Some Transcendental Elements in Positive Characteristic

Transcription

1 R ESERCH RTICE Scnca 6 ( : So Trancndna En n Pov Characrc Vchan aohaoo Kanna Kongaorn and Pachara Ubor Darn of ahac Kaar Unvry Bango 9 Thaand Rcvd S 999 ccd 7 Nov 999 BSTRCT Fv rancndna n n funcon fd of ov characrc ar conrucd bracng ho rvouy drvd by Wad durng Th conrucon ru fro a carfu anay of h orgna wor of Wad and ndca ha h hod and aocad chnu worhy of condraon KEYWORDS: rancndna characrc INTRODUCTION Th udy of rancndnc n fd of nonzro characrc nown o bgn n 94 hrough h ar of Wad Snc hn varou aroach oo and ru g ou Brownaw Da and Hgouarch 4 Dn 5 78 o ahan Wad -6 Yu 7-3 hav bn nvgad Though h odrn ran of h ubc va h conc of Drnfd odu (Drnfd 6 donang h rn day rarch h od and caca da and aroach of Wad n 94 rov o b a drc and a vry owrfu chnu of abhng cran cfc rancndnc ru a vdncd n o rcn wor of Daa and Hgouarch 4 co anay of h wor of Wad -6 rva ha h da con of fr aung agbracy cond fndng ur hrd arang arora xron no h o-cad ngra and randr ar fourh ang ur and owr bound for h ngra and randr ar and ffh and fnay drvng a conradcon fro h ur and owr bound or h Th an obcv of h wor o ubana h bf by rovng fv hor gnrazng corrondng arr ru of Wad -6 bang on a carfu anay of h orgna hod of Wad nond abov Th foowng rnoogy and noaon ao 9 ar andard hroughou h nr ar F [x] h rng of oynoa ovr h ao (fn fd F of characrc wh bng r bng a owr of F (x h uon fd of F [x] dg or (or h nonarchdan vauaon (a norazd o ha x = dg x = F (x h coon of F (x (a wh rc o whch oorhc o F ((/x h fd of fora aurn r n /x F (x co h agbrac cour of F (x Ω h coon of F (x co For N [] = x - x [] = = = [][-][] F = F = [][-] [] I nown (Carz 3 ha h a coon u of a oynoa of dgr n F [x] and F h roduc of a onc oynoa of dgr n F [x] Th noon of ngray and dvby rfr o ho n h ngra doan F [x] W rcord hr auxary a whch w b ud n h roof of our an hor Th roof of h a can b found n Wad 3 a Evry oynoa n F [x] dvd a nar oynoa F [x] a Th xron [ ] a [ ] ar r [ ] [ ] whr > > r > ; ; > ar ngr ndndn of can b wrn a a u of r ha ar ngra or ar of h for P [ ] f [ ] fu b bu whr b > > b u < f - [ ] [ v] v b < v + and P a oynoa ndndn of Furhr f a ( = v hn b u n ( + + c and h rdu od [] of h = r ngra r of ar dgr han h dgr of h rdu od [] of h nuraor n h orgna xron

2 4 Scnca 6 ( THE FIRST IN THEORE Thor b a ov ngr and b a unc of n n F (x u ha ( ach P and ( F [x] ( P for nfny any ( dg P ( ( - - b whn uffcny arg and b ( (v hr ar ony fny any dnc rrducb facor conand n a (v hr a non-dcrang unc of ov ngr (d uch ha d d - whn uffcny arg and dg d = O( a Thn h r α := whnvr convrgn = F rancndna ovr F (x Proof Suo on h conrary ha α agbrac ovr F (x Thn a roo of an agbrac uaon wh coffcn fro F (x By a w can u h agbrac uaon n h for = whr F (x Drc ubuon yd = = F = = F (whr w dfn = f > or < = = F (whr w dfn h r wh ngav owr ndc o b = = F = = F += F whr D := += F F P D = F = F ( Fro (v b h roduc of a dnc rrducb facor aard n a h and b a uffcny arg ov ngr o b uaby chon ar Fro ( w g = I + R ( whr I := d F D = F R := d F D F W now ubdvd h roof no S W ca ha I ngra Each r of I ha h for d F D F F = F Snc F dvb by F d and F dvb by F d F P F ( = += dvb by hn I ngra S W ca ha dg R ( F Each r of R ha h for N := d F D whn Ung h dfnon of D w g dg D ax {dg + (dg P dg + (dg F += dg F } a := ax (dg Snc dg F = ung ( for uffcny arg and o ao w ha dg D a + ax { dg P - + ( (- - } a + ax { (-(-- -- b } a + ax { ( / - c b' + c } whr b := n (b - ( c and c ar + ov conan ndndn of Thu dg N = d dg + dg D + (dg F - dg F d + a + {( / c b ' + c } + ( - whr = dg d + a + {( / ( + c b + c } + for uffcny arg + = a + d ' + ( / ( c b + c whr c 3 c 4 ar conan ndndn of - ( by ( and (v Conuny dg R - ( Fro ( and and w g I = R = whn uffcny arg S 3 W ca ha = I d D (od F F whn uffcny arg Fro h dfnon of I w hav I = d {D + D F - ++ D F F } (3 F

3 Scnca 6 ( 4 Fro h roof n w hav d D F [x] ( = F dvb by F and F /F F /F F /F - ar dvb by F /F - Thrfor (3 gv = I d arg D (od F F whn uffcny S 4 W ca ha D = whn uffcny arg Whn uffcny arg w hav F dg F - dg d D = { -(- }- d dg D { - (- - } c 5 a - {(- / c b ' + c } whr c 5 a ov conan fro (v and -a + {( - ( - / c 6 + c b ' } whr c 6 a ov conan a Thu fro 3 nc w g D = whn uffcny arg and o hr an ndx > a := ax (dg uch ha D = for a + S 5 W ca ha d+ = F D P d+ + (od[ ] F + Fro 4 and h dfnon of D + w hav = F F + d+ d D+ = F + +=+ F By hyoh (v d + dvb by + + F F and no ha ar a congrun o + F F + (od [] Th ca hu foow S 6 W ca ha for and []:= [] -] [] w hav + P d+ (od [] Procd by nducon on Th ca = 5 u h ca hod for = + - ( > Now by 4 = ( d D = F F + ( F P d += F := + Obrv ha F F + d ( = + T - + T - ++ T - ay (od[] and fro (v w g d -+ d whn uffcny arg Thrfor T - := + ( d d + F + F P d+ (od [-] + [] (od [] bcau [] [-] = [] Sary T - := + ( d d + F + F P d + (od ([- ] + [ -] + [ ] (od [ ] T - := + P + + ( ( d d + d + F F + (od [-] [-+] [] + [ ] (by (v (od []

4 4 Scnca 6 ( S 7 Fnay w now drv a drd conradcon Fro ( and 6 w ha for nfny any and uffcny arg dg P dg []- + dg + dg - d + dg ( a-c 7 + whr c 7 a ov conan ndndn of Thu whn uffcny arg dg P ( - + c 8 + whr c 8 a ov conan ndndn of > (-(- - b Th conradc ( un P = for a uffcny arg y h ony ohr oby n urn conradc ( Hnc α no agbrac ovr F (x THE SECOND IN THEORE Thor b a nonngav ngr and b a unc of n n F (x u ha ( ach P and ( F [x] ( P (od [] for nfny any ( dg P = o ( f ( / o( f = (v hr ar ony fny any dnc rrducb facor conand n a (v hr a nondcrang unc of nonngav ngr (d uch ha dg d = o ( f ( / o( f = Thn h r α := whnvr convrgn = [] rancndna ovr F (x Proof u o h conrary ha α agbrac ovr F (x Thn by a α a roo of an agbrac uaon of h ha = whr F [x] Drc ubuon yd a = + = [] uyng h uaon by d + whr by (v h roduc of a dnc rrducb facor aard n a h and an ngr o b chon uffcny arg and arang r w g an uaon of h ha = T + T + T 3 ( whr T := d T := d T 3 := d = P + = W T by nong ha [ + ] [] [ + ] + = + + [] + [ ] ( = ( [ + ] [] = = ( [ + ] + + = + + [ + ] Now u I = T + T whr T = T + T + + = + T := [ ] T := + d + + ( [ + ] [ + ] = = = = d P (- --- [ + ] = P + d [ + ] + S W ca ha I ngra whn uffcny arg + For =nc and + [] ar ngra hn T ngra d

5 Scnca 6 ( 43 Snc - and d hn h fr u n T ngra wh h cond u ao ngra bcau for uffcny arg -- Thu I = T + T ngra whn uffcny arg Nx condr := T + T 3 S W ca ha dg ( Snc dg [ + ] = c whr c a ov conan ndndn of and dg P d dg P + d dg o( by ( and (v hn dg T ( On h ohr hand whn + and uffcny arg w ary hav + d dg o( dg + -c [] whr c a ov conan ndndn of o dg T 3 ( Thu dg ( Fro ( w g I + = and by and w dduc ha I = = S 3 W now how ha I ; h gv a drd conradcon Wr I = T + T = T + E + E 3 whr T = E + E = E := [ + ] E 3 := = = + d (- + [ + ] = d P [] W now fro ha E ngra and nc (od [-] o E Sary for E 3 Now fro T ngra and h r n T conan h facor + [] + Thu I (od[ ] f (od[ ] f = < + (od[ ] f = + [ ] + P [ ] + P d (od [ -] d + (od[ -] (od [ ] by ( and for uffcny arg Th how ha I and h roof co THE THIRD IN THEORE Thor 3 b a ov ngr and b a unc of n n F (x u ha ( ach P and ( F [x] ( P for nfny any ( dg P = o( ( (v hr ar ony fny any dnc rrducb facor conand n a (v hr a non-dcrang unc of nonngav ngr (d uch ha dg d = o( ( (v dg dg + for uffcny arg + Thn h r α 3 := whnvr convrgn rancndna ovr F = (x Proof uu on h conrary ha α 3 agbrac ovr F (x Thn by a α 3 a roo of an agbrac uaon of h ha = whr F [x] ( Whou o of gnray a > +3 r J r := = = ( + [] ( r K := J whr a ov ngr o b uaby chon

6 44 Scnca 6 ( and by (v b h roduc of a dnc rrducb facor aard n a h Subung for α 3 no ( and uyng by d K w g = I + + R whr I := d K + ( = := d K = + P + + R := d K = ( + ha n R h u ovr ar fro = S W ca ha I ngra ; no Snc d and K ( = (- ; = hn I ngra S W now anayz R Condr ach r n h u of R Th r wh = T := d K = P + + H( := K [ + ] = = ( + = (= [ ] [ ] - Thn ach r of T can b wrn a d P + H( ( = - ( Th r wh = - T - := d K - P + = + K [ + ] = = ( + Snc = (= + [ + ] [ + ] -+ hn ach r of T - of h for d P H(- [ + ] = = ( + whr H(-:= [ + ] [ + ] (3 (= -+ Th r wh - ar of h for T - := d J = ( + P + + J J [ + ] = + = ( + Snc = {[ ( + ][ ( + ]} ( + ( = - hn ach r of T - of h for d P + ( + + ( + ( = + = ( + H( J -- (4 [ + ] whrh(:= {[ ( + ][ ( + ]} = ( + { [ ] }{ [ ] } = ( + = ( + {[ ( + ][ ( + ]} ( = - Fro a w ha ach of H( H(- and H( can b wrn a u of ngra r and of non-ngra r wh h ror gvn n h a Thu ach r of T of T - and of T - can b xrd a u of ngra r and of non-ngra r wh h ror nond n a I := I + u of ngra r fro R := + u of non-ngra r fro R Thu I + = S 3 W ca ha dg ( Snc := + u of non-ngra r fro R N b a r n Thn for + N := d P K J + + d P = + +

7 Scnca 6 ( 45 dg N dg + dg J + dg dg + d dg + dg P < + ((-++ + ( a - (+ -+ d + dg P whr a := ax dg := dg + + {- + c o( } + a co ( whr c and c ar ov conan ndndn of ( Snc R = T + T - + T - condr an arbrary r n h u of non-ngra r n R by a H( = can b wrn a a u of ngra r and non-ngra r of h for g P[ ( + g+ ] [ ( + ] NIH( := δ [ ( + ] [ ( + g+ ] [ ( + g] whr - g δ - ( = g and P ndndn of b : = ax dg P; h vau ndndn of P No ha dg NIH( axa whn ach = - g = δ = and o dg NIH( b+(-( ( (-+ - (-++ =b- (-+ Thrfor by ( (3 and (4 dg ( non-ngra r n T d + + a + (dg P dg + dg NIH( = d + a + o( + b - d = a + b - o ( + ( (= - dg (non-ngra r n T - d + a + dg NIH(- + ( - - dg + - dg P + = d + a + b (- + - o( + = a + b - - d + ( o ( + ( dg (non-ngra r n T - d + a + dg NIH( + dg dg J -- - dg (-- + dg P (-+ < d + a + b - ( ( (--(-++ + (--(-++ - ( + + o( (-+ < a + b - (-+ d ( + o ( + ( + ( + 3 ( + ( Thu dg (ach r n h u of non-ngra r n R and o dg ( Fro I + = and 3 w concud ha I = = S 4 W drv a drd conradcon by howng ha I Snc I = I + u of ngra r fro R hn condr I := d J Obrv ha J + ( = (od[ ] ; ( ; for = = and for = = J (od[ ] for = ; = Thn ung [ + ] [] (od [] w g I d [] (od [] = = ( + Now condr ngra r fro R = T + T - + T - Th ngra r fro T ar of h for d H( ( = - Th ngra r fro T - ar of h for d - H(- (od [] ( = -+ P

8 46 Scnca 6 ( Th ngra r fro T - ar of h for d J ( P H( + ( + + ( + (od [] ( = - Thu ngra r fro R ngra r of h for d H( (od [] By ( and ( w hav P and P ar boh (od [] for uffcny arg Th oghr wh (v and (v y ha h rdu od [] of d n I and of d ar boh (od [] for uffcny arg Thu dg (rdu od [] of ngra r fro R = dg (rdu od [] of h ngra r of h for d P H( + + < dg( d + dg + dg (nuraor of H( od [] (by a = dg ( d + dg +dg ( [ ] = = ( + od [] = dg ( d + dg +dg ( [] od [] = = ( + dg( d + dg +dg ( [] = = ( + od [] (by (v = dg (rdu od [] of I Hnc for uffcny arg I (od [] whch yd I THE FOURTH IN THEORE Thor 4 F [x] dg > N > no a owr of Suo ha a unc of n n F (x wh h foowng ror ( ach P ( F [x] ( P for nfny any ( dg P = o( ( (v hr ar ony fny any dnc rrducb facor conand n a (v hr a nondcrang unc of nonngav ngr (d uch ha dg d = o( ( P Thn α 4 := whnvr convrgn = rrn an n rancndna ovr F (x Proof by (v b h roduc of a dnc rrducb facor of a b a ov ngr o b uaby chon ar and ( b a dcrang unc of ngr dfnd by - / < - (- og - < ( - og No ha whn = w hav / < and o = Whn < w hav - bcau no a owr of r whch yd - / < < - u on h conrary ha α 4 whn rrnd r convrgn agbrac ovr F (x Snc F (x an agbrac xnon of F (x hn α 4 agbrac ovr F (xand o a roo of a nar uaon of h for = F [x] Subung for by α 4 and uyng by d w g = I + whri := d := d = + S W ca ha I ngra Th rva bcau of (v (v and h dfnon of ( µ := n ( = O( by h dfnon of ( Thrfor I = d + d = = + d

9 Scnca 6 ( 47 d (od µ for uffcny arg by ( and (v S W ca ha dg ( N b an arbrary r of Thn N := d ( ; + dg N = ( - dg + dg + d dg + (dg P - dg ( g + a + o( + o( whr g = dg a = dg = dg o g o a + ( + g + ( ( bcau - + far han Thu dg ( Snc = I + hn Ca and oghr y ha I = = whch conradc h fac ha I (od µ for uffcny arg Th conradcon rov h hor THE FIFTH IN THEORE Thor 5 F [x] dg > N > Suo ha a unc of n n F (x wh h foowng ror ( ach P ( F [x] ( P for nfny any whch ar no congrun o (od ( dgp = o( - ( (v hr ar ony fny any dnc rrducb facor conand n a (v hr a nondcrang unc of nonngav ngr (d uch ha dg d = o( - ( P Thn α 5 := whnvr convrgn = rrn an n rancndna ovr F (x Proof by (v b h roduc of a dnc rrducb facor of a b a ov ngr no dvb by whch o b uaby chon ar u on h conrary ha α 5 whn rrnd r convrgn agbrac ovr F (x Snc F (x an agbrac xnon of F (x hn α 5 agbrac ovr F (x and o a roo of a nar uaon of h for = F [x] Subung for by α 5 and uyng by d I := := w g = I + whr d = d + and := a non-ncrang funcon of S W ca ha I ngra Snc h owr of aarng n I - - ( ( = and dg d d = d hn I ngra S W ca ha I whn uffcny arg and no dvb by µ := n ( - (- + Now := = η whr η - - ( = + Condr ach r n h dfnon of µ W ha - = (-η - ( - Thu µ = - ( - = Ο ( - ( bcau > Fro h dfnon of I w g I = d + + = = Obrv ha afr drbung h ur nd h ow owr of n ach u whn h brac na whn = rcvy If = h ow owr of - = n h fr u If = + h ow owr of - + ( + µ (by h dfnon of n h cond u

10 48 Scnca 6 ( If = h ow owr of - µ n h a u No ao ha n h fr u h owr of nx o h ow owr ( - = ( - µ Thrfor I d (od µ by ( ( (v and (v for nfny any no dvb by Th yd I whn uffcny argr and no dvb by S 3 W ca ha dg ( N b an arbrary r of Thn N = d ( = ; + and o dg N d + g + a + (dg P dg g whr g := dg a := ax dg := dg a + g + o( - + o( - - g o( - + g - g ( + o ( = o( - + g - g ( - η + - o( o ( - + g - g ( + o( o( - - g - o( { ( + owr r } ( Fro = I + and 3 w concud ha I = = whch conradc Hnc α 5 rancndna ovr F (x a o b rovd REFERENCES ou (996 rca ru on rancndnc aur n cran fd J Nubr Thory Da W Brownaw (998 Trancndnc n ov characrc Coorary ah Carz (935 On cran funcon conncd wh oynoa n a ao fd Du ah J Daa and Hgouarch Y (99 Trancndnc of h vau of Carz za funcon by Wad hod J Nubr Thory Dn (993 Théorè d Bar odu d Drnfd J Nubr Thory Drnfd V (974 Ec odu ah USSR Sborn J (973 Schndr hod n Fd of Characrc ah Cnr Ror ZW 7/73 rda 8 J (979 Trancndnc n Fd of Pov Characrc ah Cnr Trac Vo 9 ah Cn rda 9 o D (996 Bac Srucur of Funcon Fd rhc Srngr Brn D o (998 Ror on rancndncy n h hory of funcon fd c No n ah #383 Srngr Brn 59-6 d ahan B (995 Irraonay and rancndnc n ov characrc J Nubr Thory Wad I (94 Cran uan rancndna ovr F( n x Du ah J 8 7-3Wad I (943 Cran uan rancndna ovr F( n xii Du ah J Wad I (944 Two y of funcon fd rancndna nubr Du ah J Wad I (946 Rar on h Carz ψ-funcon Du ah J Wad I (946 Trancndnc ror of h Carz ψ- funcon Du ah J Jng Yu (983 Trancndna nubr arng fro Drnfd odu ahaa Jng Yu (985 Trancndnc hory ovr funcon fd Du ah J Jng Yu (985 x xonna hor n fn characrc ah nn Jng Yu (986 Trancndnc and Drnfd odu Invn ah Jng Yu (989 Trancndnc and Drnfd odu: vra varab Du ah J Jng Yu (99 On rod and ua-rod of Drnfd odu Cooo ah Jng Yu (99 Trancndnc and ca za vau n characrc nn ah 34-3