A PROOF OF THE THUE-SIEGEL THEOREM ABOUT THE APPROXIMATION OF ALGEBRAIC NUMBERS FOR BINOMIAL EQUATIONS

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1 A PROO O THE THUE-SIEGEL THEOREM ABOUT THE APPROXIMATION O ALGEBRAIC NUMBERS OR BINOMIAL EQUATIONS KURT MAHLER, TRANSLATED BY KARL LEVY I 98 Thue () showed that algebraic ubers of the special for = p a b ca, for every positive, oly be sharply approxiated by fiitely ay ratioal ubers p q with the followig iequality holdig p q apple q ( ++ ). The proof uses, if perhaps i a soewhat hidde way, the cotiued fractio expasio of the bioial series ( )!. I further work about the approxiatio of algebraic ubers (,) faously Thue used istead a copletely di eret tool, the drawer ethod of Dirichlet, ad showed further that the above stateet holds for ay algebraic uber. Thue s ethods were later geeralied by Siegel(4,5,6,7) who showed, aog other thigs, that for every algebraic uber i the above iequality the expoet ++ could be replaced by + +, where is soe atural uber. This ote deostrates a geeraliatio of Thue s ethods i (); Like Thue I restrict yself to the roots = p a b of the bioial equatios. The cotiued fractio expasio of the bioial series is geeralied ad algebraic approxiatio fuctios are give istead of ratioal approxiatio fuctios. I doig so I proceed exactly as i y work o the expoetial fuctio(8). Itegrals are set up for the approxiatio fuctios; thus the estiates becoe uch easier ad you ca prove Thue s theore with Siegel s Expoets for the bioial algebraic equatios without di culty ad without use of the pigeohole priciple. I.. Let,,..., be atural ubers ad let!,!,...,! be coplex ubers such that o pairwise di erece! h! k (h, k =,,...,); h 6= k is a iteger. ro kow theores about hoogeeous liear equatios there are polyoials A k! (k =,,...,) that do ot siultaeously ad idetically vaish ad that are respectively of degree at ost,,...,, so that i the power series expasio of the expressio A k! X ( )! k = a... l l all coe ciets a l with k= apple l< are ero. We rewrite these expressios as R!... k! The it ca easily be show that d d k= ( )! R! l= ( )! k. Date: Jue5,5.

2 KURT MAHLER, Traslated by Karl Levy is the sae as R!!!... ad thus that cosequetly the coe ciet of the ( )-th power of i the power series expasio of R! is ot equal to ero. We choose the coe ciet to be:! ( )... ( ) = k, ( ) k= ad thus R! is uiquely deteried. Below R! is always to be uderstood as such. Thus we have the idetity d +! d ( )! d d ( )! +! d R! R!!!... ( ) = d ( )... ( )) ( ) ad sice we have we ca write R! ( )... ( ) where J stads for the operatio R!! ( ) ( ) =!! ( ) = {( )! J }{( )!! J }... ( ) {( )!! J }!! ( ) J =...d This ultiple itegral ca easily be used i the followig for: R! t = dt... dt... t dt R( t t...t ) R( t t...t )=( t ) (t t )...(t t ) (t ) ( )! ( t )!!...( t )!! ( t )!! ). Let s also give a siple Cauchy itegral for R!.Itis R! = ( ) ( )... ( ) ( ) d... Q i Q k C k= h= (! k h) which is itegrated i the positive directio o a big eough circle about the origi. Because there is a expasio i decreasig powers Y Y k! k + h X = b l l (b = ) k= h= l= See the proof i y paper (8) wherei the aalogous cosideratios for the expoetial fuctio are goe through copletely.

3 A PROO O THE THUE-SIEGEL THEOREM about the Approxiatio of Algebraic Nubers for Bioial Equatios therefore by the theore of residues we get R! =( ) ( )... ( ) = ( )... ( ) ( ) X l= +... b l (log( )) +l ( + l) O the other had, suig over the residues of the poles of the itegrals we get R!... k! ( )! k A k! k= =( ) () = Y k k= h= X k ( )... ( ) Y (! k h), where the polyoial A k! is exactly degree... k R! are fulfilled.. Usig the abbreviatio we have for k =,,..., ad h =,,..., k! () = Y = (! h) = h= Y k=! k, k (! k + h) =! k + h! k k. urther, we have Y x= ( k )! k + h! =( ) k h k k whereas fro the well-kow gaa forula we have t x ( + t) y dt = i G h= ( ) h (! k + h). Thereby the clais of the defiitio of (! + ) (! + ),! k + h! x x k, h si x ( + x) (y) ( + x + y) for <(y) > where G is the uit-circle i the positive directio ad the itegral is the priciple value. Thus it follows that for h =,,..., k, x 6= k ( x ) i( )! k + h! x h = t! k! x+h x ( + t) x dt x si(! k! x ) G x Thus the variables t,t,...,t k,t k+,...,t (itegrated respectively i the positive directio o the uitcircles G,G,...,G k,g k+,...,g i their plaes) are writte i the abbreviated for as follows dt... dt k dt k+... dt = dt. G G k G k+ G (G) So ow with Q k, the fiite ad o-ero costat Q k = Y x= i si(! k! x )

4 4 KURT MAHLER, Traslated by Karl Levy we arrive at the followig itegral forula by eas of a siple calculatio A k! Y = Q... k ( ) ( + ( ) t...t k t k+...t ( )) k. We defie the sybol hk for (h, k =,,...,) as follows for h = k hk = for h 6= k ad for (h, k =,,...,)set R h! = R (G) A hk!... k! + h... + h x=!. + h... + h Thus betwee the deteriat!... hk! ad the ior there is the idetity! ( )! k = for (k =,,...,). Thus polyoial of order exactly i which the costat hk!... h k! ( ) h+k hk h=!, the followig ust hold! =! polyoial i of degree exactly k + hk vaishes if ad oly if =. h 6=h k 6=k t! k! x x x (+t x ) x dt.! R... h! has a root of order!, at = ; Now sice it is cleary also a is idepedet of ; urthore sice A hk! is a!...!, it follows that is ot ero; Thus the deteriat! 4. Let be a atural uber such that ad ad R h () =R h..., A... hk () =A hk for h, k =,,...,, so that R h () = k= II. A hk ()( ) k ad so that R h () has root of order at =. With the ew variables x = x() =( ), x() = Carryig out the calculatio yields the value! see (8). = Y h,k= h6=k (! h! k ) ( k ) ( k +! h! k ) 6=

5 A PROO O THE THUE-SIEGEL THEOREM about the Approxiatio of Algebraic Nubers for Bioial Equatios 5 we copose ad rewrite the previous fuctios i the followig aer R h (x) =R h ( x ), A hk (x )=A hk ( x ). The eighborhood of = is apped to the eighborhood of x = ; R h (x) has therefore at x = a root of order. Settig S h (x) =(x ) R h (x), we the have that S h (x) is regular i a eighborhood of x = ; Thus we ca easily see that S h (x) is a polyoial. Itroducig yet aother idepedet variable y ad settig T h (xy) hk (x ) yk x k, y x k= U h (xy) hk (x )y k, we the have the idetity k= U h (xy)=(x ) S h (x)+(y x)t h (xy) for (h =,,...,). ro subsectio. The deteriat A hk (x ) is o-ero, if x is ot a -th root of uity. So followig fro this coditio for every value of y at least oe of the ubers U h (x, y) for (h =,,...,) is o-ero. 5. Let a, b be two atural ubers such that = r a b is a algebraic uber of degree exactly. Ay two ratioal ubers with positive deoiators p satisfy the iequalities to x ad y apple p q apple, apple p q apple q ad p q, that ay be used to assig the followig values x = q p, y = q p p q. As log as x is ot a -th root of uity the at least oe of the U h (x, y) s is o-ero. So for soe h we have U h (x, y) 6=. Clearly U h (x, y) is a ratioal uber whose deoiator ca be estiated to its upper liits. It was claied earlier that with i which + X hk ( ) A hk () =( ) ( + )!(!) h k + l = ( + hk )! l= k + l k + hk ( + hk )! k + l k ( + )!(!) h k + l ( ) l Y x= + hk eds up beig etirely ratioal, where o the other had for x 6= k we have k with K = l + x k. ( + hx )! k + l x + hx, ( + hx )! =( ) + ()()...(( + hx hx)) + l x ( + K)( + K)...(( + hx ) + K) + hx

6 6 KURT MAHLER, Traslated by Karl Levy Accordig to a theore of Maier the lowest coo deoiator of the coe ciets of all the polyoials A hk () ust be saller tha the -th power of a costat that depeds oly o ad. O accout of aq k q p U h (xy) hk bp p q k= we have the deoiator of the ratioal uber U h (xy). Therefore through ultiplicatio with b p + q, ad sice U h (xy) is ot equal to ero, there exists the iequality U h (xy) with positive costat c that depeds oly o, ad b. urther fro subsectio. we have R h () = apple c p+ q t t ( t ) + h (t t ) + h...(t t ) + h( ) (t ) + h dt dt dt ( t ) + h ( t ) + h...( t ) + h( ) ad with the ew variables of itegratio t k = u k (k =,,...,) u u R h () = ( u ) + h (u u ) + h...(u u ) + h( ) (u ) + h du du du ( u ) + h ( u ) + h...( u ) + h( ) = J whereby due to = x apple the factors of the deoiators are greater tha.urther,sice x x S h (x) = J, x x = +x + + x apple, we have the followig iequality S h (x) apple c, wherei the positive costat c depeds oly o ad. ially, it follows fro the itegral forula i subsectio. that ( ) A hk () = Y i si k x= x ad fro the defiitio of T h (xy) that (G) +( ) ( ) T h (xy) apple c Y + hk Y t x t k x hx x ( + t x ) + hk dt x= x= wherei the positive costat c depeds oly o ad. All ebers of the idetity q U h (xy) = S h p (x) p + q p T h (xy) q p q have their values derived either above or below ad fro the follows the existece of two positive costats c 4 ad c 5, which i tur oly deped o, ad so that the su of the two ubers # = c 4 q+ q p q, # = c 5 q+ q is greater tha two ad at least oe of the ubers is also greater tha oe. p q See the works (9) of Maier ad (7) of Siegel, where the proofs are carried out

7 A PROO O THE THUE-SIEGEL THEOREM about the Approxiatio of Algebraic Nubers for Bioial Equatios 7 6. Now we easily succeed i provig the Thue-Siegel theore for the specific algebraic ubers 4 : With " a arbitrary costat ad a arbitrary atural uber it follows that the iequality p q apple q ( + ) has oly fiitely ay ratioal solutios p q with positive deoiator. It su ces to liit the proof of to the ubers,,...,. Oly those solutios of the previous iequality eed be cosidered that also satisfy the followig iequality p apple q apple ; which is a cosequece of the first if the deoiator q is su ad p q, the atural uber is deteried by the coditio A siple calculatio yields the iequalities # apple q q " c 4 q " ( ) q <q apple q. Now if there were ifiitely ay solutios for the iequality the it could be show that apple p apple q apple p q apple cietly large. or ay two such ratioal ubers p q ; # apple q ++" q " p q apple q ( + ), q ax c " 4,c " 5,, q ax q " ( ),q " c 5 q ". (++" ) but this i tur would ea that # apple ad # apple, which cotradicts what has bee show above. Göttige, ebruary 4 th 9. ; Bibliography () A. Thue: Beerkug über gewisse Nährugsbrüche algebraischer ahle, Videskapsselskapets- Skrifter Christiaia (98). () O e geeral i store hele tal uløsbar ligig, Videskapsselskapets-Skrifter Christiaia (98). () Über Aäherugswerte algebraischer ahle, Joural für die reie ud agewadte Matheatik 5 (99). (4) C. Siegel: Approxiatio algebraischer ahle, Math. eitschr. (9). (5) Näherugswerte algebraischer ahle, Math. Aale 84 (9). (6) Über de Thuesche Sat, Videskapsselskapets-Skrifter Christiaia (9). (7) Über eiige Aweduge diophatischer Approxiatioe, Abh. d. Preuß. Akad. d. Wissesch. (99). (8) K. Mahler: ur Approxiatio der Expoetialfuktio ud des Logarithus, Joural für die reie ud agewadte Matheatik. (At the priter). (9) W. Maier: Potereihe irratioale Grewertes, Joural für die reie ud agewadte Matheatik 56 (95). urther work of Thue o Diophatie approxiatio is cited i the bibliography of (4). (Subitted o ebruary 6 th 9.) 4 Refer to works () though (7)