Al-Zangana Iraqi Journal of Science, 2016, Vol. 57, No.2A, pp:

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1 Results n Projetve Geometry PG( r,), r, Emd Bkr Al-Zngn* Deprtment of Mthemts, College of Sene, Al-Mustnsryh Unversty, Bghdd, Ir Abstrt In projetve plne over fnte feld F, on s the unue omplete ( ) r nd ny rs on on re nomplete r of degree less thn These rs orrespond to sets n the projetve lne over the sme feld In ths pper, The number of neuvlent nomplete k rs; k 5,,,, on the on n PG (, ) nd stblzer group types re found Also, the projetve lne PG (, ) hs been splttng nto two -sets nd prttoned nto sx dsjont tetrds Keywords: Projetve plne, Projetve lne, k Ar, Complete rs ISSN: GIF: 085 r, نتائج في الهندسة االسقاطية (,r PG( الخالصة عماد بكر عبد الكريم الزنكنة * قسم الرياضيات, كلية العلوم, الجامعة المستنصرية, بغداد, الع ارق في المستوي االسقاطي على الحقل المنتهي F واي قوس اخر على المخروط هو غير تام من الدرجة اقل من االسقاطي على نفس الحقل في هذا البحث عدد االقواس الغير على المخروط في المخروط هوالقوس الوحيد التام من الدرجة هذه االقواس تقابل مجاميع في الخط تامة من الدرجة k حيث والزمر المثبتة لها قد وجد كذلك الخط االسقاطي PG(, ) قد جزء الى مجموعتين من الدرجة و قسم ايضا الى ستة مجاميع من الدرجة ال اربعة k 5,,, PG(, ) - Introduton Let PG( r, ) be projetve geometry of dmenson r over the Glos feld F of elements If r, PG(, ) s lled projetve lne nd f r, PG(, ) s lled projetve plne Defnton []: A k r n projetve plne s set of ponts, no three of them re ollner A k r s omplete f t s not ontned n ( k ) r A k set n projetve lne PG(, ) s set of dstnt ponts Defnton []: In PG( r, ), frme s set of n ponts, no n n hyperplne; tht s, every subset of n ponts s lnerly ndependent The set { U0, U, U, U} n projetve plne nd the set {, 0, } n projetve lne PG(, ) re lled the stndrd frmes, where 9

2 U [,0,0], U [0,,0], U [0,0,], U [,,] 0 Defnton []: Let F be form of degree two; tht s, F X X, 0 j j j Wth not ll j 0 n F, then the set C v( F) { P( X ) PG(, ) F( X ) 0} s lled udr plne The set v( F) s lled non-sngulr f F rreduble over F A nonsngulr plne udr C s lled on whh s formed unue omplete ( ) r Lemm : Any on form through the stndrd frme hs the followng form F X 0X bx 0X XX Theorem 5[]: In wth, there s unue on through 5-r Defnton []: The ross-rto T { P, P, P, P } of four ordered ponts P, P, P, P PG(, ) wth oordntes t, t, t, t s ( t t)( t t) { P, P ; P, P } { t, t; t, t,} CR( T) ( t t )( t t ) Defnton 7[]: Let T be tetrd(-set) wth ross-rto Then T s lled () rmon, denoted by, f or ( ) or ( ) ; () Eunhrmon, denoted by E, f ( ) or euvlently, ( ) ; () Nether hrmon nor eunhrmon, denoted by N, f the ross-rto s nother vlue Remrk 8: () The ross-rto of ny hrmon tetrd hs the vlues,, () The ross-rto of tetrd of type E stsfes the euton 0 In wth 5 odd, n r not ontned n on n hs t most ( ) ponts n ommon wth on [] Therefore, ny nomplete r n on s t most of degree ( ) ; ere, there re two uestons () Wht s the mxmum sze of omplete r other thn the on hs ( ) ponts n ommon wth on? () Wht s the number of nomplete r n the on? In [], the frst ueston hs been nswer for some In [], ueston two hs been nswered for 9 The m of ths pper s nswered ueston two for, before tht, the ons formed through the stndrd frme hve been reprmetrzed Also, the projetve lne over F hs been splts nto two -sets nd prttoned nto sx dfferent tetrds For the group types whh pper n ths pper see [] The mn omputng tool s the mthemtl progrmmng lnguge GAP [5] - Con Representton Through 5-r Aordng to Lemm nd Theorem 5, to gve on wth dfferent form through the stndrd frme, t hs to be fndng the neuvlent 5-rs 95

3 Theorem : In PG (, ), there re sx projetvely neuvlent 5-rs through the stndrd frme s gven n Tble- Tble -Ineuvlent 5-rs n PG (,) A A A A A A 5 A 5-r {7} {8} {0} {} {} {8} In the followng, the on form through eh 5-r hs been gven C X X 9X X 0 X X {P(9( t t),9( t), t) t F } A A A * 0 0 Stblzer A tht lsted n Tble nd ts prmetrzon C C X X X X X X {P(0( t t),( t), t) t F } A * 0 0 C X X X X 5 X X {P(( t t),( t), t) t F } () A5 * 0 0 C X X 0X X X X {P(( t t),0( t), t) t F } A * 0 0 C X X 9X X 8 X X {P(( t t),( t), t) t F } Where * 0 0 F * F { } Sne there s unue on up to projetvty, so t s enough to fxed one on form to fned the number of k rs on the on Theorem : The number of neuvlent nomplete k rs; k 5,,,, on the on n PG (, ) nd stblzer group types re gven n Tble- Tble -Ineuvlent, nomplete k rs on the on -r 5-r -r 7-r 8-r 9-r 0-r -r -r N = N 5 = N = N 7 = N 8 =8 N 9 =5 N 0 =9 N =7 N =8 : I : I 5:I : I : I 9: I : I 85: I 90: I :V 5:Z 9:Z 5: Z 9: Z 7 : Z 5: Z : Z 57 : Z :V 7:V :Z 0 :V : D :Z :S :D :S 0 :V : D : D :Z - Projetve Lne PG (,) 8 I Z Z Z Z Z :S : D :Z : A : D : D 8 9

4 Eh pont P( xy, ) wth y 0 n PG(, ) s determned by the non-homogeneous oordnte xy; the oordnte for P(,0) s So, the ponts of PG(, ) n be represented by the set F { } {, t, t,, t t F } () On PG (, ), the projetve lne over Glos feld of order, there re ponts The ponts of PG (, ) re the elements of the set F { } {, 0,,,,, 5,, 7, 8, 9, 0, } A tetrd s of type f the ross-rto s, or Sne the euton 0 hs no soluton n F, so there s no tetrd of type E Therefore, there re three types of tetrds of type N Let tetrds of ross rto,8,,, 7, 0 denote by N, tetrds of ross-rto,,9,, 5, 8 denote by N nd tetrds of ross-rto 5,7,0,,, 9 denote by N Let ( k, ) be the set of ll neuvlent k sets through the stndrd frme n PG(, ) nd C ( k, ) be the set of ll neuvlent k rs on the on through the stndrd frme n PG(, ) It s ler tht from () nd (), there s one to one orrespondng between projetve lne nd on s gven below * PG(, ) F C [ x, y] t P( t) Therefore; there s one to one orrespondng between the neuvlent k sets through the stndrd frme n PG(, ) nd nomplete k rs on the on through the stndrd frme up to projetvty, where k ( ) Let denote these bjetvty by the mp k : ( k,) C( k,), then 5 : ( k,) C( k,) s defned s follows: ( A ) P; 5( A) P ; 5 5( A) P ; 5 A5 P 5( A) P5 ; 5( A) P ( ) ; Corollry : The number of neuvlent k sets n PG(,) nd ts stblzer group types s the sme s gven n Tble- Exmple : In Tble- nd Tble-, the neuvlent pentds ( 5 sets) nd hexds ( sets) through the stndrd frme {, 0, } nd ts prtton to s tetrds (pentds) wth stblzer group types hve been gven Tble - Ineuvlent pentds n PG (,) 97

5 P P P P P P 5 P The pentd P {,} P {,} P {,5} P {,} P5 {,7} P {,} Tble - Ineuvlent hexd n PG (,) Type of Tetrds N NN N N N N N N N N N N N N N N N N N N N N N N N Stblzer The hexd Types of pentds Stblzer {, 0,,,, } PPPP PP V {, 0,,,, } PP PP P5 P I {, 0,,,, 5 } PP P P PP Z {, 0,,,, } PP P P P P 5 I {, 0,,,, 7 } 5 {, 0,,,, 9 } 7 {, 0,,,, 0 } 8 {, 0,,,, } {, 0,,,, } Z Z I Z Z Z PP P P P P I PP PP P P Z 5 5 PP P P P P 5 PPP P P P Z PPPPPP D 9 0 {, 0,,,, 5 } P P P P P P Z P P P P P P V {, 0,,,, } {, 0,,,, 9 } Z {, 0,,,, 0 } P P P P5 P5 P I P P P P P P Z {, 0,,,, } 5 {, 0,,,, 7 } {, 0,,, 5, 7 } 7 P P P P P P V P P P P P P Z {, 0,,, 5, 9 } PPPPPP S P P P P P P Z 8 {, 0,,, 5, } {, 0,,, 5, 8 } P P P P P P V {, 0,,,, 9 } PPPPPP S {, 0,,,, } P P5 P P P5 P Z {, 0,,, 7, 0 } PPPPPP I S 98

6 - Splttng PG (,) Eh -set, nd ts omplement lso fxes the omplement then the stblzer group of the prtton s: () If projetvely neuvlent to ts omplement group of the prtton s lso G () If projetvely euvlent to ts omplement prtton PG (,) Clerly, the stblzer group G of So, f PG (,) s prtton nto two -sets { ; },, then G s G therefore; the stblzer, then the stblzer group of the prtton s And n ths se, the stblzer group of G unon of ll lner trnsformtons between nd the prtton generted lwys by two elements one of them belong to the projetvty between G nd Theorem: The projetve lne PG (,) hs () 90 projetvely dstnt prttons nto two euvlent -sets (EQ); () 78 projetvely dstnt prttons nto two neuvlent -sets (NEQ) Tble 5- Prtton of PG(,) nto two -sets NEQ:{ ; } EQ:{ ; } Totl:78 : I : Z 8:V :S Exmple: () The unue -set type D, nd ts omplement lne suh tht j j j = P Totl:90 8: Z :Z 0 :V :S : D : D : D : D : D :S : D 8 G nd the other s {, 5,,,, 9, 0}, whh hs stblzer group of j ={, 7, 8, 9, 0,,, 5,, 7, 8, } prtton the projetve The stblzer group of the prtton s of type D s gven bellow: D t, b (8t 0) (t 9) b, b b () The -set k = P {,, 5, 7, 8, 7, 9}, whh hs stblzer group of type S, nd ts omplement k ={, 9, 0,,,,, 5,, 8, 0, }re prtton the projetve lne suh tht The stblzer group of the prtton s lso S s gven bellow: k k S (5 t) (t ), b (9t 8) b, b b 99

7 Theorem: The projetve lne PG(,) splt nto sx dsjont hrmon tetrds nd sx dsjont tetrds of type N,,, These prttons re not unue Proof : The GAP progrmmng hs been used to splttng the projetve lne nto sx dsjont tetrds () Prttons nto rmon tetrds; {, 0,, - }, CR( ) ; {,,, }, CR( ) ; {,, 5, 5 }, CR( ) ; {,, 7, 8 }, CR( ) ; 5 { 7, 9, 0, }, CR( 5 ) ; { 8, 9, 0, }, CR( ) () Prttons nto tetrds of types N ; {, 0,, }, CR( ) ; {,,, }, CR( ) 8; {,, 5, 5 }, CR( ) 0; {,, 7, 7 }, CR( ) ; { 8, 8, 9, }, CR( ) 0; 5 5 { 9, 0, 0, }, CR( ) 8 () Prttons nto tetrds of types N ; {, 0,, }, CR( )= ; {,,, 7 }, CR( )=9; {,,, 5 }, CR( )= 8; { 5,,, 8 }, CR( )=; 5 { 7, 8, 9, 0 }, CR 5 ( )=9; { 9, 0,, }, CR( )= 5; (v) Prttons nto tetrds of types N ; {, 0,, }, CR( )= ; {,,, }, CR( )=7; {,, 5, 7 }, CR( )=7; { 5,,, 8 }, CR( )= ; {7, 0, 0, }, CR( )=0; 5 5 {8, 9, CR 9, }, ( )=0 Referenes rshfeld, J W P 998 Projetve geometres over fnte felds, nd Edton, Oxford Mthemtl Monogrphs, The Clrendon Press, Oxford Unversty Press, New York Brtol, D, Dvydov, A A, Mrugn S nd Pmbno, F 0 A -yle onstruton of omplete rs shrng (+)/ ponts wth on, Advnes n Mthemts of Communtons, 7(), pp: 9-970

8 Al-Zngn, E B 0 The geometry of the plne of order nneteen nd ts pplton to errororretng odes, PhD Thess, Unversty of Sussex, Unted Kngdom Thoms, A D nd Wood, G V 980 Group tbles Shv Mthemts Seres, Seres 5 GAP Group 0 GAP Referene mnul, URL 97