern only quadratic polynomials. For example, the sequence u n becomes eventually periodic when d = 2. If f is an irreducible polynomial of degree k, then G n (f 0 , . . . , f d ) = Norm F q k /Fq (f (n) (γ)) is the norm of f (n) (γ) in F q . Apart from these two cases, very little is known about the sequence u n for general polynomials f .
The sparsity, or number of monomials, is another important characteristic of polynomials and it is certainly interesting to obtain lower bounds on the number of monomials of the iterates f (n) . For iterates of polynomials and even rational functions over a field of characteristic zero such bounds can be derived from the results of [5].
Finally, we note that similar questions can also be asked for iterates of rational functions, which is yet another challenging direction of research.