A STRONG BOUND FOR THE NUMBER OF SOLUTIONS OF THUE EQUATIONS

Transcription

1 Shukhat Alladustov A STRONG BOUND FOR THE NUMBER OF SOLUTIONS OF THUE EQUATIONS Maste s thesis, Jul 05 Supeviso: Pof. Yui Bilu Leiden Univesit Univesit of Bodeau

2 Contents Intoduction 3 Additional theoems and facts 5. The Thue theoem about finiteness of solutions Disciminant of the polnomial Simplification of the fom F and nomalization Decomposition fo Z Classification of polnomials Mahle height and the Mahle inequalit Hadama s inequalit The Lewis and Mahle estimation The Thue-Siegel pinciple Some definitions and facts on numbe fields Admissible tiple An uppe bound fo the numbe of solutions 0 3. Lage solutions Stong gap pinciple Small solutions

3 Chapte Intoduction AThueequationisaDiophantineequationofthefom F, m (.0.) whee F, Z, is an ieducible homogeneous polnomial in two vaiables of degee 3 and m Z is an intege numbe. It is named afte Ael Thue, who in 909 has shown that equations of this kind has onl finitel man intege solutions. We will discuss the poof late on. Othe appoaches to the poblem wee pefomed also b Th.Skolem and A.Bake. Th.Skolem used p-adic powe seies, unde weak estictions on F and A.Bake used lowe bounds fo linea foms in logaithms. Late, thee have been put anothe question: Does thee eist an uppe bound fo the numbe of solutions not depending on the polnomial F but onl on the degee and intege m? The fist esults wee obtained b C.L.Siegel fo a specific case when 3 and F, a 3 b 3 Z. But, the question was answeed b J.H.Evetse ( ) in the geneal case. Evetse s method was based on the ideas of Siegel and Bake to educe the poblem to the specific cases which ae moe comfotable to wok with. Fistl, he consideed the equation of the fom a b c whee he obtained w c 4 as an uppe bound fo the numbe of solutions with w c is an intege numbe depending on c N. Also, the equation F, with F is a cubic polnomial with positive disciminant was investigated. As a esult in this case, it was obtained that the equation has at most twelve solutions. Finall, some appoaches in equations in numbe fields and techniques of Pade appoimation wee used to obtain the following uppe bound fo the numbe of pimitive solutions of the equation t whee t is the numbe of pime divisos of m. In ou wok we conside a paticula tpe of the Thue equations F, (.0.) unde the same conditions fo F, Z, as above and we deal with the poblem of finding an uppe bound fo the numbe of intege solutions, we descibe the method of E.Bombiei and W.M.Schmidt. 3

4 4 It is clea that each intege solution of.0., if eists, consist of copime numbes,. Also, if, is a solution then, is so too. Theefoe fo simplicit we conside onl one of these solutions, moe pecisel thoughout the thesis we onl wok with solutions such that 0withoutmentioning. As a main esult we deive the following elation fo the numbe of solution of.0.. Theoem.0.. Let F, Z, be an ieducible homogeneous polnomial of degee 3. Then fo c, the numbe of intege solutions of the equation F, which solutions, and, ae egaded the same, does not eceed 5, whee c 0 is an absolute constant. Fistl, we classif the polnomials satisfing the above conditions accoding to the numbe of solutions of.0..thenitallowsustoconsideoneepesentativefomeach class instead of woking with all polnomials. We also show that as such a epesentative we can take the polnomials whose leading coe cient is, fo this pupose we constuct the so-called auilia foms. Also, we deive a decomposition of Z into finitel man smalle sets Z n i 0Z i, using this decomposition we can onl deal with polnomials whose disciminants ae big enough. Hee, a ke point is the fact that the numbe of solutions of.0. does not eceed the sum of the numbes of solutions in each Z i, i 0,...,n. The poof of the main theoem is based on two steps, fistl we deal with finding a bound fo the numbe of solutions, whose heights H, ma, is not less than some fied numbe M 0 and then fo the solutions, with M,which we call lage and small solutions espectivel. Obviousl, lage and small solutions cove all the solutions. In the fist case, we need some additional theoems. We state the Lewis and Mahle estimation and the Thue-Siegel pinciple in specific cases ( 5 ) with poofs. Also, we develop the so-called Stong gap pinciple. Using these appoaches we conclude that 3 is an uppe bound povided su cientl lage. In the second case, we use an auilia polnomial and some ticks to show that the bound is. Actuall, this esult can be applied to obtain a bound fo the numbe of solutions of.0. too, moe pecisel if the numbe of solutions of (.0.) does not eceed N then N t is an uppe bound fo the numbe of solutions of.0.,whee t is the numbe of pime factos of intege m.

5 Chapte Additional theoems and facts. The Thue theoem about finiteness of solutions As we mentioned befoe, we fist show that.0. has onl finitel man solutions unde fied coe cients. Fo this we need the following theoem due to Thue. Theoem.. (Thue). Let C be an algebaic intege numbe of degee, then fo an " 0 and c 0 the inequalit p q c q " can have onl finitel man solutions with p Z and q N. This esult will be diectl used in the poof of the following theoem. Theoem... Let F, a 0 a... a Z,, 3 be an ieducible polnomial with a 0 0 and let m Z be an intege, then the numbe of intege solutions of the equation is finite. F, m (..) Poof. We conside the following two cases sepaatel: ) m 0 then the equation has no intege solutions ecept tivial one, because F is an ieducible homogeneous polnomial of degee 3. ) Suppose m 0. We conside the polnomial f z F z, then we have that f z a 0 z a z n... a z a Z z is also an ieducible polnomial in one vaiable, let,..., n be its oots. Then we have the factoization F, f a 0... So, solutions of (..) satisf the elation... m a 0 (..) 5

6 6.. The Thue theoem about finiteness of solutions Poduct of numbes equals to m a 0 theefoe thee eists k, k with k m a 0 (..3) Take 0 such that min i j,i j,itispossiblesince,..., ae the dieent oots of the ieducible polnomial. The inequalit (..3) then implies i k i k k i k m a 0 (..4) fo all i,...,, i k. It is clea that if is bounded then the numbe of solutions is finite. Theefoe, thowing awa finitel man solutions, if needed, one can assume that m a 0 Then fom (..4) we obtain that Thus, fo the poduct of i,i,...,,i k. componentsthefollowingelationholds i k i Now, appling this fact in.. we obtain k c,c m a 0 o equivalentl, fo almost all the solutions, we have that k c (..5) But, b the Thue theoem the last inequalit has onl finitel man solutions, Z, 0 fo an intege n and c 0. We have that all, but finitel man intege solutions of.. satisf the elation..5, which implies that the numbe of solutions cannot be infinite. Coolla... Let F be as in the definition of the last theoem. Then the equation has onl finitel man solutions. Poof. The poof follows fom consideing the two equations F, (..6) F,. Each of these equations is a paticula case of the Theoem.. with m m, theefoe the numbe of solutions of..6 isalsofinite. and

7 7.. Disciminant of the polnomial. Disciminant of the polnomial Let F, a 0 a... a be an ieducible homogeneous polnomial with intege coe cients, of degee 3. Suppose,..., be the oots of the polnomial then we have Now, we define the disciminant D F the Vandemonde mati as D F a 0 det V,moepecisel f F, a 0 a... a F, a 0... V of the polnomial F using the deteminant of... n... n n D F a 0 i j i j (..) Let us see what happens if we change, b A,,whee A is a mati with a b non-zeo deteminant of the fom A GL c d Z and A, a b, c d. Conside the polnomial F A, F a b, c d.then F a b, c d b 0 b... b fo some integes b 0,...,b. Evaluating F A at, 0 we can see that Conside the equation b 0 F a, c (..) F A, a 0 a b a a b c d... a c d 0 We note that we can assume c d 0, othewise the equation has a solution onl if a b 0, but this contadicts to the assumption that det A 0. So, dividing both sides of the equalit b non-zeo tem c d we obtain a b a 0 c d a b a c d... a F Accoding to ou assumption, the equation F z, solving the equations a b i c d fo i,..., we obtain that the oots of..3 ae a b c,..., a a b c d, 0 (..3) 0 has solutions,...,,then b c

8 8.3. Simplification of the fom F and nomalization Then fo the dieence of these oots we have i j a i b i c a j b det A j c j i a i c a j c and we obtain the following elation i j i j det A i j j i a i c a j c det A i j j i F a, c Taking into account the fact (..) we get D F A b 0 i j i j b 0 a 0 det A i j j i det A D F F a, c Thus, the change of vaiables, A, gives the following elation between the disciminants of the polnomials F A and F D F A det A D F (..4).3 Simplification of the fom F and nomalization.3. Decomposition fo Z Fo simplicit we need to conside the foms satisfing cetain popeties. Moe pecisel, we want to wok with polnomials which have quite big disciminants. Lemma.3.. Fo an pime p we have the following decomposition of Z, Z p j 0 A j Z whee A 0 p 0 0,A j 0 p j,j,...,p Poof. Indeed, take an pai, Z.If 0 mod p,then p,some Z, then A 0 p 0 0 p,, Now let 0 mod p.weagainconsidetwocases: ) 0 mod p,then p fo some Z then taking, Z we obtain A p, 0 p p, p p, ) 0 mod p.thenwewanttofind, Z and j, j p suchthat A j, 0 p j,p j,

9 9.3. Simplification of the fom F and nomalization Take,thenitisenoughtoshowthattheequation p j has a solution,j Z, j p. It is clea that j mod p has such a solution j j p since, 0 mod p,i.e., j 0 mod p.choose, p then A j,,. Thus, fo an pai, Z thee eist j, 0 j p and, Z such that, A j, i.e., the decomposition above holds. This decomposition allows us to estict ouselves with finding an uppe bound fo the numbe of solutions in each A i Z.Then,thenumbeofallsolutionsof.0. in Z does not eceed the sum of numbe of solutions in all these sets. We note that if, A j, is a solution of.0. then F, F A j, F Aj, and fom..4 we have that D F Aj p D F,since det A j p fo an 0 j p. Now denote b N p the numbe of solutions of.0. fo the polnomials with D F p Then the numbe of solutions in each set A j Z does not eceed N p,theefoefo N, the numbe of all the solutions in Z we have N p N p We note that p is an pime numbe that we have not put an additional conditions et, it will be fied at the end of ou discussions..3. Classification of polnomials Let F and G be two ieducible homogeneous polnomials in two vaiables of the same degee. Then these polnomials ae called to be equivalent, if the equations F, and G, have the same numbe of solutions, and fo equivalent polnomials F and G we use the notation F G.Thefisteaseampleofequivalentpolnomialsis F and F,whee F satisfies the conditions above. Also, we note that if A SL Z,then F A F. B this definition of equivalence the set of ieducible homogeneous polnomials of degee 3 is divided into seveal classes. Using, this classification we can eplace polnomials with equivalent foms which is easie to wok with. Now suppose that.0. has a solution 0, 0 fo some fied F.Then,theeeists A Sl Z,suchthat A 0, 0, 0.Theefoe,, 0 is a solution of of the equation F A,. We have that F A, 0 b 0,whee b 0 is the leading coe cient of the polnomial F A,hence b 0. As F, F A and F A ae equivalent, theefoe consideing an equivalent fom F A we can estict ouselves to foms which ae nomalized, that is with the leading coe cient equal to. Theefoe, without loss of genealit we can assume that the polnomial F is nomalized if needed.

10 0.4. Mahle height and the Mahle inequalit.4 Mahle height and the Mahle inequalit Let F, Z, be a polnomial as in.0.,denote f F,,thenitis ieducible polnomial in one vaiable and let,..., be its oots. We define the Mahle height of the polnomial F as M F a 0 with a 0 is the leading coe cient of F..4. Hadama s inequalit i ma, i Is the Mahle height and disciminant of the polnomial F elated to each othe, if so, what is the elation? To answe this question we need the following theoem b Hadama. Theoem.4.. Let A a a... a n a a... a n a n a n... a nn be a mati with comple enties. Then we have the following inequalit det A n i a i a i... a in Now, we appl this theoem to the Vandemonde mati, V Then we obtain that det V i i... i i ma, i i ma, i That is, a 0 det V a 0 i ma, i So, fom the definitions of D F and M F we obtain the following elation between the Mahle height and the disciminant of the polnomial F D F M F (.4.)

11 .5. The Lewis and Mahle estimation.5 The Lewis and Mahle estimation We ecall that in this and all the net sections we assume that F is ieducible homogeneous polnomial in two vaiables with intege coe cients, of degee 3. Let and be two copime intege numbes, then we define the height of the point, b H, ma, Then we have the following estimation b Lewis and Mahle. Lemma.5.. Fo an pai of copime integes,, with 0 we have that min min, M F F, H, whee uns though all the oots of the polnomial f F,. Poof. Let g f whee i fo some i,..., and f a 0....Then g is a polnomial of degee. We use the same notation fo the disciminant of f as fo F,i.e., D f D F. Then we have D f a 0 i j f D g i j Also, fom (.4.) we have that D F M F. Then using the last fact fo G, F, taking into account the fact that M G M F we obtain f D F D G D F M F As D F is a non-zeo intege we have D F, then f M F (.5.) Let, be a fied solution, then without loss of genealit one ma assume that min,..., Also, eodeing the oots if necessa we suppose that i, if i,..,n i, if i N,.., fo some N. Then N i N N N N N i i (.5.)

12 .5. The Lewis and Mahle estimation Fo, i N,..., wehavethat i Theefoe i i i i Then taking the poduct we obtain i N i N i N i (.5.3) We have that F, a 0... a 0 N i i i N i Then (.5.) and (.5.3) impl that using the fact we obtain F, a 0 i f a 0 i i i F, f Now we conside two cases on H,. Fist case, H,.Thenfomtheelation.5.4 we have that (.5.4) F, H, f Taking into account the elations.5. and the last inequalit we obtain F, M F F, H, f H, M F F, H, Theefoe Second case, H, min,.wehavethat M F F, H, F, a 0... a

13 3.5. The Lewis and Mahle estimation Again without loss of genealit one ma assume min whee minimum is taken though all the oots of f. Denote, ' F,.Then,..., ae its oots, epeating the same pocedue as we followed fo f in this case we obtain that Again using.5. fo ' we have F, ' M F F, M F F, (.5.5) H, Now, denote If we have ma,,..., then appling this diectl to (.5.5) we get M F F, H, min, (.5.6) Othewise if then accoding to the definition of theefoe we have that Then Theefoe fom (.5.5) we obtain M F F, H, min, (.5.7) Fom the elations (.5.6) and (.5.7), taking into account the fact that conclude that we can

14 4.6. The Thue-Siegel pinciple Also, we note that min, Theefoe we obtain M F F, H, ma,,..., M F M F F, min, H, Thus, in both cases the desied esult is poven..6 The Thue-Siegel pinciple.6. Some definitions and facts on numbe fields Let K be a numbe field with K : Q 3. Denote b M K the set of places on K,thenfoan K we have the following poduct fomula M K d whee d K : Q with Q and K completions of Q and K espectivel, with espect to an absolute value M K. Definition.6.. The height of a numbe and logaithmic height is H h K is M K ma, d K : Q MK d log whee log ma 0, log assuming that log 0: 0 and d K : Q as above. We note that in the geneal case we can define the height of an algebaic numbe Q in the same wa, with K is an numbe field containing,thenthisdefinitionis well defined and does not depend on the choice of K. Definition.6.. Fo a vecto,..., n A n K the a ne height h A is h A whee : ma, 0,..., n. K : Q MK d log We also define the logaithmic height h P of a polnomial P, K, as the a ne height of the vecto consisting of the coe cients of P. Let K and suppose that fosomeplace M K. We sa that a ational numbe Q appoimates if. K:Q

15 5.6. The Thue-Siegel pinciple.6. Admissible tiple To descibe the Thue-Siegel pinciple coesponding to ou case, fist we give a definition of admissible tiple. Also, fom now on we alwas assume that the polnomial F is nomalized accoding to the discussions in the Section.3.. Definition.6.3. Let t 0 be an positive numbe and let, be two ational numbes that appoimate K. We sa that A,A, is an admissible tiple fo,,,t,, whee 0 is an abita numbe, if fo all positive numbes d,d with d d thee eists a polnomial P, Q, that satisfies the following thee conditions (i) deg P d,deg P d whee deg i P is the highest degee of i in the polnomial P, i, and fo all i,i with (ii) thee eists j,j with i i i i P, 0 i d i d t and j j (iii) the following elation is satisfied j j P, 0 j d j d when d,d ae big enough with d d. h P A d A d o d o d Theoem.6. (Thue-Siegel pinciple). Let 0 t and let,,, be the same as in the definition above, and let A,A, be an admissible tiple fo this data. Suppose also 0 t, and If then we have that o whee t. t and 3e A h 3e A h t log 3 A h log 3 A h

16 6.6. The Thue-Siegel pinciple Poof. As A,A, is an admissible tiple fo,,,t,,fomthedefinitionit follows that thee eists a polnomial P, Q, satisfing the popeties i, ii and iii. Accodingl, thee eist j,j, such that j!j! j j j d j d j j P, 0 we denote this non-zeo value b. Then fom the poduct fomula we have that M K d log d log d log 0 Using the Talo epansion aound a point, we have the following elation fo the value of i i,i!i!j!j! i i j j i j i j P, i i Accoding to the choice of P, we have that also we have that j j i i i j i j theefoe it holds fo all i,i with P, 0, fo j i d j i d j d j d t i d i d t Now, we estimate the values of log fom above fo all M K. Fo this, we conside the following cases sepaatel: ), i.e., absolutevaluethat is esticted fom the usual achimedian absolute value on C,) achimedian ecept the fist case 3) is non-achimedian. ).ThenfomtheTaloepansionaboveweobtainthat log ma log i,i i!i!j!j! i i j j i j i j P, i i o d o d Let deg P d and deg P d, then non-zeo coe cients in the epansion of ae of the fom k!k! j!j!i!i! k i j! k i j! fo some k,k such that i j k d and i j k d. Now, we want to find a bound fo the logaithmic absolute values of these coe the following lemma descibes this elation. cients,

17 7.6. The Thue-Siegel pinciple Lemma.6.. The following inequalit holds k!k! ma log k d,k d i!i!j!j! k i j! k i j! whee F u, v u log u v log v F u, v u log u Appling this lemma we obtain d F i d, j d d F i d, j d u v log u v if u v v log v if u v log log P d log d log ma d F, j d F, j d log d log, d d whee maimum is taken ove t,0 Dieentiating and taking into account the facts that j d, 0 t and t and afte some calculations we get that the maimum is eached fo t,then t and t,then j d. F, j d F t, j d log 3 and F, j d log 3 Also, consideing the maimum ove t we obtain ma d log d log t min d log So, we have that d log log log P d log d log d log 3 d log 3 t min d log d log ) achimedian ecept the fist case. Then, using the fact that ma k d log k i d log 3 we obtain log log j!j! j j j j P, log P d log d log d log 3 d log 3 o d o d 3) is non-achimedian. Then we have that n foalln Z,theefoe log log j!j! j j j j P, log P d log d log

18 8.6. The Thue-Siegel pinciple Then, and 3 impl that K d v log d v log, d v log d v log K d v log P d K d v log d K d v log d log 3 d log 3 t min d log,d log o d o d h P d h d h d log 3 d log 3 t min d log,d log On the othe hand, fo Q fom poduct fomula we have d v log 0 K Accoding to the choice of A and A,fo d d we have h P A d A d o d o d. Now, choose and d d D log 3 A h D log 3 A h whee D R 0 lage enough. If d d then log 3 A h log 3 A h Othewise, if d d then, h P A d A d o d o d fo D big enough, appling this we obtain o d o d 0 d log 3 A h d log 3 A h t min d log,d log Then, the condition and t min d log impl,d log 3e A h 3e A h d log 3 A h d log 3 A h

19 9.6. The Thue-Siegel pinciple Eample of an admissible tiple To appl the Thue-Siegel pinciple we need some paticula admissible tiple fo,,,t,. Fo this pupose we descibe two facts below fom the discussions in 4. Lemma.6.. Let t and let Q be an algebaic numbe of degee 3 ove Q, if A,A, is admissible then A,A, 0 with 0 t is also admissible. Lemma.6.3. We have that t A A h t with 0 t is an admissible tiple fo,,,t,. Coolla.6.. Let t, t t t and A log M F. Also t suppose that. t Let be an algebaic numbe of degee, and then whee 4e A H, and log 4e A log H, Poof. Fo the chosen value fo A and 4e A H, log 4e A log H, t we alead have that 4e A H, t 4e A H, t Then, if, the esult immediatel follows fom the Thue-Siegel pinciple above. Othewise, if then 4e A H, and the fact that appoimates stongl (i.e., the dieence is small enough) impl 4eA H, 3e A H, then again Thue-Siegel pinciple fo,, gives the desied esult, since Mahle heights of F, and F, ae the same.

20 Chapte 3 An uppe bound fo the numbe of solutions 3. Lage solutions We fist conside the solutions of the equation F, (3..) which ae lage, i.e., solutions, such that H, M fo some M 0 big enough, which will be fied late. 3.. Stong gap pinciple Now, classif the oots of ( 3..) dividing them into classes, we call that two solutions, and, belong to the same class if min 0 and min 0 fo some 0,aootofthepolnomial f,whee uns though all the oots. Now, take an such a class, then we can numeate the elements as,,,,... and eodeing if necessa suppose that H, H,... Then fo two consequent solutions n, n and n, n fom one class we have the following elation n n n n n n n n 0 0 n n n n 0 0 n n Then Lewis and Mahle estimation stated befoe implies n n 0 n n 0 C F n, n H n, n C F n, n H n, n whee C M F. 0

21 3.. Lage solutions Accoding to ou assumption H n, n H n, n and F n, n F n, n as n, n and n, n ae solutions, theefoe n n n n n n C H n, n (3..) Theoem 3.. (Stong gap pinciple). Suppose that H, n,,... we have C. Then fo each H n, n C n H, Poof. Fistl, taking into account the elation (3..) we have H i, i i i i i H i, i C i H i, i C fo all i,,... Appling this fact seveal times we obtain H n, n H n, n C We have that H n, n C... H, n C... n C... n C n C n Thus, H n, n H, n C... n C n H, Now, we combine the stong gap pinciple, the coolla of the Thue-Siegel pinciple and the Lewis and Mahle estimation. Fo this pupose we choose M such that,,... appoimate 0 good enough, 0 is some oot of the polnomial f. Fom the Lewis and Mahle estimation we have that fo eve i.so,ifweassumethat 0 i i C H i, i then M C 4e A C H i, i 4e A H i, i Then the coolla of the Thue-Siegel pinciple implies log 4e A log H n, n log 4e A log H, as H n, n C n H,

22 3.. Lage solutions we have that log H n, n n log C H, n log H, log C Theefoe, log 4e A n log H, log C log 4e A log H, o n log M log 4e A log M log C when log M log C,whichholdsinoucaseaccodingtoequiementsfo M and the choice of A.Ifwechoose then we get M C 4e A (3..3) n log C log 4e A log C log 4e A log C log 4e A log C log 4e A since accoding to ou assumption.then log C log 4e A log C log 4e A n log log n log log log log log So, the numbe of copime solutions,,povided, and, ae counted as one, of the equation F, such that H, M,whee M is the same in 3..3 does not eceed log log since we divided the solutions into dieent classes and in each such a class thee is no log moe than elements. log As this fact tue fo all t and satisfing the conditions t, t t and Now, we choose t a, bt with 0 a b then the satisf the above conditions. Fo this data we have that

23 3 3.. Small solutions and Then we obtain t b a b t b b a that is log log log log log b a log fo big enough, theefoe the uppe bound is 3 in this case. 3. Small solutions Now, we deal with small solutions, i.e., the solutions, with M, whee M is the same as in the last section. We emind that we classified the polnomials in the pat.3.. and in this section we alwas assume that the polnomial F has the smallest Mahle height in the class that it belongs. The idea we ae going to follow is changing the polnomial F with anothe polnomial which has the same numbe of solutions as F which is easie to wok with. We note that if A SL Z,then F A and F ae equivalent, that is the equations F, and G, (3..) have the same numbe of solutions, whee G, F A, F a b, c d with a b A SL c d Z. Theefoe fo simplicit we can change F with F A, A SL Z if needed. And we also assume that the leading coe cient of F is. Auilia polnomial As befoe, let,..., be the oots of the polnomial f F, then F,... We use the notations L i, : i, i,..,,then F, L,... L, Then, if 0, 0 is an intege solution of the equation F, then L i 0, 0 0,foall i,...,. As gcd 0, 0,theeeistsapai 0, 0 Z with , i.e., 0 0 SL 0 Z that is 0, 0 and 0, 0 is a basis fo 0 Z.Thenfoan, Z we have a decomposition, a 0, 0 b 0, 0,fo some a, b Z.Infact, a 0 b 0 0 a 0 b 0 b b

24 4 3.. Small solutions Then,, a 0, , 0 Theefoe fo all solutions, and fo all i,..., we have that L i, L i 0, 0 a 0 0 L i 0, 0 L i 0, 0 a 0 0 i whee i L i 0, 0 L i 0, 0.Then, L i, L i 0, 0 L j, L j 0, i j (3..) Now, let 0, 0 be a fied solution, then define an auilia polnomial G,as G v, w v w... v w v L 0, 0 L 0, 0 w... v L 0, 0 L 0, 0 w We have that L 0, 0... L 0, 0 theefoe G v, w i L i 0, 0 v L i 0, 0 w i 0 v 0 w 0 v 0 w i F 0 v 0 w, 0 v 0 w F X v, w whee X 0 0 SL 0 Z.Theefoe G is equivalent to F,wehavethat G u, v 0 implies G u, v, we choose a sign in such a wa that G, 0, i.e., leading coe cient equal to. As G is well defined fo all the solutions 0, 0,wecanconside the specific case when 0, 0, 0 which is cleal a solution. In this case we have that L i 0, 0 L i, 0 foall i and 3.. can be ewitten as L i, L j, i j (3..3) fo all solutions,.iffothepoductof positive numbes we have a... a then thee is at least one with a i. Using this fact fo L,... L, we can conclude that i, i such that L i,. Then 3..3 implies that L j, L j, i j i j (3..4) fo all j.foan, R and C we have the equivalence theefoe 3..4 is equivalent to L j, i j (3..5) Fom 3..4 and 3..5 we obtain the following L j, i j i j i j j i Re j

25 5 3.. Small solutions that is we have L j, i Re j Fo all j i thee eists an intege m,dependingon j,thatisonthesolution, such that m Re j,then L j, i m m Re j i m (3..6) As we mentioned befoe, we onl deal with solutions,, 0. Now, we again classif the solutions, of.0.,with M.Wenotethatthesetofallthesolution, with H, M is coveed in this wa. Denote X i the set of solutions, such that M and L i,. Among the elements of X i we have the following elations. Lemma 3... Let,,, X i be two dieent elements and suppose. Then 7 ma, i m (3..7) Poof. As,,, X i ae dieent solutions we have that 0, also it is an intege numbe, so. On the othe hand det det i i i i L i, L i, L i, L i, Theefoe L i,.thenthiselationtogethewith 3..6 impl that That is o L i, L i, ma, i m i m i m Also, it can be easil checked that ma, z 3 4 z 0. Thus, i m ma,z fo an eal numbe 7 L i, i m i m 7 ma, i m

26 6 3.. Small solutions Accoding to this fact, if a solution, with 0 belongs to a class X i,fosome i i.e., L i,,thenfootheelements, X i,with we have ma, 7 i m. Now we want to investigate the case when, is a solution with M not belonging to X i,i.e., L i,.inthiscase,fom 3..6 we obtain that i m L j, Then dividing both sides b positive intege we obtain 7 i m (3..8) We conside the set X,whichconsistofallsolutionsoftheequation F,, with M,buttheelementswiththelagest fom each class X i, i,..., is ecluded if X i is not empt, i.e., at most elements emoved. At the beginning we took a pime p without an conditions on it. Now, we conside 7 apimenumbe p, such that p.thenwehavethefollowingtheoem. Theoem 3... Fo an " 0 and p, " the cadinalit of the set X satisfies the inequalit X " log 7 log p fo some p, " 0, depending on pime numbe p and ". Poof. Take some X i and ode the elements,,..., v, v,suchthat... v. Then accoding to the definition, the solution v, v is not included in X.Thenfo these elements fom 3..7 we have that fo k,...,v. Theefoe, k k 7 ma, i k, k m k, k (3..9) X i X 7 ma, i k, k m k, k v v v v... v v M and fo othe elements, in X but not in X i,fom 3..8 we have that Theefoe X 7 ma, i k, k m k, k 7 ma, i k, k m k, k M (3..0) We had that G v, w i v iw is equivalent to F. Also, we should note that G is equivalent to Ĝ i v i m w,theefoe F Ĝ.Weassumedthat F has the smallest height in its equivalence class, theefoe M Ĝ M F. Hence ma, i, m, M Ĝ M F i

27 7 3.. Small solutions Using the fact (3..0) fo all i,... and taking poduct of all of them we obtain 7 M F X M Since p 7 we have that M F 7 log M X log M F log 7 Accoding to the choice of A,t, and we have that fo some 0 p big enough and then A a log M F Also, we chose M as theefoe, M C 4e A log M log M F log log log 4 a log M F Accoding to the choice of t and we have t, a bt with 0 a b theefoe. Using these fact we obtain b t log M O log M F O Taking into account the fact that log M F log 7 we obtain " log M log M F fo " su cientl lage satisfing the condition 0 p too. Then X " log M F log M F log 7 " log 7 log M F (3..) Note that we ae consideing the polnomial with D F p also fom the elation D F M F we have that M F p Then appling this to (3..) we obtain X " log 7 log M F " log 7 log p log " log 7 log p

28 8 3.. Small solutions Now we sum up and make the things moe concete choosing a pime p.infact, b consideing the solutions, with H, M and the solutions with M (which contains all the solutions with H, M of the equation F, we cove all the solutions. Theefoe it is enough to obtain uppe bounds fo the numbe of the solutions in each of these sets. We have shown that in the fist case the bound is 3. In the second case, the set X contains all the solutions, with M,eceptat most numbes which has highest heights fom each set X i, i,..., and the solution, 0. Theefoe we have an uppe bound If p 7 X 4 and is lage enough, then we obtained that X " log 7 log p Now choose p 9, then we have that X. In conclusion, fo c,whee c big enough, which also satisfies all the conditions we equied fo in the last sections, we have that N 5,thatisthenumbeof intege solutions of the equation (.0.) is bounded b 5 fom above.

29 Bibliogaph [] E.Bombiei, W.M.Schmidt: On Thue s equation. Invent. math. 88, 69-8(983) [] Evetse, J.H.: Uppe bounds fo the numbe of solutions of Diophantine equations. Math. Centum. Amstedam, -7 (983) [3] Yui Bilu: Lectue notes on p-adic numbes and Diophantine equations and e- a [4] E.Bombiei, J.Muelle: On eective measues of iationalit fo b lated numbes. J.Reine Angew. Math. 34, 73-96(983) [5] Lewis, D., Mahle, K.: Repesentation of integes b bina foms. Acta Aith. 6, (96) [6] Evetse, J-H.: Lectue notes on Diophantine Appoimations deliveed in the Sping tem in Leiden Univesit, 04. [7] Shidlovski A.B.: Diophantine appoimations and tanscendental numbes / Diofantov piblizhenia i tanstsendentne chisla (Russian), (98) 9