Diophantine equations over global function fields I: The Thue equation

ARTICLE IN PRESS UNCORRECTED PROOF S0022-314X(05)00211-8/FLA AID:3277 Vol.•••(•••) [DTD5] P.1 (1-17) YJNTH:m1 1.50 P n:17/11/2005; 9:56 yjn h3277 by:Vi a p. 1 Jou nal o Numbe Theo y ••• (••••)•••–••• www.else ie .com/loca e/jn 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 10 10 11 11 12 12 13 13 14 14 15 15 16 16 17 17 18 18 19 19 20 20 21 21 22 22 23 23 24 24 25 25 26 26 27 27 28 28 29 29 30 30 31 31 32 32 33 33 34 34 35 35 36 36 37 37 38 38 39 39 40 40 41 41 42 42 43 43 44 44 45 45 Diophan ine equa ions o e global unc ion ields I: The Thue equa ion Is án Gaála,1, Michael Pohs b,∗,2 aUni e si y o Deb ecen, Ma hema ical Ins i u e, H-4010 Deb ecen P . 12, Hunga y bTechnische Uni e s ä Be lin, Ins i u ü Ma hema ik, S aße des 17. Juni 136, Be lin, Ge many Recei ed 19 Janua y 2004; e ised 5 July 2005 Communica ed by Da id Goss Abs ac We sol e comple ely Thue equa ions in unc ion ields o e a bi a y ini e ields. In he unc ion ield case such equa ions we e o me ly only sol ed o e algeb aically closed ields (o cha ac e - is ic ze o and posi i e cha ac e is ic). Ou me hod can be applied o simila ypes o Diophan ine equa ions, as well. 2005 Published by Else ie Inc. MSC: 11D59; 11Y50; 11R58 Keywo ds: Thue equa ions; Global unc ion ields 1. In oduc ion Classical Diophan ine equa ions like Thue equa ion (c . Thue [10]) a e adi ionally sol ed o e he ings o a ional in ege s o o e he ing o in ege s o a numbe ield *Co esponding au ho . E-mail add esses: [email p o ec ed] (I. Gaál), [email p o ec ed] (M. Pohs ). 1Resea ch suppo ed in pa by G an s T 037367 and T 042985 om he Hunga ian Na ional Founda ion o Scien i ic Resea ch. 2Resea ch suppo ed by he Deu sche Fo schungsgemeinscha . 0022-314X/$ – see on ma e 2005 Published by Else ie Inc. doi:10.1016/j.jn .2005.10.009 ARTICLE IN PRESS UNCORRECTED PROOF S0022-314X(05)00211-8/FLA AID:3277 Vol.•••(•••) [DTD5] P.2 (1-17) YJNTH:m1 1.50 P n:17/11/2005; 9:56 yjn h3277 by:Vi a p. 2 2I. Gaál, M. Pohs / Jou nal o Numbe Theo y ••• (••••)•••–••• 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 10 10 11 11 12 12 13 13 14 14 15 15 16 16 17 17 18 18 19 19 20 20 21 21 22 22 23 23 24 24 25 25 26 26 27 27 28 28 29 29 30 30 31 31 32 32 33 33 34 34 35 35 36 36 37 37 38 38 39 39 40 40 41 41 42 42 43 43 44 44 45 45 (c . Bake [1]). Se e al au ho s conside ed he analogous p oblem o e unc ion ields: mo e exac ly o e he ing o in ege s o a unc ion ield o e an algeb aically closed ield, see, e.g., [6,8], bo h in case he g ound ield is o ze o o posi i e cha ac e is ic. These esul s ha e also common gene aliza ions (c . Gy˝ o y [5]). Ou pu pose is now o in es iga e his p oblem in unc ion ields o e a bi a y ini e ields. I is well known ha Diophan ine equa ions o e ini e ields play an impo an ole in c yp og aphy, c . Niede ei e and Xing [7]. Also, om a p ac ical poin o iew his case is much mo e s aigh o wa d han he case o algeb aically closed g ound ields (which mos ly occu in heo y only). As i will u n ou , also in his case he uni equa ion plays a c ucial ole and he so- lu ions can be compu ed easily. Howe e , o cons uc ing he app op ia e unc ion ields, pe o ming calcula ions in hem, de e mining heigh s, e c. we in ensi ely use he compu e algeb a package KASH [2]. 2. Global unc ion ields In he ollowing we shall s ongly ely on he a gumen used by Mason [6] o unc ion ields o e algeb aically closed ields. We show how his ideas can be ans e ed o ou si ua ion. A gene al desc ip ion o p ope ies o unc ion ields (also o e ini e ields) can be ound in he book o S ich eno h [9]. We in oduce some no a ions. k=Fqdeno es a ini e ield wi h q=pdelemen s. The a ional unc ion ield o kis k( ) as usual, and Kis a ini e ex ension o k( ) o deg ee n0 and genus g0. The in eg al closu e o k[ ]in Kis deno ed by oK. We assume ha K is sepa ably gene a ed o e k( ) by an elemen ybelonging o oKand ha kis he ull cons an ield o K. Any elemen ∈Khas a unique p esen a ion = n0  i=1 hiyi−1,h i∈k( ). Conjuga es o elemen s ( ields) a e deno ed by uppe case indices. Le A:= ((y(j))i−1)1⩽i,j⩽n∈Kn×nha e de e minan D. We no e ha Dis he disc iminan o y. I is nonze o since Kis sepa ably gene a ed. We ob ain he sys em o linea equa ions:  (1),..., (n)=(h1,...,h n)A. Hence, he hia e a ional unc ions in he (j),(y(j))i−1. The se o all (exponen ial) alua ions o Kis deno ed by V, he subse o in ini e al- ua ions by V∞. By abuse o no a ion we do no dis inguish be ween places and alua ions. Fo example, we w i e deg o he deg ee o he di iso belonging o he alua ion ∈V. Fo a nonze o elemen ∈Kwe deno e by ( ) he alue o a . Fo in eg al elemen s his is he highes powe o he di iso belonging o ha di ides he di iso ( ), and his ARTICLE IN PRESS UNCORRECTED PROOF S0022-314X(05)00211-8/FLA AID:3277 Vol.•••(•••) [DTD5] P.3 (1-17) YJNTH:m1 1.50 P n:17/11/2005; 9:56 yjn h3277 by:Vi a p. 3 I. Gaál, M. Pohs / Jou nal o Numbe Theo y ••• (••••)•••–••• 3 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 10 10 11 11 12 12 13 13 14 14 15 15 16 16 17 17 18 18 19 19 20 20 21 21 22 22 23 23 24 24 25 25 26 26 27 27 28 28 29 29 30 30 31 31 32 32 33 33 34 34 35 35 36 36 37 37 38 38 39 39 40 40 41 41 42 42 43 43 44 44 45 45 concep is ex ended o a ional elemen s in he usual way. Fo he no malized alua ions N( ) = ( ) ·deg he p oduc o mula holds:  ∈V N( ) =0∀ ∈K {0}. The heigh o a nonze o elemen o Kis de ined o be H( ):=  ∈V max0, N( ). Because o he p oduc o mula his is an amoun o H( )=− ∈V min0, N( ) which hen holds o all elemen s o Kincluding 0. 3. Uni equa ions Le V0be a ini e subse o V. Then he nonze o elemen s γ∈Ksa is ying (γ) =0 o all /∈V0 o m a mul iplica i e g oup in K. These elemen s a e called V0-uni s.Fo V0=V∞ he V0-uni s a e jus he uni s o he ing oK. The esolu ion o Thue equa ions (as well as se e al o he ypes o classical Diophan ine equa ions) is usually educed o equa ions o he o m γ1+γ2+γ3=0(1) whe e he γia e V0-uni s o a sui able se V0. The c ucial inequali y o Mason on he abo e equa ion becomes in ou case: Lemma 3.1. Le V0be a ini e subse o Vand le γi(1⩽i⩽3)be V0-uni s sa is ying (1). Then ei he γ1 γ3is in Kpo i s heigh is bounded: Hγ1 γ3⩽2g−2+ ∈V0 deg . (2) P oo . The p oo is along he lines o he p oo o Lemma 2 in [6] (see [6, p. 14]). Since in ou case he ield o cons an s kis no algeb aically closed we encoun e a ew addi ional di icul ies which we will poin ou in wha ollows. ARTICLE IN PRESS UNCORRECTED PROOF S0022-314X(05)00211-8/FLA AID:3277 Vol.•••(•••) [DTD5] P.4 (1-17) YJNTH:m1 1.50 P n:17/11/2005; 9:56 yjn h3277 by:Vi a p. 4 4I. Gaál, M. Pohs / Jou nal o Numbe Theo y ••• (••••)•••–••• 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 10 10 11 11 12 12 13 13 14 14 15 15 16 16 17 17 18 18 19 19 20 20 21 21 22 22 23 23 24 24 25 25 26 26 27 27 28 28 29 29 30 30 31 31 32 32 33 33 34 34 35 35 36 36 37 37 38 38 39 39 40 40 41 41 42 42 43 43 44 44 45 45 We may assume ha :=γ1/γ3and he e o e γ2/γ3=− −1 do no belong o k.Le V1:= ∈V: ( ) < 0,V 2:= ∈V: ( ) > 0, V3:= ∈V: ( ) =0∧ ( +1)>0. These a e disjoin subse s o he se V0. Then we ha e H( )= ∈V1− ( )deg = ∈V2 ( )deg = ∈V3 ( +1)deg . This is ue because o 1 =(1+ )+(− ), he p oduc o mula, and he p ope y ( ) < 0 ⇔ (1+ )<0. I zdeno es a p ime elemen o he alua ion ∈V hen he di e en ial 1d sa is ies (1d ) = (d /dz) (see [9, Chap e IV]). I is no a p h powe hen he di iso 1d is canonical, i.e., deg(1d ) =2g−2, since he ield ko cons an s is comple e (see [9, Chap e I.5]). This yields 2g−2= ∈V (d )deg = ∈V0 (d )deg = ∈V1∪V2∪V3 (d )deg ⩾ ∈V1∪V2 ( ) −1deg + ∈V3 (1+ )−1deg =−H( )+H( )− ∈V1∪V2 deg +H( )− ∈V3 deg ⩾H( )− ∈V1∪V2∪V3 deg ⩾H( )− ∈V0 deg whence he asse ion ollows. 2 Taking Φ=−γ1/γ3,Ψ=−γ2/γ3Eq. (1) gi es he uni equa ion in wo a iables Φ+Ψ=1(3) whe e Φ,Ψ a e V0-uni s. Because o cha ac e is ic p he numbe o solu ions o such a uni equa ion can be in ini e. Fo example, i V0is jus he se o in ini e alua ions and η,1−ηa e bo h uni s o oK hen also ηκ,(1−η)κis a solu ion o (3) o e e y exponen κ=p. Hence, he e exis solu ions o a bi a y la ge heigh s in his si ua ion. The subsequen lemma shows ha o any ini e subse V0o V, he g oup o V0-uni s o Kcon ains only a ini e numbe , say s,o V0-uni s ηwhich a e no p h powe s and o which also 1 −ηis a V0-uni . We deno e he se o hese uni s by {η1,...,η s}. Lemma 3.2. Le V0be a ini e subse o V. Assume ha a V0-uni Φin Kis a solu ion o (3).I Φis no a p h powe o ηi(1⩽i⩽s,  ∈Z⩾0) hen Φbelongs o a ini e subse o Kwhich can be calcula ed. ARTICLE IN PRESS UNCORRECTED PROOF S0022-314X(05)00211-8/FLA AID:3277 Vol.•••(•••) [DTD5] P.5 (1-17) YJNTH:m1 1.50 P n:17/11/2005; 9:56 yjn h3277 by:Vi a p. 5 I. Gaál, M. Pohs / Jou nal o Numbe Theo y ••• (••••)•••–••• 5 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 10 10 11 11 12 12 13 13 14 14 15 15 16 16 17 17 18 18 19 19 20 20 21 21 22 22 23 23 24 24 25 25 26 26 27 27 28 28 29 29 30 30 31 31 32 32 33 33 34 34 35 35 36 36 37 37 38 38 39 39 40 40 41 41 42 42 43 43 44 44 45 45 Fo he p oo we e e o [6, Lemma 11, p. 98]. The p oo s a s by assuming ha Φis no a p h powe in Kand he e o e also p o ides he means o calcula e he ηi. 4. Applica ion o Thue equa ions 4.1. P elimina ies We wan o apply hese esul s o ( ela i e) Thue equa ions o e K.Le F(X,Y):= n  i=0 AiXn−iYi∈oK[X, Y ] be a bina y homogeneous o m o deg ee a leas 3. Wi hou loss o gene ali y (c . [4, p. 20]) we can assume ha Fis monic in X, i.e., A0=1. The polynomial F(X,1)∈oK[X] is equi ed o be sepa able and i educible. Then o a bi a y m∈oK he equa ion F(x,y) =min x,y ∈oK is called a Thue equa ion (o e K). Deno e by αa ze o o F(x,1)in K,le L=K(α) and oL he in eg al closu e o k[ ]in L. Assume ha kis he ull cons an ield o L, oo. I x,y ∈oKis a solu ion o he Thue equa ion hen F(x,y) =NL/K (x −αy) =m. (4) Deno e by γ(j) (j =1,...,n) he conjuga es o any γ∈Lo e K. Assume ha (x, y) ∈o2 Kis a solu ion o (4). Then β=x−αy is o no m m, ha is βcan be ep e- sen ed in he o m β=x−αy =µ·η(5) whe e ηis a uni in Land µis an elemen o a ini e se So non-associa ed elemen s o L o no m mo e K. Fo he solu ion o he co esponding no m equa ion we use he usual me hods om algeb aic numbe heo y, i.e., calcula e sui able S-uni s [3]. Those, oge he wi h Di ichle ’s uni heo em (c ., e.g., [11]), can be easily ans e ed o he unc ion ield case, oo. Deno e by η1,...,η a se o undamen al uni s in L( ha can be calcula ed by he compu e algeb a sys em KASH [2]). Se ing k∗=η0, he e a e in ege exponen s a0,a 1,...,a such ha β=x−αy =µ·ηa0 0·ηa1 1···ηa .(6) ARTICLE IN PRESS UNCORRECTED PROOF S0022-314X(05)00211-8/FLA AID:3277 Vol.•••(•••) [DTD5] P.6 (1-17) YJNTH:m1 1.50 P n:17/11/2005; 9:56 yjn h3277 by:Vi a p. 6 6I. Gaál, M. Pohs / Jou nal o Numbe Theo y ••• (••••)•••–••• 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 10 10 11 11 12 12 13 13 14 14 15 15 16 16 17 17 18 18 19 19 20 20 21 21 22 22 23 23 24 24 25 25 26 26 27 27 28 28 29 29 30 30 31 31 32 32 33 33 34 34 35 35 36 36 37 37 38 38 39 39 40 40 41 41 42 42 43 43 44 44 45 45 Fo ixed dis inc i, j, k (wi h 1 ⩽i, j, k ⩽n)se Lij k =L(α(i),α(j),α(k))wi h genus g. Deno e by V0a ini e se o alua ions o Lij k con aining he in ini e alua ions and such ha α(i) −α(j)=0, α(j) −α(k)=0, α(k) −α(i)=0i /∈V0 and µ(i)=0, µ(j)=0, µ(k)=0i /∈V0. Siegel’s iden i y (holding i ially o any solu ion, see [4, Chap e 3]) gi es α(i) −α(j)β(k) +α(j) −α(k)β(i) +α(k) −α(i)β(j) =0.(7) By he undamen al Lemma 3.1 τij k =(α(j) −α(k))β(i) (α(i) −α(j))β(k) is ei he o bounded heigh o is con ained in Lp ij k . In he ollowing he heigh unc ion is applied always in Lij k . 4.2. E ec i e uppe bounds o he solu ions o Thue equa ions In case Eq. (4) has only ini ely many solu ions we de i e an uppe bound o he heigh s o he solu ions. I he equa ion has only ini ely many solu ions, hen he e mus be i, j, k such ha τij k is no a p h powe in Lij k . We keep he abo e no a ion and se A=max(H(α(i),H(α(j)), H (α(k))). Theo em 4.1. I τij k is no a p h powe , hen Eq. (4) has only ini ely many solu ions and o all solu ions (x, y) we ha e maxH(x),H(y)⩽11A+1 nH(µ)+4g−4+2 ∈V0 deg . P oo . Applying Lemma 3.1 we ge H(τij k )⩽2g−2+ ∈V0 deg =c1. This implies Hβ(i) β(k) =Hx−α(i)y x−α(k)y⩽H(τij k )+Hα(i) −α(j) α(j) −α(k) ⩽c1+4A=c2. ARTICLE IN PRESS UNCORRECTED PROOF S0022-314X(05)00211-8/FLA AID:3277 Vol.•••(•••) [DTD5] P.7 (1-17) YJNTH:m1 1.50 P n:17/11/2005; 9:56 yjn h3277 by:Vi a p. 7 I. Gaál, M. Pohs / Jou nal o Numbe Theo y ••• (••••)•••–••• 7 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 10 10 11 11 12 12 13 13 14 14 15 15 16 16 17 17 18 18 19 19 20 20 21 21 22 22 23 23 24 24 25 25 26 26 27 27 28 28 29 29 30 30 31 31 32 32 33 33 34 34 35 35 36 36 37 37 38 38 39 39 40 40 41 41 42 42 43 43 44 44 45 45 Using an a gumen o Mason [6, Chap e II.1] o y=0weha e x y=α(k)β(i)/β(k) −α(i) β(i)/β(k) −1 whence Hx y⩽2A+2c2. By yn=µ n h=1x y−α(h) we de i e nH(y) ⩽H(µ)+nHx y+A whence he asse ion ollows o y. The bound o xcan be ob ained simila ly. 2 4.3. An algo i hm o calcula ing he solu ions o Thue equa ions We now u n o inding he solu ions o Eq. (4). Case I. Conside i s he case when τij k is o bounded heigh . Simila ly as in he p oo o Theo em 4.1 we ob ain Hx−α(i)y x−α(k)y⩽H(τij k )+Hα(i) −α(j) α(j) −α(k) ⩽c1+Hα(i) −α(j) α(j) −α(k) =c 2.(8) By (6) we ha e x−α(i)y x−α(k)y=µ(i) µ(k) η(i) 1 η(k) 1a1 ···η(i) η(k) a whence using (8) we ob ain Hη(i) 1 η(k) 1a1 ···η(i) η(k) a ⩽c 2+Hµ(k) µ(i) =c3. This means o any in ini e alua ion o Lij k we ha e a1· η(i) 1 η(k) 1+···+a · η(i) η(k) ⩽c3. ARTICLE IN PRESS UNCORRECTED PROOF S0022-314X(05)00211-8/FLA AID:3277 Vol.•••(•••) [DTD5] P.8 (1-17) YJNTH:m1 1.50 P n:17/11/2005; 9:56 yjn h3277 by:Vi a p. 8 8I. Gaál, M. Pohs / Jou nal o Numbe Theo y ••• (••••)•••–••• 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 10 10 11 11 12 12 13 13 14 14 15 15 16 16 17 17 18 18 19 19 20 20 21 21 22 22 23 23 24 24 25 25 26 26 27 27 28 28 29 29 30 30 31 31 32 32 33 33 34 34 35 35 36 36 37 37 38 38 39 39 40 40 41 41 42 42 43 43 44 44 45 45 No e ha by in e changing iand kwe ge he same exp ession on he le -hand side wi h opposi e sign: o his eason he inequali ies a e also alid wi h absolu e alues:  a1· η(i) 1 η(k) 1+···+a · η(i) η(k)  ⩽c3.(9) No e ha he uni s in he abo e o mula ha e ze o alues a ini e alua ions. The in- equali ies o ype (9) (ob ained o di e en choices o i, k) can be used o de e mine all possible alues o he exponen s a1,...,a . Fo any possible exponen ec o a1,...,a we can de e mine η=ηa1 1···ηa in (5). Then he sys em o equa ions x−α(1)y=µ(1)·η(1),x−α(2)y=µ(2)·η(2) can be used o de e mine he co esponding x,y. Case II. I in (7) we ha e τij k =(α(j) −α(k))β(i) (α(i) −α(j))β(k) ∈Lp ij k , hen using (5) we ob ain (α(j) −α(k))µ(i) (α(i) −α(j))µ(k) ·η(i) η(k) ∈Lp ij k . He e he las e m is a uni in Lij k hence o any ini e alua ion o Lij k (α(j) −α(k))µ(i) (α(i) −α(j))µ(k)  mus be di isible by p. This usually does no hold and he e is no Case II solu ion. O he - wise, τij k is a p h powe , say τij k =ψp ij k , we eplace τij k by ψp ij k and epea he a gumen . Rema k. In he abo e calcula ions se e al elemen s (e.g., α(i) −α(j)) a e con ained in sub ields o ype Lij =K(α(i),α(j))o Lij k . Since o elemen s in Lij he alues a any alua ion o Lij k can be easily calcula ed om he alues o he co esponding alua ions o Lij , hence in ac almos all calcula ions can be pe o med in he sub ields Lij which a e much easie o deal wi h, especially o la ge deg ees n. 5. Examples Example 1. In he i s example we do no need o apply he undamen al lemma. ARTICLE IN PRESS UNCORRECTED PROOF S0022-314X(05)00211-8/FLA AID:3277 Vol.•••(•••) [DTD5] P.9 (1-17) YJNTH:m1 1.50 P n:17/11/2005; 9:56 yjn h3277 by:Vi a p. 9 I. Gaál, M. Pohs / Jou nal o Numbe Theo y ••• (••••)•••–••• 9 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 10 10 11 11 12 12 13 13 14 14 15 15 16 16 17 17 18 18 19 19 20 20 21 21 22 22 23 23 24 24 25 25 26 26 27 27 28 28 29 29 30 30 31 31 32 32 33 33 34 34 35 35 36 36 37 37 38 38 39 39 40 40 41 41 42 42 43 43 44 44 45 45 Le k=F5,le K=k( ),le αbe a oo o y3+ 7+ =0 and le L=K(α). Conside he Thue equa ion NL/k( )(x −αy) =1inx,y ∈k[ ].(10) Deno e by α(1),α(2),α(3) he conjuga es o α. Using symme ic polynomials we ha e α(3)=−α(1)−α(2) and subs i u ing i in o α(1)α(3)+α(2)α(3)+α(1)α(2)=0 we ob ain α(2)2+α(1)α(2)+α(1)2=0 whence α(2)=4α(1)±α(1)√−3 2=34α(1)±α(1)√2.(11) Obse e ha √2 is con ained in F25, a quad a ic ex ension o K, hence in his case M=Lα(1),α(2),α(3)=F25( )(α). Deno e by wa gene a ing elemen o he mul iplica i e g oup F∗ 25 o F25 wi h 2=w6,3=w18,4=w12. By (11) we ha e α(2)=w16α(1),α (3)=w8α(1). Siegel’s iden i y ge s he o m w8x−α(1)y+x−α(2)y+w16x−α(3)y=0.(12) In ou case Mhas one in ini e alua ion. In he abo e equa ion all e ms a e uni s ha - ing ze o alues a all ini e alua ions. By he p oduc o mula hei alue a he in ini e alua ion is also 0, hence hey a e con ained in he cons an ield F25. Equa ion (12) leads o he uni equa ion w4x−α(1)y x−α(3)y+w20 x−α(2)y x−α(3)y=1 ARTICLE IN PRESS UNCORRECTED PROOF S0022-314X(05)00211-8/FLA AID:3277 Vol.•••(•••) [DTD5] P.16 (1-17) YJNTH:m1 1.50 P n:17/11/2005; 9:56 yjn h3277 by:Vi a p. 16 16 I. Gaál, M. Pohs / Jou nal o Numbe Theo y ••• (••••)•••–••• 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 10 10 11 11 12 12 13 13 14 14 15 15 16 16 17 17 18 18 19 19 20 20 21 21 22 22 23 23 24 24 25 25 26 26 27 27 28 28 29 29 30 30 31 31 32 32 33 33 34 34 35 35 36 36 37 37 38 38 39 39 40 40 41 41 42 42 43 43 44 44 45 45 γ1=α(2)−α(3)x−α(1)y, γ2=α(3)−α(1)x−α(2)y, γ3=α(1)−α(2)x−α(3)y, hen we ha e γ1+γ2+γ3=0.(26) The ield Mhas genus 13. I has eigh in ini e alua ions, all o deg ees 1. The x−α(i)y a e uni s, ha ing nonze o alues only a he in ini e alua ions. The quo ien s α(1)−α(2) α(1)−α(3),α(2)−α(3) α(2)−α(1),α(3)−α(1) α(3)−α(2) ha e nonze o alues all oge he a ou ini e alua ions, wo o hem being o deg ee 4, he o he wo o deg ee 6. Deno e by V0 he se o he eigh in ini e and hese ou ini e alua ions. Then by he undamen al Lemma 3.1 γi/γjis ei he o bounded heigh , o is con ained in M5. Case I. Assume Hγ1 γ3⩽2·13 −2+(8+12 +8)=52. This implies Hx−α(1)y x−α(3)y⩽52 +Hα(1)−α(2) α(2)−α(3)=65.(27) As we men ioned abo e, Khas uni ank 3. We deno e by ε1,ε2,ε3 he undamen al uni s. Conside ing he alues o ε(i) h ε(k) h (a in ini e alua ions) o h=1,2,3, by (27) and (9) we become x−αy =µ·εa1 1·εa2 2·εa3 3 wi h a oo o uni y µin kwhe e among o he s he exponen s sa is y |50a1+6a2+54a3|⩽65, |51a1+5a2+54a3|⩽65, |49a1+3a2+54a3|⩽65. ARTICLE IN PRESS UNCORRECTED PROOF S0022-314X(05)00211-8/FLA AID:3277 Vol.•••(•••) [DTD5] P.17 (1-17) YJNTH:m1 1.50 P n:17/11/2005; 9:56 yjn h3277 by:Vi a p. 17 I. Gaál, M. Pohs / Jou nal o Numbe Theo y ••• (••••)•••–••• 17 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 10 10 11 11 12 12 13 13 14 14 15 15 16 16 17 17 18 18 19 19 20 20 21 21 22 22 23 23 24 24 25 25 26 26 27 27 28 28 29 29 30 30 31 31 32 32 33 33 34 34 35 35 36 36 37 37 38 38 39 39 40 40 41 41 42 42 43 43 44 44 45 45 The e a e abou 8000 solu ions (a1,a 2,a 3)o he abo e sys em o linea inequali ies. Tes ing all possible exponen ec o s we ound ha Eq. (24) has only he i ial solu- ions (x, y) =(1,0), (2,0), (3,0), (4,0)( hese yield in ac he mul iplies o x−αy o x=1,y=0 wi h oo s o uni y in k). Case II. To exclude γ1 γ3∈M5we conside γ1 γ3=α(2)−α(3) α(1)−α(2)·x−α(1)y x−α(3)y. The second e m on he igh -hand side is a uni , hence α(2)−α(3) α(1)−α(2) should be di isible by 5 a all ini e alua ions . This is no sa is ied, howe e . (Simila ly o γ1/γ2and γ2/γ3.) Compu a ional expe iences. All compu a ions used in he examples we e pe o med by using he compu e algeb a sys em KASH [2], unning on 1 GHz PC-s. The calcula ions ook jus some seconds wi h he excep ion o he es o abou 8000 possible exponen ec o s in Example 4 which ook abou 90 minu es. Acknowledgmen The au ho s a e hank ul o he e e ee o his/he aluable ema ks ha lead o an im- p o emen o he pape . Re e ences [1] A. Bake , T anscenden al Numbe Theo y, Camb idge Uni . P ess, Camb idge, 1990. [2] M. Dabe kow, C. Fieke , J. Klüne s, M. Pohs , K. Roegne , K. Wildange , KANT V4, J. Symbolic Com- pu . 24 (1997) 267–283. [3] C. Fieke , Übe ela i e No mgleichungen in algeb aischen Zahlkö pe n, PhD hesis, Be lin, 1997. [4] I. Gaál, Diophan ine Equa ions and Powe In eg al Bases, Bi khäuse Bos on, Bos on, 2002. [5] K. Gy˝ o y, Bounds o he solu ions o no m o m, disc iminan o m and index o m equa ions in ini ely gene a ed in eg al domains, Ac a Ma h. Hunga . 42 (1983) 45–80. [6] R.C. Mason, Diophan ine Equa ions O e Func ion Fields, Camb idge Uni . P ess, Camb idge, 1984. [7] H. Niede ei e , C. Xing, Ra ional poin s on cu es o e ini e ields, in: London Ma h. Soc. Lec u e No e Se ., ol. 285, Camb idge Uni . P ess, Camb idge, 2001. [8] W.M. Schmid , Thue’s equa ion o e unc ion ields, J. Aus al. Ma h. Soc. Se . A 25 (1978) 385–422. [9] H. S ich eno h, Algeb aic Func ion Fields and Codes, Sp inge , Be lin, 1993. [10] A. Thue, Übe Annähe ungswe e algeb aische Zahlen, J. Reine Angew. Ma h. 135 (1909) 284–305. [11] E. Weiss, Algeb aic Numbe Theo y, New Yo k, 1963.