{VERSION 3 0 "IBM INTEL NT" "3.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 }{CSTYLE " " -1 256 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 1" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 }1 0 0 0 8 4 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading \+ 2" 3 4 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 8 2 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 " " 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 11 12 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 } 1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Title" 0 18 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 1 0 0 0 0 0 0 }3 0 0 -1 12 12 0 0 0 0 0 0 19 0 }} {SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 44 "Typical problem about uni variate polynomials" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 24 "Ideal mem bership problem" }}{PARA 0 "" 0 "" {TEXT -1 5 "Let " }{XPPEDIT 18 0 " P := x^5+3*x^4-17*x^3-51*x^2+16*x+48;" "6#>%\"PG,.*$%\"xG\"\"&\"\"\"*& \"\"$F)*$F'\"\"%F)F)*&\"#%\"MG,**$)%\"xG\"\"#\"\"\"\"\"\"F(\"\"&*&)%\"kGF)F*F(F+! \"\"*$F.F*!\"&" }{TEXT -1 9 " and " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }{XPPEDIT 18 0 "N := x^2-5*x-k^2*x+5*k^2;" "6#>%\"NG,**$%\"xG\"\"#\" \"\"*&\"\"&F)F'F)!\"\"*&%\"kG\"\"#F'F)F,*&\"\"&F)*$F.\"\"#F)F)" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 35 "For which values of the parameter " }{XPPEDIT 18 0 "k" " 6#%\"kG" }{TEXT -1 20 " the polynomial " }{XPPEDIT 18 0 "P" "6#%\" PG" }{TEXT -1 48 " is in the ideal generated by the polynomials " } {XPPEDIT 18 0 "M" "6#%\"MG" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "N" " 6#%\"NG" }{TEXT -1 1 "?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 8 "Solution" }}{PARA 0 "" 0 "" {TEXT -1 29 "D efine the polynomials first:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 87 "P := x^5+3*x^4-17*x^3-51*x^2+16*x+48;M := x^2+5*x-k^2*x-5*k^2;N := x^2-5*x-k^2*x+5*k^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"PG,.*$)% \"xG\"\"&\"\"\"\"\"\"*$)F(\"\"%F*\"\"$*$)F(F/F*!#<*$)F(\"\"#F*!#^F(\"# ;\"#[F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"MG,**$)%\"xG\"\"#\"\"\" \"\"\"F(\"\"&*&)%\"kGF)F*F(F+!\"\"*$F.F*!\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"NG,**$)%\"xG\"\"#\"\"\"\"\"\"F(!\"&*&)%\"kGF)F*F(F+ !\"\"*$F.F*\"\"&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 16 "The polynomial " }{XPPEDIT 18 0 " P" "6#%\"PG" }{TEXT -1 32 " is in the ideal generated by " } {XPPEDIT 18 0 "M" "6#%\"MG" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "N" " 6#%\"NG" }{TEXT -1 29 " if and only if the gcd of " }{XPPEDIT 18 0 " M" "6#%\"MG" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "N" "6#%\"NG" } {TEXT -1 11 " divides " }{XPPEDIT 18 0 "P" "6#%\"PG" }{TEXT -1 60 ". Let us compute the gcd using Maple's built in procedure. " }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "gcd(M,N);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#,&%\"xG\"\"\"*$)%\"kG\"\"#\"\"\"!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 17 "The gcd divides " }{XPPEDIT 18 0 "P" "6#%\"PG" }{TEXT -1 55 " \+ is and only if the root of the gcd, i.e., the value " }{XPPEDIT 18 0 "x=k^2" "6#/%\"xG*$%\"kG\"\"#" }{TEXT -1 37 ", is also a root of th e polynomial " }{XPPEDIT 18 0 "P" "6#%\"PG" }{TEXT -1 36 ". To find \+ out for which values of " }{XPPEDIT 18 0 "k" "6#%\"kG" }{TEXT -1 28 " this is true, make first " }{XPPEDIT 18 0 "P" "6#%\"PG" }{TEXT -1 18 " into a function:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "P :=unapply(P,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"PGR6#%\"xG6\"6$ %)operatorG%&arrowGF(,.*$)9$\"\"&\"\"\"\"\"\"*$)F/\"\"%F1\"\"$*$)F/F6F 1!#<*$)F/\"\"#F1!#^F/\"#;\"#[F2F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 39 "Then we compu te the desired values of " }{XPPEDIT 18 0 "k" "6#%\"kG" }{TEXT -1 13 " by solving:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "solve(P(k ^2)=0,k);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6,\"\"\"\"\"#!\"#!\"\",$%\" IGF$,$F(F%*&F(F#-%%sqrtG6#\"\"$\"\"\",$F*F&F(,$F(F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 79 "O nly real solutions can be considered. Hence we conclude that for the \+ values " }{XPPEDIT 18 0 "k" "6#%\"kG" }{TEXT -1 31 "= 1, -1, 2, -2 th e polynomial " }{XPPEDIT 18 0 "P" "6#%\"PG" }{TEXT -1 32 " is in the ideal generated by " }{XPPEDIT 18 0 "M" "6#%\"MG" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "N" "6#%\"NG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 18 "Radicals of ideals" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "Let " }{XPPEDIT 18 0 "I" "6#%\"IG" } {TEXT -1 38 " be an ideal in the polynomial ring " }{TEXT 256 1 "Q" }{TEXT -1 54 "[x]. The radical of I consists of all polynomials " }{XPPEDIT 18 0 "Q" "6#%\"QG" }{TEXT -1 12 " such that " }{XPPEDIT 18 0 "Q^n" "6#)%\"QG%\"nG" }{TEXT -1 13 " belongs to " }{XPPEDIT 18 0 "I " "6#%\"IG" }{TEXT -1 29 " for some positive integer " }{XPPEDIT 18 0 "n" "6#%\"nG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 132 "In view of the above definition, show th at the polynomial P=x-1 belongs to the radical of the ideal generate d by the polynomials " }{XPPEDIT 18 0 "M := 77-524*x+1570*x^2-2636*x^ 3+2395*x^4-322*x^5-1744*x^8-2093*x^6+2800*x^7-8*x^11-45*x^10+530*x^9; " "6#>%\"MG,:\"#x\"\"\"%\"xG!$C&*$)F(\"\"#\"\"\"\"%q:*$)F(\"\"$F-!%OE* $)F(\"\"%F-\"%&R#*$)F(\"\"&F-!$A$*$)F(\"\")F-!%W<*$)F(\"\"'F-!%$4#*$)F (\"\"(F-\"%+G*$)F(\"#6F-!\")*$)F(\"#5F-!#X*$)F(\"\"*F-\"$I&" }{TEXT -1 9 " and " }{XPPEDIT 18 0 "N := 66*x^8-342*x^7+661*x^6-480*x^5-1 65*x^4+514*x^3-333*x^2+84*x-5;" "6#>%\"NG,4*$)%\"xG\"\")\"\"\"\"#m*$)F (\"\"(F*!$U$*$)F(\"\"'F*\"$h'*$)F(\"\"&F*!$![*$)F(\"\"%F*!$l\"*$)F(\" \"$F*\"$9&*$)F(\"\"#F*!$L$F(\"#%)!\"&\"\"\"" }{TEXT -1 5 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 8 "So lution" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 53 "Let us define the above polynomials for computations." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 173 "M := 7 7-524*x+1570*x^2-2636*x^3+2395*x^4-322*x^5-1744*x^8-2093*x^6+2800*x^7- 8*x^11-45*x^10+530*x^9;N := 66*x^8-342*x^7+661*x^6-480*x^5-165*x^4+514 *x^3-333*x^2+84*x-5;P:=x-1;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"MG, :\"#x\"\"\"%\"xG!$C&*$)F(\"\"#\"\"\"\"%q:*$)F(\"\"$F-!%OE*$)F(\"\"%F- \"%&R#*$)F(\"\"&F-!$A$*$)F(\"\")F-!%W<*$)F(\"\"'F-!%$4#*$)F(\"\"(F-\"% +G*$)F(\"#6F-!\")*$)F(\"#5F-!#X*$)F(\"\"*F-\"$I&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"NG,4*$)%\"xG\"\")\"\"\"\"#m*$)F(\"\"(F*!$U$*$)F(\" \"'F*\"$h'*$)F(\"\"&F*!$![*$)F(\"\"%F*!$l\"*$)F(\"\"$F*\"$9&*$)F(\"\"# F*!$L$F(\"#%)!\"&\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"PG,&%\" xG\"\"\"!\"\"F'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 102 "Now the ideal generated by M and N is generated by the gcd of M and N, i.e., it is generated by" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "MN:= gcd(M,N);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#MNG,0*$)%\"xG\"\"'\"\" \"\"\"\"*$)F(\"\"&F*!\"'*$)F(\"\"%F*\"#:*$)F(\"\"$F*!#?*$)F(\"\"#F*F3F (F/F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 413 "Since we were told to show that the pol ynomial P is in the radical of the ideal generated by M and N, w e next try to find, by explicit computation, a suitable power of P s uch that the polynomial MN (the gcd of M and N) divides that power o f P. We can check whether a given polynomial divides another given p olynomial by Maple command divide. We apply it to the present situati on in the following way" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 90 "for i \+ from 1 do if divide((x-1)^i,MN) then print(\"P is in the radical.\"); break; fi; od; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#Q5P~is~in~the~radi cal.6\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "i;" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#\"\"'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 18 "We conclude that " } {XPPEDIT 18 0 "(x-1)^6" "6#*$,&%\"xG\"\"\"\"\"\"!\"\"\"\"'" }{TEXT -1 30 " is a multiple of the gcd of " }{XPPEDIT 18 0 "M" "6#%\"MG" } {TEXT -1 7 " and " }{XPPEDIT 18 0 "N" "6#%\"NG" }{TEXT -1 10 ". Hen ce " }{XPPEDIT 18 0 "P" "6#%\"PG" }{TEXT -1 47 " is in the radical o f the ideal generated by " }{XPPEDIT 18 0 "M" "6#%\"MG" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "N" "6#%\"NG" }{TEXT -1 1 "." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 34 "We can see that also by factoring:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "factor(MN);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*$) ,&%\"xG\"\"\"!\"\"F'\"\"'\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}}{MARK "1" 0 }{VIEWOPTS 1 1 0 1 1 1803 }