{VERSION 3 0 "SGI MIPS UNIX" "3.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 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 }} {SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 31 "Solving equations by resul tant." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "f:=x^2+y^2+z^2-3; " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "g:=x*y+y*z+z*x-3;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "h:=x*y*z+x+y+z-4;" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 75 "What are the common solutions (x,y ,z) =(alpha,beta,gamma) of the equations:" }}{PARA 0 "" 0 "" {TEXT -1 3 "f=0" }}{PARA 0 "" 0 "" {TEXT -1 3 "g=0" }}{PARA 0 "" 0 "" {TEXT -1 4 "h=0." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 104 "Well, for f and g to have a common root x=alpha, we need that the following vanishes for y=beta,z=gamma:" }}{PARA 0 "" 0 "" {TEXT -1 21 "Rfg=resultant(f,g,x)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 97 "Furthermore g and h must have a common root x=alph a, and h and f must have a common root x=alpha." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 55 "So y=beta,z=gamma must sa tisfy the following equations:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 16 "resultant(f,g,x)" }}{PARA 0 "" 0 "" {TEXT -1 16 "resultant(g,h,x)" }}{PARA 0 "" 0 "" {TEXT -1 17 "resultan t(h,f,x)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 100 "Note that these are only necessary conditions on y and z, they're not sufficient conditions. Namely:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 149 "f,g,h have common root x=something im plies that f,g have common root, g,h have common root and h,f have com mon root. The converse is not true, see:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 42 "f:=x*(x-1) g:=(x-2)*(x-1) and h:= x*(x-2)." }}{PARA 0 "" 0 "" {TEXT -1 136 "Now f,g have a common root \+ x=1, g,h have a common root x=2, and h,f have a common root x=0, but n evertheless f,g,h have no common root." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 99 "Anyway, we obtain the following neces sary (but maybe not sufficient) conditions on y=beta, z=gamma." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "Rfg:=resultant(f,g,x);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "Rgh:=resultant(g,h,x);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "Rhf:=resultant(h,f,x);" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 82 "Now from these equations we comput e the following necessary conditions on z=gamma." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "Rfg_gh:=resultant(Rfg,Rgh,y);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "Rgh_hf:=resultant(Rgh,Rhf,y);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "Rhf_fg:=resultant(Rhf,Rfg,y) ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "gcd(Rfg_gh, Rgh_hf);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "gcd(%,Rhf_fg);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "factor(%);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "v:=\{solve(%)\};" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 77 "So for any solution (alpha,beta,gamma) of \{f,g,h\} we have that gamma must be:" }}{PARA 0 "" 0 "" {TEXT -1 19 "1, -2 +/- sq rt(-3)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 160 "If we select one of these values, say: -2+sqrt(-3), how can we ch eck if there exists a solution (alpha, beta, -2+sqrt(-3)), and if it e xists, how do we find it?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "gamma:=-2+sqrt(-3);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 " Gamma:=-2+sqrt(-3);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "subs (z=Gamma,\{Rfg,Rgh,Rhf\});" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "gcd(%[1],%[2]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "gcd( %,%%[3]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "factor(%);" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "solve(%);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 98 "So, when z=-2+sqrt(-3), then we must have eithe r y=1 or y=-2-sqrt(-3). Lets just take one of them:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "beta:=-2-sqrt(-3);" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 29 "subs(z=Gamma,y=beta,\{f,g,h\});" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "gcd(%[1],%[2]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "gcd(%,%%[3]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 103 "If z=-2+sqrt(-3) and y=-2-sqrt(-3) then f,g,h indeed hav e a common solution x=1. So we find a solution:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 29 "(1, -2-sqrt(-3), -2+sqrt( 3))." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 118 " Of course, we had different choices for z and for y. If we try all pos sibilities, we will find all solutions of f,g,h." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "solve( \{f,g,h\} );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "_EnvExplicit:=true;" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 17 "solve( \{f,g,h\} );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "Now lets try a harder example" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "f:=x^2+y^2+x*z^2-3;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "g:=y^2+z^2+y*x^2-3;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "h:=z^2+x^2+z*y^2-1;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "R1:=resultant(f,g,x); R2:=re sultant(g,h,x); R3:=resultant(h,f,x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "T1:=resultant(R1,R2,y); T2:=resultant(R2,R3,y); T3:=r esultant(R3,R1,y);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "facto r(gcd( gcd(T1,T2) ,T3));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "r1, r2:=RootOf(op(1,%)) , RootOf(op(2,%));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "subs(z=r1,[R1,R2,R3]);" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 25 "gcd(gcd(%[1],%[2]),%[3]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "su bs(z=r2,[R1,R2,R3]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "gcd (gcd(%[1],%[2]),%[3]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "49" 0 }{VIEWOPTS 1 1 0 1 1 1803 }