We give improved upper bounds for the number of primitive solutions of the Thue inequality

Transcription

1 NUMBER OF SOLUTIONS OF CUBIC THUE INEQUALITIES WITH POSITIVE DISCRIMINANT N SARADHA AND DIVYUM SHARMA Abstact Let F(X, Y) be an ieducible binay cubic fom with intege coefficients and positive disciminant D Let k be a positive intege satisfying k < (3D)/4 π We give impoved uppe bounds fo the numbe of pimitive solutions of the Thue inequality F(X, Y) k Intoduction Let F(X, Y) be an ieducible binay fom with intege coefficients Let and D denote the degee and disciminant of F, espectively Suppose that 3 Let k be a positive intege In a pioneeing wok in 909, Thue poved that the equation F(X, Y) = k has only finitely many solutions in integes x and y Such equations ae now called Thue equations A pai (x, y) of integes is said to be pimitive if gcd(x, y) = In 983, Evetse [6] obtained an uppe bound fo the numbe of pimitive solutions of the above equation, which depended only on and k and was othewise independent of F, theeby poving a conjectue of Siegel Late, the uppe bound of Evetse was geatly impoved by Bombiei & Schmidt [3] Let N F (k) denote the numbe of pimitive intege solutions of the inequality () F(X, Y) k Hee, (x, y) and ( x, y) ae counted as one solution Theefoe if (x, y) is a solution of () with y 0, we can assume that y > 0 Hence if (x, y) is a pimitive solution of (), then eithe () gcd(x, y) =, y > 0 o (x, y) = (, 0) Thoughout this pape, we assume () without any mention Many mathematicians have consideed inequality () when k is small in compaison 00 Mathematics Subject Classification Pimay D45, D75; Seconday E76, D5 Key wods and phases Thue inequalities, binay cubic foms

2 N SARADHA AND DIVYUM SHARMA with D and obtained bounds fo N F (k) which involve only (See [7] and [8]) Fo instance, in [0, p 53], we showed that if then D(F) (7 k 44 ), N F (k) 7 In fact, if is lage, bette bounds wee found by Győy [7, see Coollay 3 and emaks afte Theoem ] He showed that if then D(F) (35 k ) ( ) ϑ with 0 < ϑ <, N F (k) ϑ, fo sufficiently lage We now estict to the case = 3 Futhe, suppose that the disciminant D of F is positive (We efe to Wakabayashi [] fo the case of negative disciminant) By the esult of Evetse mentioned above, we know that thee is an absolute constant C such that N F () C fo all such foms F Infact, in [5], Evetse developed on the method of Siegel [9] and Gel man (See [4, Chapte 5]) to show that N F () Fo poving this, Evetse fist showed the following theoem Theoem E Let H(X, Y) be the Hessian of the fom F The numbe of solutions (x, y) of () with (3) H(x, y) 3 3D is at most 9 (See Section fo the definition of Hessian) When k =, he showed that thee ae at most 3 solutions satisfying H(x, y) < 3 3D Recently, using Theoem E, Akhtai [, Theoem ] showed that if k is a positive intege satisfying (4) k < (3D)/4 π, then () has at most 9 + log( 3 8ɛ + ) log solutions in copime integes x and y with y 0, whee k 3 ɛ = 4 log(πk) log(3d)

3 NUMBER OF SOLUTIONS OF CUBIC THUE INEQUALITIES 3 We point out hee that by following he agument on [, p 737], the above bound must be coected as log( 3 8ɛ ) log (Hee and elsewhee, the symbol x denotes the smallest intege geate than o equal to the eal numbe x) Adding to this coected estimate fo the possible solution (, 0), he esult yields log( 3 8ɛ N F (k) ) log (5) = log log ( ) log(3d) + log k + log(π) log(3d) 4 log k 4 log(π) In 00, Bennett [] used extensive computation and made the wok of Evetse moe pecise in the case k = to show that N F () 0 In fact, accoding to [, Sections 8 & 9], N F () 9 if D 0 6 Since the smallest positive disciminant of an ieducible cubic fom is 49, we will assume fom now onwads that { 0 6 if k = (6) D 49 if k We shall use calculations of Bennett to obtain the following esult analogous to Theoem E Theoem Let F(X, Y) be an ieducible binay cubic fom with intege coefficients and positive disciminant D Let k be a positive intege Then thee ae at most 6 solutions (x, y) of () with (7) H(x, y) 8(3D) 5/6 k 4 Complementing Theoem, we show the next esult Theoem Let F(X, Y) be an ieducible binay cubic fom with intege coefficients and positive disciminant D Let k be a positive intege satisfying (4) Then the numbe of solutions (x, y) of () with y 0 and is at most H(x, y) < 8(3D) 5/6 k 4 ( ) 5 log(3d) + log k log log 3 log(3d) log k 556 As an immediate consequence of Theoems, and including the possible solution (, 0), we get the following coollay

4 4 N SARADHA AND DIVYUM SHARMA Coollay 3 Let F(X, Y) be an ieducible binay cubic fom with intege coefficients and positive disciminant D Let k be a positive intege satisfying (4) Then (8) N F (k) log log ( ) 5 log(3d) + log k log(3d) log k 556 Remak 4 (i) Note that N F () 0, thus etieving Bennett s esult (ii) We now show that the uppe bound fo N F (k) in (8) is bette than the bound in (5) Put Υ = χ = Now Υ = 5 log(3d) + log k log(3d) + 6 log k + 05 χ 5 log(3d) + log k log(3d) log k 556, log(3d) + log k log(3d) 4 log k log(3d) + 6 log k log(3d) log k 556 By (8), we have N F (k) log Υ log χ log log log χ 0 + 3, log poving ou claim (iii) Ou poof of Theoem depends on the method given in the papes of Evetse and Bennett They assumed that thee wee at least fou solutions of () elated to any pai of esolvent foms (See Section fo definition) This enabled them to get a good gap pinciple and deive a contadiction To obtain Theoem, we assume that thee ae at least thee solutions of () elated to any pai of esolvent foms To get a good gap pinciple, we need to assume that (7) is valid, which is weake than (3) In fact, if k satisfies (4), then it is enough to assume that H(x, y) 8(3D) 5/6 k 7/ and this leads to an estimate ( ) 5 log(3d) + 9 log k + 3 (9) N F (k) log log 3 log(3d) log k 556 in Coollay 3 It is possible to cay this method futhe to get that thee ae at most thee pimitive solutions (x, y) of () satisfying H(x, y) 3(3D) 5/3 k 6

5 NUMBER OF SOLUTIONS OF CUBIC THUE INEQUALITIES 5 This yields ( ) 0 log(3d) + 4 log k + 5 N F (k) log log 3 log(3d) log k 556 One can easily see that this bound is no bette than the bound given in (9) Thus it may not be possible to significantly impove (9) by this method Coollay 3 eadily yields an estimate fo the numbe of all intege solutions (pimitive and non-pimitive) to () When k =, all the solutions ae pimitive So we assume that k Coollay 5 Suppose that πk = (3D) /4 δ with 0 < δ < /4 Then the numbe of intege solutions to () is at most 8 k /3 Remak 6 In [], Thunde consideed the inequality () fo any positive intege k and disciminant of F positive o negative Using a diffeent method, he showed that the numbe of intege solutions is at most k/3 008 k/ k /3 D /6 D / Unde the assumptions of Coollay 5, the above bound can be majoized by 3770 k /3 Peliminaies We efe to [5] fo the ensuing facts on cubic foms Wite F(X, Y) = ax 3 + bx Y + cxy + dy 3 The quadatic covaiant, o Hessian, and the cubic covaiant of F ae defined as H(X, Y) = ( 4 F F X Y F X Y) and G(X, Y) = F H X Y F H Y X, espectively It can be checked that whee Futhe, H(X, Y) = AX + BXY + CY A = b 3ac, B = bc 9ad and C = c 3bd B 4AC = 3D,

6 6 N SARADHA AND DIVYUM SHARMA whee D is the disciminant of F, and (0) 4H(X, Y) 3 = G(X, Y) + 7D F(X, Y) F is said to be educed if C A B Since evey cubic fom of positive disciminant is GL(, Z)-equivalent to a educed fom and N F (k) = N F (k) fo equivalent foms F and F, we can assume that F is educed Let 3D be a fixed choice of the squae-oot of 3D and let M = Q( 3D) with O M denoting the ing of integes in M Then { } m + n 3D O M = : m, n Z, m nd (mod ) Put and U(X, Y) = G(X, Y) + 3 3D F(X, Y) V(X, Y) = G(X, Y) 3 3D F(X, Y) Obseve that U(X, Y) and V(X, Y) ae cubic foms in M[X, Y] having no common facto Also, the coesponding coefficients of U(X, Y) and V(X, Y) ae complex conjugates The elation (0) implies that U(X, Y)V(X, Y) = H(X, Y) 3 Hence U(X, Y) = ξ(x, Y) 3, V(X, Y) = η(x, Y) 3, whee ξ(x, Y) and η(x, Y) ae linea foms whose coesponding coefficients ae complex conjugates Futhe, ξ(x, Y) 3 η(x, Y) 3 = 3 3D F(X, Y) () ξ(x, Y) 3 + η(x, Y) 3 = G(X, Y) ξ(x, Y)η(X, Y) = H(X, Y) ξ(x, Y) ξ(, 0), η(x, Y) η(, 0) M[X, Y] Theefoe fo all integes x and y, we have () ξ(x, y) = η(x, y) = H(x, y) A pai (ξ, η) of foms satisfying the popeties () is called a pai of esolvent foms If (ξ, η) is such a pai, then thee ae pecisely two othes, namely, (ρξ, ρ η) and (ρ ξ, ρη), whee ρ is a pimitive cube oot of unity A pai (x, y) of integes is said to be elated to a pai of esolvent foms (ξ, η) if η(x, y) (3) ξ(x, y) = min η(x, y) 0 l elπi/3 ξ(x, y)

7 NUMBER OF SOLUTIONS OF CUBIC THUE INEQUALITIES 7 We need the following lemma fom [] Lemma [, Lemma 5] Let F be an ieducible, educed binay cubic fom with positive disciminant D and Hessian H Then fo all integes x, y with y 0, we have 3D H(x, y) Lemma Let (x, y), y 0, be a solution of () elated to (ξ, η) Put (, π/3) if ξ(x, y) 8 / (3D) 5/ k (µ, ν) = ( 3, ) if (4) holds Then π η(x, y)3 ξ(x, y) 3 < µ and Poof Lemma and equation () imply that (4) ξ(x, y) (3D)/4 η(x, y) ξ(x, y) < ν 3D k ξ(x, y) 3 If (4) holds, then using () and (6) we obtain η(x, y)3 ξ(x, y) 3 3 3D k ξ(x, y) 3 3 3D k 3/ < 3 (3D) 3/4 π Similaly, if ξ(x, y) 8 / (3D) 5/ k, then we have η(x, y)3 ξ(x, y) / (3D) 3/4 k < 5 This poves the fist assetion of the lemma Let θ = ag(η(x, y)/ξ(x, y)) Since η(x, y)/ξ(x, y) = and equation (3) holds, we have ag(η(x, y) 3 /ξ(x, y) 3 ) = 3θ In view of the fact that cos(3θ) = η(x, y)3 ξ(x, y) 3 we get θ < < µ, { π/9 if ξ(x, y) 8 / (3D) 5/ k 0495 if (4) holds

8 8 N SARADHA AND DIVYUM SHARMA Now η(x, y) θ η(x, y)3 ξ(x, y) θ = cos(3θ) ξ(x, y) 3 < ν η(x, y)3 3 ξ(x, y) 3 (See also Figue fo an appoximate value of ν) y Figue : y = θ cos(3θ) Using (), we obtain the second assetion of the lemma 3 Gap pinciple Let (x, y ), (x, y ) be two distinct solutions of () elated to (ξ, η) with ξ(x, y ) ξ(x, y ) In this section, we will establish cetain esults egading the gaps between such solutions Fo i =,, we denote ξ(x i, y i ) by ξ i and η(x i, y i ) by η i Since the deteminant of the linea tansfomation (x, y) (ξ, η) is ± 3D, we have ξ η ξ η = ± 3D(x y x y ) This implies that ( 3D ξ η ξ η ξ ξ η + η ) By Lemma, we obtain that 3D < νk ξ ξ 3D ( ξ 3 + ξ 3 ) Thus (5) ξ 3 + ξ 3 > νk ξ ξ implying that ξ > νk ξ We obtain a bette gap pinciple in the following lemma ξ ξ

9 NUMBER OF SOLUTIONS OF CUBIC THUE INEQUALITIES 9 Lemma 3 Let (x, y ), (x, y ) be two solutions of () elated to (ξ, η) with ξ ξ and y, y 0 Then whee τ = ξ τ k ξ, { 095 if ξ(x, y) 8 / (3D) 5/ k 089 if (4) holds Poof Let φ = k ξ ξ and h(z) = z 3 ( ) 3 k ν z + ξ Fom equation (5), we have h(φ) > 0 Obseve that h(0) > 0 Using (6) and (4), espectively, we obtain k ξ < 0047 if ξ(x, y) 8 / (3D) 5/ k if (4) holds π Hence h(/(3ν)) < 0 Futhe h(z) assumes a local maximum at z = 0 and a local minimum at z = /(3ν) Hence h(z) has two positive zeos, say φ and φ, with φ < φ ; h(z) is negative fo φ < z < φ and positive fo 0 < z < φ, z > φ If φ φ then we must have h(k/ ξ ) > 0 as φ k/ ξ But h(k/ ξ ) < 0 Theefoe φ > φ This, togethe with the fact that h(φ) > 0 implies that φ > φ Since h(τ) < 0, we have φ τ This completes the poof of the lemma 4 Poof of Theoem Let (x, y) be a solution of () with y 0 and (6) ξ(x, y) = H(x, y) < 8(3D) 5/6 k 4 Enumeate the solutions (x, y) of () with y 0 as (x, y ), (x, y ), with ξ i = ξ(x i, y i ) and ξ ξ Applying the gap pinciple stated in Lemma 3 inductively and using (4), we obtain that ( 089 ξ t k ) t Let t be the least intege such that ( ) 089 t ( (3D) /4 )t ξ t k fo all t ( ) 089 t ( (3D) /4 )t (7) 8 / (3D) 5/ k k Then thee ae at most t solutions (x, y) of () with y 0 and satisfying (6) Taking logaithms twice in (7), we obtain the assetion of the theoem

10 0 N SARADHA AND DIVYUM SHARMA 5 Padé appoximation In this section, we intoduce some auxiliay polynomials used in the poof of Theoem Let α, β and γ be complex numbes The standad hypegeometic function F(α, β, γ, z) is epesented by α(α + ) (α + n )β(β + ) (β + n ) F(α, β, γ, z) = + z n γ(γ + ) (γ + n )n! n= Let be a positive intege and let g {0, } Define ( ) g A,g = F ( ) 3 + g,, + g, z and B,g = ( ) ( ) g F g 3, + g, + g, z See [5, Lemma 3] fo the poof of the following lemma on Padé appoximation Lemma 5 (i) Thee exists a powe seies F,g (z) such that fo all complex numbes z with z <, we have (8) A,g (z) ( z) /3 B,g (z) = z + g F,g (z) and (9) F,g (z) ( g+/3 )( /3 + g ( + g ) ) ( z ) (+ g)/ (ii) Fo all complex numbes z with z, we have ( ) g (0) A,g (z) (iii) Let z be a non-zeo complex numbe and let h {0, } Then A,0 (z)b +h, (z) A +h, (z)b,0 (z) 0 Let C,g denote the geatest common diviso of the numeatos of the coefficients of A,g Note that C,g is also the geatest common diviso of the numeatos of the coefficients of B,g See Table fo the values of C,g fo some choices of and g 6 Poof of Theoem In this section, we will show that thee ae at most solutions (x, y) of () with () ξ(x, y) = H(x, y) 8(3D) 5/6 k 4 which ae elated to a given pai of esolvent foms Since thee ae exactly thee pais of esolvent foms, Theoem will follow Assume that thee ae 3 solutions (x, y ), (x 0, y 0 ), (x, y ) of () satisfying

11 NUMBER OF SOLUTIONS OF CUBIC THUE INEQUALITIES () and elated to (ξ, η) As befoe, we denote ξ(x i, y i ) by ξ i and η(x i, y i ) by η i fo i =, 0, Put z 0 = η 3 0 /ξ3 0 Σ,g = η A,g (z 0 ) η 0 B,g (z 0 ) ξ ξ 0 Λ,g = ξ 3+ g ξ 0 Σ,g C,g Fom (6), () and Lemma 3, it follows that () 0 < z 0 < 0 6 Hence fo any pimitive cube oot of unity ρ, we have ( z 0 ) /3 < ρ( z 0 ) /3 This, togethe with (3), implies that (3) η 0 /ξ 0 = ( z 0 ) /3 It follows fom the poof of [5, Lemma 5] that Theefoe if Λ,g 0, we have i e Λ,0 3D Z and Λ 3, O M \ Z Λ,0 3D and Λ 3, 3D, (4) Λ,g g/3 (3D) / g/3 Lemma 6 Let c (, g) = ( ) π C,g 3 g/3, If Σ,g 0, then c (, g) = C,g ( g+/3 )( /3 + g ( + g ) ) g/3 (300) + g, Ξ(, g) = c (, g)(3d)g/3 ξ 0 3+ g ξ k, Π(, g) = c (, g)(3d) g/6 ξ ξ 0 (3+ g) k + g Ξ(, g) + Π(, g) >

12 N SARADHA AND DIVYUM SHARMA Poof Using (3), (8), (0), (9), Lemma and (), we obtain that ( Λ,g = ξ 0 3+ g η ) ξ C,g A,g (z 0 ) + z + g F 0,g (z 0 ) ξ (( ) g ξ 0 3+ g ξ η C,g ξ ( g+/3 )( /3 ) + g z 0 + g + ( + g ) ( z 0 ) (+ g)/ (( ) g π 3D k ξ 0 3+ g ξ C,g 3 ξ 3 ( g+/3 )( /3 ) ( + g 3 3D k )+ g + ) ( 0 6 ) / ξ 0 3 ( + g < g/3 (3D) / g/3 ( c (, g)(3d)g/3 ξ 0 3+ g ξ k Now the lemma follows using (4) +c (, g)(3d) g/6 ξ ξ 0 (3+ g) k + g) Fo 7 and fo cetain choices of g, let c (, g) and c (, g) be defined by Table Fo 8, put c (, g) = 4 and c (, g) = (5) Table (, g) C,g c (, g) c (, g) (, g) C,g c (, g) c (, g) (, ) 3 3 (5, 0) (, 0) (6, ) (, 0) (6, 0) (3, 0) (7, ) (4, 0) (7, 0) The values of C,g and c (, g) ae exactly as in [] The values of c (, g) ae diffeent due to the gap pinciple in Lemma 3 We will now pove that (5) c (, g) c (, g) and c (, g) c (, g) (Then fo these choices of and g, we can use Lemma 6 with c (, g) and c (, g) eplaced by c (, g) and c (, g), espectively) It can be easily checked that (5) holds fo the choices of and g listed in Table Futhe, by [, Eq 67], we have ( ) < 4 π

13 NUMBER OF SOLUTIONS OF CUBIC THUE INEQUALITIES 3 fo all positive integes Theefoe c (, g) < 4 π π 3 < 4 N By [, Poof of Lemma 6, last inequality], Hence c (, g) < π This completes the poof of (5) ( ) + (5) + c (5) (, g) < fo 5 The next two lemmas ae Lemmas 63 and 64 fom [] Lemma 6 Σ,g 0 fo (, g) = (, ), (, 0), (, 0), (3, 0), (4, 0), (5, 0) Lemma 63 Let be a positive intege and let h {0, } Then at least one of Σ,0 and Σ +h, is non-zeo The final step Note that (6) ξ k ξ 7(3D) 5/6 k 3 We say that popety P[a, a, a 3, a 4 ] holds if ξ > a ξ 0 a (3D) a 3 k a 4 By Lemma 3, P[095,, 0, ] holds We shall show that (7) P[6, 3 +,, + ] holds fo all Hence by (6), we get that P[, 0,, 7] holds fo any This is not possible Thus thee can be at most solutions elated to (ξ, η) Now we pove (7) In the calculations below, (6), (5) and the values of c (, g), c (, g) will be epeatedly used Fist we take (, g) = (, ) Then P[095,, 0, ] and (6) yield Ξ(, ) 3 (3D) /3 k ξ 0 3 < (3D) / Hence it follows fom Lemmas 6 and 6 that Π(, ) > 0994, which shows that P[0749, 3, 5/6, ] holds We abbeviate this as {P[095,, 0, ], (, )} P[0749, 3, 5/6, ]

14 4 N SARADHA AND DIVYUM SHARMA Next, we fix (, g) = (, 0) Then aguing as above, we get {P[0749, 3, 5/6, ], (, 0)} P[0467, 5,, 3] Poceeding thus we get the following sequence {P[095,, 0, ], (, )} {P[0749, 3, 5/6, ], (, 0)} {P[0467, 5,, 3], (, 0)} {P[03573, 8,, 5], (3, 0)} {P[3333,, 3, 7], (4, 0)} {P[0555, 4, 4, 9], (5, 0)} P[48, 7, 5, ] Hence, (7) holds fo 5 We now poceed by induction Suppose that (7) holds fo some 5 Then we fix (, g) as ( +, 0) Suppose Σ +,0 0 Then we ague as ealie to get and hence giving Thus we have Ξ( +, 0) < 000 Π( +, 0) > 0999 ξ ξ (3D) + k +3 ξ (3D) + k +3 {P[6, 3 +,, + ], ( +, 0)} P[6 (+), 3 + 5, +, + 3] poving the claim If Σ +,0 = 0 then by Lemma 63, both Σ +, and Σ +, ae non-zeo Fist we fix (, g) as ( +, ) Then we get {P[6, 3 +,, + ], ( +, )} P[5, 3 + 3, + 5/6, + ] Now we fix (, g) as ( +, ) Then {P[5, 3 + 3, + 5/6, + ], ( +, )} P[6 (+), 3 + 5, +, + 3] This completes the induction 7 Poof of Coollay 5 It is easy to see that the numbe of intege solutions to () is k / ( ) 5 log(3d) + log(k/d 3 log log ) log(3d) log(k/d 3 ) 556 d= Since πk = (3D) /4 δ and δ > 0, this can be estimated as 0 k /3 + 3 ( ) k/3 (8 δ) log(3d) log(π) + 3 log log δ log(3d) + log(π) k /3 Acknowledgement We thank Pofesso K Győy and the efeee fo thei useful comments

15 NUMBER OF SOLUTIONS OF CUBIC THUE INEQUALITIES 5 Refeences [] Shabnam Akhtai, Cubic Thue inequalities with positive disciminant, Publ Math Debecen 83 (03), no 4, [] Michael A Bennett, On the epesentation of unity by binay cubic foms, Tans Ame Math Soc 353 (00), [3] E Bombiei, W M Schmidt, On Thue s equation, Invent Math 88 (987), 69 8 [4] B N Delone, D K Fadeev, The theoy of iationalities of the thid degee, Tanslation of Math Monogaphs, AMS 0 (964) [5] J-H Evetse, On the epesentation of integes by binay cubic foms of positive disciminant, Invent Math 73 (983), no, 7 38; Eatum, Invent Math 75 (984), no, 379 [6] J-H Evetse, Uppe bounds fo the numbe of solutions of Diophantine equations, Math Centum, Amstedam (983), 7 [7] K Győy, Thue inequalities with a small numbe of pimitive solutions, Peiod Math Hunga 4 (00), [8] K Győy, On the numbe of pimitive solutions of Thue equations and Thue inequalities, Paul Edős and his Mathematics I, Bolyai Soc Math Studies, (00), [9] C L Siegel, Übe einige Anwendungen diophantiche Appoximationen, Abh Peuss Akad Wiss (99), no [0] Divyum Shama, N Saadha, Numbe of epesentations of integes by binay foms, Publ Math Debecen 85 (04), no, 33 55; Coigendum, Publ Math Debecen (to appea) [] J L Thunde, On cubic Thue inequalities and a esult of Mahle, Acta Aith 83 (998), no, 3 44 [] Isao Wakabayashi, Cubic Thue inequalities with negative disciminant, J Numbe Theoy 97 (00), no, -5 School of Mathematics, Tata Institute of Fundamental Reseach, Homi Bhabha Road, Mumbai , INDIA addess: saadha@mathtifesin School of Mathematics, Tata Institute of Fundamental Reseach, Homi Bhabha Road, Mumbai , INDIA addess: divyum@mathtifesin