On the irreducibility of some polynomials in two variables

Transcription

1 ACTA ARITHMETICA LXXXII.3 (1997) On th irrducibility of som polynomials in two variabls by B. Brindza and Á. Pintér (Dbrcn) To th mmory of Paul Erdős Lt f(x) and g(y ) b polynomials with intgral cofficints in th singl indpndnt variabls X and Y. Th diophantin problm f(x) = g(y) is strongly rlatd to th absolut irrducibility and th gnus of f(x) g(y ) as pointd out by Davnport Lwis and Schinzl [DLS]: Thorm A. Lt f(x) b of dgr n > 1 and g(y ) of dgr m > 1. Lt D(λ) = disc(f(x) + λ) and E(λ) = disc(g(y) + λ). Suppos thr ar at last [n/2] distinct roots of D(λ) = 0 for which E(λ) 0. Thn f(x) g(y ) is irrducibl ovr th complx fild. Furthr th gnus of th quation f(x) g(y) = 0 is strictly positiv xcpt possibly whn m = 2 or m = n = 3. Apart from ths possibl xcptions th quation has at most a finit numbr of intgral solutions. Th purpos of this not is to handl som spcial cass. For an intgr > 1 w st f (X) = X(X + 1)... (X + 1). For svral scattrd ffctiv and inffctiv rsults on th quation (1) f (x) = f l (y) in intgrs x y w rfr to [BS] [MB] [SS] [SST1] [SST2] and [Sh]. By using an algbraic numbr-thortic argumnt w can guarant th conditions of Thorm A in crtain cass. Lt I dnot th st of intgrs for which f (X) is ithr irrducibl or it has an irrducibl factor of dgr 2. Our conjctur basd upon svral numrical xampls is that I is th whol st of positiv intgrs mor xactly ithr f (X) or (X)/(2X + 1) ar irrducibl dpnding on th parity of. Applying f 1991 Mathmatics Subjct Classification: Primary 11D41. Rsarch supportd in part by th Hungarian Acadmy of Scincs by Grants and from th Hungarian National Foundation for Scintific Rsarch. [303]

2 304 B. Brindza and Á. Pintér Eisnstin s thorm on can s that th prims blong to I and w hav chcd by computr that { } I. Thorm 1. If and l ar lmnts of I with 2 < < l thn th polynomial f (X) f l (Y ) is irrducibl (ovr C) and (1) has only finitly many solutions. Morovr som simpl inqualitis lad to Thorm 2. Lt and m b intgrs gratr than 2. Thn th quation ( ) y f (x) = in positiv intgrs x and y m has only finitly many solutions. R m a r. Similar (ffctiv) rsults in th cass = 2 l > 2; = 2 m > 2 and m = 2 > 2 wr obtaind in [Y] and [SST2] rspctivly. Ths quations can b tratd by Bar s mthod. P r o o f o f T h o r m 1. Th discriminant of th polynomial f (X)+λ is dnotd by D (λ) i.. D (λ) = C (f (x) + λ) f (x)=0 (cf. [DLS]) whr C is a non-zro absolut constant. To show that D (λ) and D l (λ) hav no common zros w ta any irrational zros α and β l of f and f l rspctivly and put K = Q(α β l ). Th crucial stp is that instad of th comparison of f (α ) and f l (β l ) w show that thir fild norms with rspct to K ar not qual. If f (X) is irrducibl thn a simpl calculation yilds ( f N K/Q (f (α )) = (0)... f ) [K:Q(α )] (1 ) ; furthrmor if is vn thn f (X) is always divisibl by th linar factor 2X + 1 and in cas I as was pointd out by A. Schinzl w gt ( 2 f N K/Q (f (α )) = (0)... f ) [K:Q(α )] (1 ). ( 1) /2 ( 1)!! According to ths formula for an intgr n > 2 w writ f n(0)... f 1/(n 1) n(1 n) n a n = n if n is odd 2 n f n(0)... f n(1 1/(n 2) n) n n (n 1)!! if n is vn.

3 Sinc For convninc st b 1 = b 2 = 1 and Irrducibility of som polynomials 305 b = f (0)... f (1 ) ( > 2). f +1(i) = (i + )f (i) i = f +1( ) =! w hav th rcursion b +1 = b (!) 2 and thrfor b = (2!... ( 1)!) 2 ( > 2). To prov that th squnc a n n = is strictly incrasing w hav two cass to distinguish dpnding on th parity of th indics. To illustrat th tndncy a 3... a 14 ar listd blow up to svral digits: a 3 = a 4 = a 5 = a 6 = a 7 = a = a 9 = a 10 = a 11 = a 12 = a 13 = a 14 = If is vn thn a < a +1 ( > 2) is quivalnt to b 2/(( 2)) < (!) 2/ /( 2) (( 1)!!) 1/( 2) ( + 1) (+1)/ 2 /( 2) and in th squl w may assum that 14. obtain < 2 By using induction w (2) b 2/(( 2)) ( 9). Indd supposing (2) and th rcursion for b +1 w hav to show ( ) 2 ( 2) ( ) ( + 1) (3) (!) <. Assuming (3) not tru and applying! < ( ) +1 2 ( > 2) w obtain ( + 1) and ( ( + 1)2 > ) which is fals for 14. Thrfor (3) and hnc (2) is provd for 14. On th othr hand ( ) 2 ( ) 1/2 < (!) 2/ < (( 1)!!) 1/( 2) and /( 2) ( + 1) (+1)/ > ( ) /( 2) + 1 ( ) /( 2) 14 > 15 ( ) 7/6 14 >

4 306 B. Brindza and Á. Pintér imply 2 / < < ( ) 1/2 ( ) /6 ( 14) hnc a < a +1 is provd if is vn. Th rmaining cas ( is odd) is simpl. W gt ( ) 1/( 1) b a = a +1 = (2 +1 (!) 2 b ( + 1) (+1) (!!) 1 ) 1/( 1). On can obsrv that! >!! and < and thus Thorm 1 is provd. ( + 1) ( + 1) +1 < 2+1! ( + 1) +1 P r o o f o f T h o r m 2. Th xcptional cas ( m) = (3 3) is covrd by a rathr gnral rsult of [S1] (cf. [ST p. 122]). St A() = ( ( 1)) 2 2 if is vn ( ( 2)) 2 2 if is odd. As a mattr of fact w prov a littl mor. Namly th quation af (x) = bf m (y) in positiv intgrs x and y with aa() > b(m 1)! has only finitly many solutions. To guarant th conditions of Thorm A it is nough to show that (4) a min f (x) > b max f m(y). f (x)=0 f m (y)=0 Obviously b(m 1)! > b max f m(y). f m (y)=0 Sinc all th zros of f (x) ar ral also all zros of f (x) ar ral and by Roll s thorm thy altrnat with th zros of f (x). Elmntary calculus yilds ( ( a min f (x) > a min f 1 ( f 2) (x)=0 f 3 (... 2) f 2 3 ) ). 2

5 Irrducibility of som polynomials 307 Sinc ( a f 2j 1 ) (2( j) 1)!!(2j 1)!! = 2 2 a (2j 1) (2 (2j + 1)) = 2 a A() (j = ) (4) is provd. Rfrncs [BS] R. Balasubramanian and T. N. Shory On th quation f(x + 1)... f(x + ) = f(y + 1)... f(y + m) Indag. Math. (N.S.) 4 (1993) [DLS] H. Davnport D. J. Lwis and A. Schinzl Equations of th form f(x) = g(y) Quart. J. Math. 12 (1961) [MB] R. A. MacLod and I. Barrodal On qual products of conscutiv intgrs [SS] Canad. Math. Bull. 13 (1970) N. Saradha and T. N. Shory Th quations (x+1)... (x+) = (y+1)... (y+ m) with m = 3 4 Indag. Math. (N.S.) 2 (1991) [SST1] N. Saradha T. N. Shory and R. Tijdman On th quation x(x + 1) (x + 1) = y(y + d)... (y + (m 1)d) m = 1 2 Acta Arith. 71 (1995) [SST2] On arithmtic progrssions with qual products ibid. 6 (1994) [S1] A. Schinzl An improvmnt of Rung s thorm on diophantin quations Commnt. Pontific. Acad. Sci. 2 (1969) no [S2] Rducibility of polynomials of th form f(x) g(y) Colloq. Math. 1 (1967) [Sh] [ST] [Y] T. N. Shory On a conjctur that a product of conscutiv positiv intgrs is nvr qual to a product of m conscutiv positiv intgrs xcpt for 9 10 = 6! and rlatd qustions in: Numbr Thory (Paris ) London Math. Soc. Lctur Not Sr. 215 Cambridg Univ. Prss Cambridg T. N. Shory and R. Tijdman Exponntial Diophantin Equations Cambridg Univ. Prss Cambridg 196. P. Z. Yuan On a spcial Diophantin quation a ( x n) = by r + c Publ. Math. Dbrcn 44 (1994) Mathmatical Institut of Kossuth Lajos Univrsity P.O. Box 12 H-4010 Dbrcn Hungary apintr@math.lt.hu Rcivd on and in rvisd form on (310)