Stable Polynomials over Finite Fields

Transcription

1 Rev. Mat. Iberoam., 1 14 c European Mathematical Society Stable Polynomials over Finite Fiels Domingo Gómez-Pérez, Alejanro P. Nicolás, Alina Ostafe an Daniel Saornil Abstract. We use the theory of resultants to stuy the stability of an arbitrary polynomial f over a finite fiel F q, that is, the property of having all its iterates irreucible. This result partially generalises the quaratic polynomial case escribe by R. Jones an N. Boston. Moreover, for p = 3, we show that certain polynomials of egree three are not stable. We also use the Weil boun for multiplicative character sums to estimate the number of stable arbitrary polynomials over finite fiels of o characteristic. 1. Introuction For a polynomial f of egree at least 2 an coefficients in a fiel K, we efine the following sequence: f (0) (X) = X, f (n) (X) = f (n 1) (f(x)), n 1. A polynomial f is stable if f (n) is irreucible over K for all n 1. In this article, K = F q is a finite fiel with q elements, where q = p s an p an o prime. Stuying the stability of a polynomial is an exciting problem which has attracte a lot of attention. However, only few results are known an the problem is far away from being well unerstoo. The simplest case, when the polynomial is quaratic, has been stuie in several works. For example, some results concerning the stability over F q an Q can be foun in [3, 4, 8, 12, 13]. In particular, by [13, Proposition 2.3], a quaratic polynomial f(x) = ax 2 + bx + c K[X] over a fiel K of o characteristic an with the unique critical point γ, is stable if the set { af(γ)} {f (n) (γ) n 2} contains no squares. In the case when K = F q is a finite fiel of o characteristic, this property is also necessary. Mathematics Subject Classification (2010): Primary 11L40; Seconary 1T55, 11R09, 37F10. Keywors: finite fiels, irreucible polynomial, iterations of polynomials, iscriminant.

2 2 D. Gómez-Pérez, A. P. Nicolás, A. Ostafe an D. Saornil In [11] an estimate of the number of stable quaratic polynomials over the finite fiel F q of o characteristic is given, while in [2] it is prove that almost all monic quaratic polynomials f Z[X] are stable over Q. Furthermore, in [2] it is shown that there are no stable quaratic polynomials over finite fiels of characteristic two. One might expect that this is the case over any fiel of characteristic two, which is not true as it is also shown in [2] where an example of a stable quaratic polynomial over a function fiel of characteristic two is given. The goal of this paper is to characterize the set of stable polynomials of arbitrary egree an to evise a test for checking the stability of polynomials. Our techniques come from theory of resultants an they use the relation between irreucibility of polynomials an the properties of the iscriminant of polynomials. Using these techniques, we partially generalize previous results known for quaratic polynomials. A test for stability of quaratic polynomials was given in [15], where it was shown that checking the stability of such polynomials can be one in time q 3/4+o(1). As in [13], for an arbitrary polynomial f over F q, the set efine by {f (n) (γ 1 )... f (n) (γ k ) n 1 }, where γ i, i = 1,..., k, are the roots of the erivative of the polynomial f, plays also an important role in checking the stability of f. In particular, we use techniques base on resultants of polynomials together with the Stickelberger s theorem to prove our results. We introuce analogues of the orbit sets efine in [13] for arbitrary egree 2 polynomials. As in [15], we obtain a nontrivial estimate for the carinality of these sets for polynomials with irreucible erivative. We also give an estimate for the number of stable arbitrary polynomials which generalises the result obtaine in [11] for quaratic stable polynomials. The outline of the paper is the following: in Section 2 we introuce the preliminaries necessary to unerstan the paper. These inclue basic results about resultants an iscriminants of polynomials. This section ens with the Stickelberger s result. Next, Section 3 is evote to proving a necessary conition for the stability of a polynomial. We efine a set, which generalizes the orbit set for a quaratic polynomial, an then we give an upper boun on the number of elements of this set. Section 4 gives a new proof of the result that appeare in [2] for cubic polynomials when the characteristic is equal to 3. Finally, in Section 5 we give an estimate of the number of stable polynomials for any egree. For that, we relate the number of stable polynomials with estimates of certain multiplicative character sums. 2. Preliminaries Before proceeing with the main results, it is necessary to introuce some concepts relate to commutative algebra. Let K be any fiel an let f K[X] be a polynomial of egree with leaing coefficient a. The iscriminant of f, enote by

3 Stable Polynomials over Finite Fiels 3 Disc(f), is efine by Disc(f) = a 2 2 (α i α j ) 2, i<j where α 1,..., α are the roots of f in some extension of K. It is wiely known that for any polynomial f K[X], its iscriminant is an element of the fiel K. Alternatively, it is possible to compute Disc(f) using resultants. We can efine the resultant of two polynomials f an g over K of egrees an e, respectively, with leaing coefficients a an b e, as Res (f, g) = a e b e (αi β j ), where α i, β j are the roots of f an g, respectively. Like the iscriminant, the resultant belongs to K. In the following lemmas we summarize several known results about resultants without proofs. The intereste reaer can fin them in [7, 14]. Lemma 2.1. Let f, g K[X] be two polynomials of egrees 1 an e 1 with leaing coefficients a an b e, respectively. Let β 1,..., β e be the roots of g in an extension fiel of K. Then, Res (f, g) = ( 1) e b e e f(β i ). The behaviour of the resultant with respect to the multiplication is given by the next result. Lemma 2.2. Let K be any fiel. Let f, g, h K[X] be polynomials of egree greater than 1 an a K. The following hol: Res (fg, h) = Res (f, h) Res (g, h), Res (af, g) = a e Res (f, g), where eg g = e. The relation between Disc(f) an Res (f, f ) is given by the next statement. Lemma 2.3. Let K be any fiel an f K[X] be a polynomial of egree 2 with leaing coefficient a, non constant erivative f an eg f = k 1. Then, we have the relation Disc(f) = C f Res (f, f ), where C f = ( 1) ( 1) 2 a k 2. One of the main tools use to prove our main result regaring the stability of arbitrary polynomials is the Stickelberger s result, see [18] or [19, Corollary 1], which gives the parity of the number of istinct irreucible factors of a polynomial over a finite fiel of o characteristic.

4 4 D. Gómez-Pérez, A. P. Nicolás, A. Ostafe an D. Saornil Lemma 2.4. Suppose f F q [X], q o, is a polynomial of egree 2 an is the prouct of r pairwise istinct irreucible polynomials over F q. Then r mo 2 if an only if Disc(f) is a square in F q. To count the number of stable polynomials of a given egree we also nee the Weil boun for character sums, see [14, Chapter 5]. Lemma 2.5. Let χ be the multiplicative quaratic character of F q an let f F q [X] be a polynomial of positive egree that is not, up to a multiplicative constant, a square polynomial. Let be the number of istinct roots of f in its splitting fiel over F q. Uner these conitions, the following inequality hols χ(f(x)) x F q ( 1)q1/2. 3. Stability of arbitrary polynomials In this section we give a necessary conition for the stability of arbitrary polynomials. For this purpose, we use the following general result known as Capelli s Lemma, see [9]. Lemma 3.1. Let K be a fiel, f, g K[X], an let β K be any root of g. Then g(f) is irreucible over K if an only if both g is irreucible over K an f β is irreucible over K(β). We prove now one of the main results about the stability of an arbitrary polynomial. We note that our result partially generalises the quaratic polynomial case presente in [13] which is known to be necessary an sufficient over finite fiels. Theorem 3.2. Let q = p s, p an o prime, an f F q [X] a stable polynomial of egree 2 with leaing coefficient a, non constant erivative f an eg f = k 1. Then the following hol: 1. if is even, then Disc(f) an a k Res ( f (n), f ), n 2, are nonsquares in F q ; 2. if is o, then Disc(f) an ( 1) 1 squares in F q. 2 a (n 1)k+1 Res ( f (n), f ), n 2, are Proof. Let f F q [X] be a stable polynomial. We assume first that is even. We have that f (n) is irreucible for any n, an thus, by Capelli s Lemma 3.1, we know that f α is irreucible over F q n 1, where α is a root of f (n 1). By Lemma 2.4 this means that Disc(f α) is a nonsquare in F q n 1. Now, taking the norm over

5 Stable Polynomials over Finite Fiels 5 F q an using Lemma 2.3, we get Nm q n 1 q Disc(f α) = Disc(f α) = C f Res (f α, f ) α F q n 1 f (n 1) (α)=0 = C n 1 f Res α F q n 1 f (n 1) (α)=0 ( f (n 1) ) (f) = Cf n 1 Res, f A α F q n 1 f (n 1) (α)=0 (f α), f ( = A k Cf n 1 Res f (n), f ), where C f is efine by Lemma 2.3, A is the leaing coefficient of f (n 1) an Nm q n 1 q is the norm map from F q to F q. n 1 Since the norm Nm q n 1 q maps nonsquares to nonsquares, we obtain that A k Cf n 1 Res ( f (n), f ) is a nonsquare. Taking into account that A = a (n 1)/( 1) an the parity of the exponents involve, the result follows. The case of o can be treate in a similar way. Theorem 3.2 is interesting because it gives a metho for testing the stability of a polynomial. Lemma 2.1 says that the resultant is just the evaluation of f (n) in the roots of f multiplie by some constants. Taking into account this fact, the quaratic character of a an the exponents which are involve in Theorem 3.2, we have the following characterisation. Corollary 3.3. Let q = p s, p an o prime, an f F q [X] a stable polynomial of egree 2 with leaing coefficient a, non constant erivative f, eg f = k 1 an a k+1 the coefficient of X k+1 in f. Let γ i, i = 1,..., k, be the roots of the erivative f. Then the following hol: 1. if is even, then { (3.1) S 1 = a k contains only nonsquares in F q ; 2. if is o, then { (3.2) S 2 = contains only squares in F q. } { k f (n) (γ i ) n > 1 ( 1) 2 a k ( 1) ( 1) 2 +k (k + 1)a k+1 a (n 1)k+1 k f(γ i ) } } k f (n) (γ i ) n 1

6 6 D. Gómez-Pérez, A. P. Nicolás, A. Ostafe an D. Saornil Proof. The result follows irectly from Theorem 3.2 an Lemma 2.1. We note that the converse of Corollary 3.3 is not true. Inee, take any with gc(, q 1) = gc(, p) = 1, F q an extension of even egree of F p an a 0 a quaratic resiue in F q. Let us consier the polynomial f(x) = (X a 0 ) + a 0 F q [X]. It is straightforwar to see that f (n) (X) = (X a 0 ) n + a 0 an that the set (3.2) is {( 1) 1 2 a 1 0 }. We note that the polynomial f is reucible. Inee, let the integer 1 e q 1 be such that e = 1 (mo q 1). Then (a e 0) = a 0, an thus a e 0 + a 0 is a root of f. On the other han, since 1 an are squares in F q because both elements belong to F p an F q is an extension of even egree, the set (3.2) contains only squares. We finish this section by showing that, when the erivative f of the stable polynomial f is irreucible, the sets (3.1) an (3.2) are efine by a short sequence of initial elements. The proof follows exactly the same lines as in the proof of [15, Theorem 1]. Inee, assume eg f = k an γ 1,..., γ k F q k are the roots of f. Using Corollary 3.3 we see that the sets (3.1) an (3.2) contain only nonsquares an squares, respectively, an thus, the problem reuces to the cases when f (n) (γ 1 )... f (n) (γ k ) are either all squares or all nonsquares for any n 1. It is clear that, when f is irreucible, taking into account that γ i = γ qi 1, i = 1,..., k 1, we get for every 1 n N, f (n) (γ 1 )... f (n) (γ k ) = f (n) (γ 1 )... f (n) (γ qk 1 1 ) = f (n) (γ 1 )... f (n) (γ 1 ) qk 1 = Nm qk qf (n) (γ 1 ). Applying now the same technique with multiplicative character sums as in [15, Theorem 1] (as the argument oes not epen on the egree of the polynomial f), we obtain the following estimate: Theorem 3.4. For any o q an any stable polynomial f F q [X] with irreucible erivative f, eg f = k, there exists N = O (q 3k/4) such that for the sets (3.1) an (3.2) we have S 1 = S 2 = { { a k } { k f (n) (γ i ) 1 < n N ( 1) 2 a k ( 1) ( 1) 2 +k (k + 1)a k+1 a (n 1)k+1 k f(γ i ) } } k f (n) (γ i ) 1 n N.,

7 Stable Polynomials over Finite Fiels 7 4. Non-existence of certain cubic stable polynomials when p=3 The existence of stable polynomials is ifficult to prove. For p = 2, there are no stable quaratic polynomials as shown in [1], whereas for p > 2, there is a big number of them as is shown in [11]. In this section, we show that for certain polynomials of egree 3, f (3) is a reucible polynomial when p = 3. This result also appears in [2], but we think this approach uses new ieas that coul be of inepenent interest. For this approach, we nee the following result which can be foun in [6, Corollary 4.6]. Lemma 4.1. Let q = p s an f(x) = X p a 1 X a 0 F q [X] with a 1 a 0 0. Then f is irreucible over F q if an only if a 1 = b p 1 an Tr q p (a 0 /b p ) 0, where Tr q p represents the trace map of F q over F p. Base on Lemma 4.1, we can present an irreucibility criterium for polynomials of egree 3 in characteristic 3. Lemma 4.2. Let p = 3 an q = 3 s. Then f(x) = X 3 a 2 X 2 a 1 X a 0 is irreucible over F q if an only if 1. a 1 = b 2 an Tr q 3 (a 0 /b 3 ) 0, if a 2 = 0 an a 1 0; 2. a 4 2/(a 2 2a a 3 1 a 0 a 3 2) = b 2 an Tr q 3 (1/a 2 b) 0, if a 2 0. Proof. The case a 2 = 0 is a irect application of Lemma 4.1. In the other case, we take the polynomial f(x + a 1 /a 2 ) = (X + a 1 /a 2 ) 3 a 2 (X + a 1 /a 2 ) 2 a 1 (X + a 1 /a 2 ) a 0 = X 3 a 2 X 2 a 0 + a 2 1/a 2 + a 3 1/a 3 2 = X 3 a 2 X 2 + (a 2 1a a 3 1 a 0 a 3 2)/a 3 2. Notice that f(x + a 1 /a 2 ) is irreucible if an only if f(x) is irreucible. We enote g(x) = f(x + a 1 /a 2 ) to ease the notation an g the reciprocal polynomial of g, i. e. ( ) 1 g (X) = X 3 g. X By [14, Theorem 3.13], g is irreucible if an only if g is. Applying Lemma 4.1, we get the result. For simplicity, we prove an irreucibility criterium for monic polynomials, however the proof hols for non-monic polynomials as well taking into account the principal coefficient. Using Lemma 4.2 an following the same lines as in [1], we can prove now the following result. Theorem 4.3. For any polynomial f F 3 [X] of the form f(x) = a 3 X 3 a 1 X a 0, at least one of the following polynomials f, f (2) or f (3) is a reucible polynomial.

8 8 D. Gómez-Pérez, A. P. Nicolás, A. Ostafe an D. Saornil Proof. Assume that f, f (2), f (3) are all irreucible polynomials. Using Lemma 3.1, f (3) is irreucible if an only if f (2) is irreucible over F q an f γ is irreucible over F q 9, where γ is a root of f (2). Thus, the monic polynomial h = f γ a 3 is irreucible over F q 9 ( ) an we can apply now Lemma 4.2 from where we get that Tr a0 γ q9 3 a 3b 0, where b 2 = a 3 1 an b F q 9. Notice that b F q. Inee, as b is the root of the polynomial X 2 a 1, then either b F q or b F q 2. Since b F q 9 we obtain that b F q. Using the properties of the trace map we obtain ( ) ( ) a0 γ γ Tr q 9 3 a 3 b 3 = Tr q 9 3 a 3 b 3, an from here we conclue that the right han sie of the last equation is non zero. Using now the transitivity of the trace, see [14, Theorem 2.26], we get ( ) ( ( )) ( ) γ γ Trq Tr q9 3 a 3 b 3 = Tr q 3 Tr q9 q a 3 b 3 = Tr 9 q( γ) q 3 a 3 b 3. Now, f (2) is an irreucible polynomial with roots γ, γ q,..., γ q8. Thus, Tr q9 q(γ) is given by the coefficient of the term X 8 in f (2), which is zero. This shows that Tr q 9 3(γ) = 0, which is a contraiction with the fact that f (3) is irreucible. We note that Theorem 4.3 cannot be extene to infinite fiels. As in [2], let K = F 3 (T ) be the rational function fiel in T over F 3, where T is transcenental over F 3. Take f(x) = X 3 + T K[X]. Then it is easy to see that f (n) (X) = X 3n + T 3n 1 + T 3n T 3 + T. Now from the Eisenstein criterion for function fiels (see [17, Proposition III.1.14], for example), it follows that for every n 1, the polynomial f (n) is irreucible over K. Hence, f is stable. 5. On the number of stable polynomials In this section we obtain an estimate for the number of stable polynomials of a given egree. We use Corollary 3.3 as our main tool. For a given, let f(x) = a X + a 1 X a 1 X + a 0 F q [X] an we efine k F l (a 0,..., a ) = f (l) (γ i ), which is a polynomial in the variables a 0,..., a an with coefficients in F q. Following [15], the number of stable polynomials of egree, which will be enote by S, satisfies the inequality (5.1) S 1 2 K a 0 F q a F q l=1 K (1 ± χ(f l (a 0,..., a ))),

9 Stable Polynomials over Finite Fiels 9 where χ is the multiplicative quaratic character of F q an K is an arbitrary positive integer. The sign of χ epens on an is chosen in orer to count the elements of the orbit of f which satisfy the conition of stability. Since the upper boun of S is inepenent of this choice, let us suppose from now on that χ is taken with +. If we expan an rearrange the prouct, we obtain 2 K 1 sums of the shape µ χ F lj (a 0,..., a ), 1 l 1 < < l µ K, a 0 F q j=1 a F q with µ 1 plus one trivial sum correponing to 1 in (5.1). The upper boun for S will be obtaine using Lemma 2.5. This result can only be use when µ j=1 F l j (a 0,..., a ) is not a square polynomial with respect to some variable. The next lemmas are use to estimate the number of values for a 0,..., a i 1, a i+1,..., a such that the resulting polynomial in some variable a i is a square. The first lemma is a boun on the number of common zeros of two multivariate polynomials. For a proof, we refer the reaer to [10]. Lemma 5.1. Let F (Y 0, Y 1,..., Y ), G(Y 0, Y 1,..., Y ) be two polynomials of egree 1 an 2, respectively, in + 1 variables with gc (F (Y 0, Y 1,..., Y ), G(Y 0, Y 1,..., Y )) = 1. Then, the number of common roots in F q is boune by 1 2 q 1. Base on Lemma 5.1, the next result shows that, if the egree of a polynomial G(Y 0, Y 1,..., Y ) in a variable Y i is greater than 1, then it is possible to boun the number of ba choices for a 0,..., a i 1, a i+1,..., a, that is, the number of choices for a 0,..., a i 1, a i+1,..., a such that G(a 0,..., a i 1, Y i, a i+1,..., a ) is a square polynomial in Y i. Lemma 5.2. Let G F q [Y 0,..., Y ] be a polynomial of egree D, which is not a square polynomial in the algebraic closure of F q. Then there exists i {0,..., } such that G(a 0,..., a i 1, Y i, a i+1,..., a ) is not a square polynomial in Y i for all but at most O(D 2 q 1 ) values of a 0,..., a i 1, a i+1,..., a F q. Proof. Let G(Y 0,..., Y ) = G 1 (Y 0,..., Y ) 1 G h (Y 0,..., Y ) h the ecomposition of the polynomial in a prouct of irreucible polynomials. Without loss of generality, 1 is o because G is not a square of a polynomial up to a multiplicative constant. Moreover, because G 1 is an irreucible factor of G, then eg G 1 D. We suppose that G 1 (Y 0,..., Y ) epens on some variable Y i an use it to count the number of choices for a 0,..., a i 1, a i+1,..., a such that G(a 0,..., a i 1, Y i, a i+1,..., a ) is a constant polynomial, G(a 0,..., a i 1, Y i, a i+1,..., a ) is a nonconstant square polynomial up to a multiplicative constant in the variable Y i.

10 10 D. Gómez-Pérez, A. P. Nicolás, A. Ostafe an D. Saornil There are at most Dq 1 ifferent choices of a 0,..., a i 1, a i+1,..., a when the polynomial can be a constant. Now, we consier in which cases the polynomial is a square of a polynomial when we substitute a 0,..., a i 1, a i+1,..., a an how these cases will be counte. We have the following two possible situations: G 1 1 is a square, nonconstant polynomial, an because 1 is not even, then we must have that G 1 has at least one multiple root as a polynomial in Y i. This is only possible if G 1 an the first erivative with respect to the variable Y i of G 1 have a common root. Since G 1 is an irreucible polynomial, Lemma 5.1 applies. We remark that the first erivative is a nonzero polynomial. Otherwise G 1 is a reucible polynomial. This can only happen in (eg G 1 )(eg G 1 1)q 1 cases. G 1 an G j have a common root for some 1 j h. In this case, using the same argument, there are at most (eg G 1 )(eg G j )q 1 possible values for a 0,..., a i 1, a i+1,..., a where it happens. This conclues the proof. From Lemmas 2.5 an 5.2, we have the following corollary. Corollary 5.3. If G(Y 0,..., Y ) is a polynomial of egree D, which is not a square polynomial in the algebraic closure of F q, then χ(g(a 0,..., a )) = O (Dq +1/2) a 0,...,a F q where χ is the quaratic character of F q. Proof. The proof follows irectly by applying Lemma 2.5 for those polynomials which are nonsquares in some variable a i. Since these polynomials have egree at most D in the ineterminate a i (see the proof of Lemma 5.2), we obtain O(Dq +1/2 ) for this part. For the rest, that is, the square polynomials in the variable a i, we can apply Lemma 5.2 an use the trivial boun for O(D 2 q 1 ) values of a 0,..., a i 1, a i+1,..., a. So the total boun becomes O ( Dq +1/2 + D 2 q ). Noticing that for D > q 1/2 the claime result is weaker that the trivial boun q +1, we conclue the proof. To use Corollary 5.3 in counting the number of stable polynomials of egree, we nee the following lemma. Lemma 5.4. There exists i = 1,..., m such that, for fixe integers l 1,..., l µ with 1 l 1 < < l µ K, there are at most O( 2K q 1 ) choices for a 0,..., a i 1, a i+1,..., a such that the polynomial µ F lj (a 0,..., a i 1, A i, a i+1,..., a ) j=1 is a square polynomial in the variable A i up to a multiplicative constant.

11 Stable Polynomials over Finite Fiels 11 Proof. The proof follows from Lemma 5.2. polynomial (5.2) µ F lj (A 0,..., A ) j=1 For this we have to prove that the is not a square polynomial as a multivariate polynomial, up to a multiplicative constant, an, to obtain this, it is enough to prove it for particular choices of the variables. If the egree of f is even an coprime to p, we consier the polynomial f = (X B) +C +B, where B, C are consiere as variables. Then f = (X B) 1 (here f represents the erivative with respect to the variable X) an f (n) (B) = B + H n (C), where eg H n (C) = n 1. Thus, as is even, we have µ F lj (A 0,..., A ) = j=1 µ (B + H lj (C)) 1, which is not a square polynomial as a multivariate polynomial up to a multiplicative constant. If is o, coprime to p, we consier the following polynomial f = (X B) 1 (X B + 1) + C + B with the erivative f = (X B) 2 ((X B) + 1). Notice that, if the egree of this polynomial is coprime to the characteristic p, then f has two ifferent roots B, B + (1 ) 1. Substituting these in the polynomial f, we get j=1 f (n) (B) = B + H n (C), f (n) (B + (1 ) 1 ) = B + L n (C), where L n (C) H n (C) an eg L n (C) = eg H n (C) = n 1. The fact that L n (c) H n (c) comes from the following observation: (5.3) H n (C) = (H n 1 (C)) 1 (H n 1 (C) + 1) + C, L n (C) = (L n 1 (C)) 1 (L n 1 (C) + 1) + C, where H 1 (C) = C, an L 1 (C) = C + (1 ) 2. It is clear that H 1 (C) L 1 (C), so now we suppose that H n (C) = L n (C) an using the equation (5.3), we get an thus (H n 1 (C)) 1 (H n 1 (C) + 1) + C = (L n 1 (C)) 1 (L n 1 (C) + 1) + C (C + C 1 ) H n 1 (C) = (C + C 1 ) L n 1 (C). Applying now the Ritt ecomposition theorem, see [16], we obtain H n 1 (C) = L n 1 (C).

12 12 D. Gómez-Pérez, A. P. Nicolás, A. Ostafe an D. Saornil In this case, as is o, we have µ F lj (A 0,..., A ) = j=1 µ (B + H lj (C)) 2 (B + L lj (C)), j=1 which is not a square polynomial as a multivariate polynomial up to a multiplicative constant. When the egree of f is not coprime to the characteristic of the fiel, take f = (X B) + (X B) 2 + C + B an one can prove following the same path as for the last two cases that the polynomial (5.2) is not a perfect square as a multivariate polynomial up to a multiplicative constant. The result now follows by applying Lemma 5.2 to the polynomial (5.2). In this case, as in the proof of Lemma 5.2, because G 1 is an irreucible factor of the polynomial (5.2), then there exists 1 j µ such that G 1 is an irreucible factor of the polynomial F lj (A 0,..., A ), which implies that eg G 1 K. Now we are able to fin a boun for S, the number of stable polynomials of egree. Theorem 5.5. The number of stable polynomials of egree is O ( q +1 1/2 log(2)). Proof. The trivial summan of (5.1) can be boune by O(q +1 /2 K ). other terms, we apply Corollary 5.3 an Lemma 5.4. Then, For the S = O(q +1 /2 K + K q +1/2 ). Choosing K = log q/2 log(2) the result follows. Unfortunately we have not been able to give a lower boun for S similar to the quaratic case obtaine in [11, Theorem 1]. This is because we o not have a necessary an sufficient conition for the stability of polynomials of egree > 2. We can however show that very frequently we have S ϕ(q 1), where ϕ(k) is the Euler function, which comes from the following construction. Assume that a positive integer an b F q are such that the binomial X + b is irreucible over F q. By [14, Theorem 3.75], we know that X + b is irreucible if an only if each prime factor of ivies the orer e of b, but not (q 1)/e, an q 1 (mo 4) if t 0 (mo 4). If (q 1) an b is a primitive element of F q, then X + b, an thus also f = (X b) + b F q [X], are irreucible. Furthermore, one can easily prove that f (n) = (X b) n + b is also irreucible for every n 2. Since there are ϕ(q 1) primitive elements in F q we obtain S ϕ(q 1). Acknowlegement The authors woul like to thank Igor Shparlinski for useful iscussions an comments which improve the presentation of the paper an for several corrections in Section 5. We are also grateful to the referee for the useful suggestions.

13 Stable Polynomials over Finite Fiels 13 A. N. was supporte by MTM C04-01, A. O. was supporte by SNSF Grant an D. S. was supporte by MTM C02-02 an MTM References [1] O. Ahmai. A note on stable quaratic polynomials over fiels of characteristic two. Preprint, 2010 (available from [2] O. Ahmai, F. Luca, A. Ostafe, I. Shparlinski. On stable quaratic polynomials. Glasgow Math. J., 54:59-369, [3] N. Ali. Stalilité es polynômes. Acta Arithmetica, 119:53 63, [4] M. Aya D. L. McQuillan. Irreucibility of the iterates of a quaratic polynomial over a fiel. Acta Arithmetica, 93(1):87 97, [5] E. R. Berlekamp. Algebraic coing theory. McGraw-Hill Book Co., New York, [6] I. Blake, X. Gao, A. Menezes R. Mullin. Application of finite fiels. Kluwer, [7] D. Cox, J. Little, D. O Shea. Ieals, varieties, an algorithms. An introuction to computational algebraic geometry an commutative algebra.. Unergrauate Texts in Mathematics. Springer, New York [8] L. Danielson B. Fein. On the irreucibility of the iterates of x n b. Proc. Am. Math. Soc., 130(6): , [9] B. Fein M. Schacher. Properties of iterates an composites of polynomials. J. Lonon Math. Soc., 54(3): , [10] J. von zur Gathen J. Gerhar. Moern computer algebra. Cambrige University Press, [11] D. Gómez A. P. Nicolás. An estimate on the number of stable quaratic polynomials. Finite Fiel an their Applications, 16(6): , [12] R. Jones. The ensity of prime ivisors in the arithmetic ynamics of quaratic polynomials. J. Lon. Math. Soc., 78: , [13] R. Jones N. Boston. Settle polynomials over finite fiels. Proc. Amer. Math. Soc., 140: , [14] R. Lil H. Nieerreiter. Finite fiels, volume 20 of Encyclopeia of Mathematics an its Applications. Cambrige University Press, Cambrige, [15] A. Ostafe I. Shparlinski. On the length of critical orbits of stable quaratic polynomials. Proc. Amer. Math. Soc., 138(8): , [16] J. Ritt. Prime an composite polynomials. Trans. Amer. Math. Soc., 23:51 66, [17] H. Stichtenoth. Algebraic function fiels an coes. Springer-Verlag, Berlin, [18] L. Stickelberger. Uber eine neue eigenschaft er iskriminanter algebraischer zahlkorper. Verh. 1 Internat. Math. Kongresses, [19] R. G. Swan, Factorization of polynomials over finite fiels, Pacific J. Math., 12: , Receive??

14 14 D. Gómez-Pérez, A. P. Nicolás, A. Ostafe an D. Saornil Domingo Gómez-Pérez: Department of Mathematics, University of Cantabria, Santaner 39005, Spain Alejanro P. Nicolás: Departamento e Matemática Aplicaa, Universia e Vallaoli, Spain anicolas@maf.uva.es Alina Ostafe: Department of Computing, Macquarie University, Syney, NSW 2109, Australia alina.ostafe@mq.eu.au Daniel Saornil: Department of Mathematics, University of Cantabria, Santaner 39005, Spain saornil@unican.es