ON POWER VALUES OF POLYNOMIALS. A. Bérczes, B. Brindza and L. Hajdu

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1 ON POWER VALUES OF POLYNOMIALS ON POWER VALUES OF POLYNOMIALS A. Bérczes, B. Brindza and L. Hajdu Abstract. In this paper we give a new, generalized version of a result of Brindza, Evertse and Győry, concerning superelliptic equations. Let f(x) Z[x] be a polynomial of degree n and b be a nonzero integer. For effective upper bounds obtained by Baker s method for the exponent z in the equation () f(x) = by z, x, y, z Z with y >, z > we refer to [T], [ST], [Tu], [Tu2], [ShT], [B], [BEGy], [Bu]. For a polynomial P let M(P ) denote the Mahler height of it (cf. [M]). The purpose of this paper, which is related to a recent observation of Brindza on the number of solutions of generalized Ramanujan - Nagell equations [B3], is to derive a bound for z which is polynomial in M(f). For brevity write M = M(f). Theorem. If f has at least two distinct zeros, then z < cm 3n log 3 2b, where c is an effectively computable constant depending only on n. Remarks. If f is an irreducible monic and b = then this inequality was proved by Brindza, Győry and Evertse with different constants (see [BEGy], Th. 4). Moreover, if n > 2 and f is irreducible then a profound result of Győry (cf. [Gy] or [Gy2]) makes it possible to substitute cm 3n by an effective constant depending only on the discriminant of f. Mathematics Subject Classification: D4 Keywords and phrases: Diophantine equations, superelliptic equations. Research supported in part by Grant from the Hungarian National Foundation for Scientific Research and by the Universitas Foundation of Kereskedelmi Bank RT. Research supported in part by the Hungarian Academy of Sciences and by Grant D from the Hungarian National Foundation for Scientific Research. Research supported in part by the Hungarian Academy of Sciences and by Grants and T from the Hungarian National Foundation for Scientific Research. Typeset by AMS-TEX

2 2 A. BÉRCZES, B. BRINDZA AND L. HAJDU Auxiliary results To prove our Theorem, we need two lemmas. In what follows, for any non-zero algebraic number α, h(α) and H(α) denotes the logarithmic height and the classical (ordinary) height of α, respectively. Lemma. Let K be an algebraic number field of degree n and denote by R and r the regulator and the unit rank of K, respectively. There exists a fundamental sytem of units ε,..., ε r for K so that h(ε i ) c R, i =,..., r where c is an effectively computable constant depending only on n. Proof. This statement is a consequence of Lemma in [BGy]. For other versions of this result cf. [B2] or [H]. Lemma 2. Let α,..., α n be nonzero algebraic numbers and let A,..., A n be positive real numbers with A i max{h(α i ), e} for i =,..., n. Furthermore, let b,..., b n be rational integers with α b... αbn n and suppose that B is a positive real number satisfying B max b i and B e. Now we have i=,...,n α b... αbn n B c log A... log A n, where c is an effectively computable constant depending only on n and on the degree of Q(α,..., α n ) over Q. Proof. This is Theorem.2 in [PW]. Proof of the Theorem We have two cases to distinguish. First we assume that f has an irreducible factor P Z[x] of degree t 2. Let α be a zero of P, moreover, let R, h, D and r be the regulator, class number, discriminant and unit rank of the field K = Q(α), respectively. In the sequel c, c 2,... will denote effectively computable positive constants depending only on n. The well-known inequalities hr D (log D ) n, (cf. e.g. [L]) and imply D n n M(P ) 2n 2 n n M 2n 2 (cf. [M]) (2) hr < c M n. Let a denote the leading coefficient of f and β,..., β n be the zeros of g(x) = a n f( x a ). Set (g) = β i β j (β i β j ) 2,

3 ON POWER VALUES OF POLYNOMIALS 3 and write g in the form g(x) = P k (x)p 2(x) where P and P 2 are relatively prime polynomials in Z[x] and P is an irreducible monic of degree t; (actually P (x) = a t P ( x a )). Let β,..., β t be the zeros of P and (x, y) be an arbitrary, however, fixed solution to (). The g.c.d. of the principal ideals ax β and g(ax)(ax β ) k divides n (g), therefore, there are integral ideals A, B, C in K so that (3) A ax β = BC w where w = z (z, k ), furthermore, max{n K/Q (A), N K/Q (B)} a b (g) n2. Hence, by a well-known inequality (cf. for example [Gy3], Lemma 3) and by (2), the ideals A h and B h have generators α and β, respectively, with The relation (3) can be written as max{ α, β } exp(c 2 M n (log M) n log 2b ). α(ax β ) h = εβγ w where γ is a generator of C h and ε is a unit. Let ε,..., ε r be a fundamental system of units for K satisfying Lemma. Then we can express ε as ε = ρε l... εlr r where ρ is a root of unity and we may assume that max l i < w (the remaining factors, i r if any, are incorporated in γ). If ax M(g) + then 2 z y z (2M(g) + ) n and the Theorem is proved. Otherwise, ax > M(g) + and ax β i >, i =,..., n implies and ax β i a n by z, i =,..., n, a n by z h max i t ax β i h ε nw... εr nw α n β n γ w γ a n b h w y nh α n w β n w r n ε i. If w < nh then by < R (cf. [Z]) we obtain w < 20nhR and In case of w nh and we get i= z < c 3 M n (log(2m)) n. γ M b n y nh α β r n ε i, i= ( ) γ log H c γ (2) 4 log 2b M n (log(2m)) n log y.

4 4 A. BÉRCZES, B. BRINDZA AND L. HAJDU We may assume that ax 2 y z n. Indeed, otherwise max i n ax β i y z n and the Theorem is proved. Supposing ax y z n M(g) yields β i β j ax β i β 2 β ax β 2, i, j t, i j we have Then i,j t β i β j β i β j ax β i (g) 2n y z. β 2 β ax β 2 y z 4, or else we can derive a bound for z better than stated in the Theorem. Avoiding ( ) h the trivial case ax β ax β 2 =, whenever 2 y z n (g) n2 we obtain ( ) h ax β ( ) log ax β 2 log h ax β ax β 2 z log y. 8 Finally, Lemma 2 yields ( ) h ax β ( ) lh ( ) 0 ax β 2 = lrh ( ) wh ε εr β/α γ... ε (2) ε (2) r β (2) /α (2) γ (2) exp ( c 5 log 2b M 3n 3 (log 2M ) 3n log y log w ) and the comparision of the upper and lower bounds completes the proof (in the first case). In the easier second case all the zeros of g are integral. Let k i denote the multiplicities of β i, i =, 2. Repeating the argument one can have u i (ax β i ) = v i y w i z where w = (a,k and u k 2) i, v i, y i Z, y i >, i =, 2. To derive a bound for w from the equation Ay w By w 2 = C (A = u 2 v, B = u v 2, C = u u 2 (β 2 β )) one can apply Lemma 2 again, and we have z log z c 6 log M log 2b, and the Theorem is proved.

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