{VERSION 6 0 "IBM INTEL LINUX" "6.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 257 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Text Output" -1 2 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 0 0 0 0 0 1 3 0 3 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Warning" 2 7 1 {CSTYLE "" -1 -1 " " 0 1 0 0 255 1 0 0 0 0 0 0 1 0 0 0 }0 0 0 -1 -1 -1 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 1 }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 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 256 14 "The result ant." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 169 " Suppose you have two polynomials f and g in x. Say one is of degree 2 \+ and one is of degree 3. Suppose the highest coefficient of both polyno mials is 1, so we can write: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "f:=add(a[i]*x^i,i=0..1)+x^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%\"fG,(&%\"aG6#\"\"!\"\"\"*&&F'6#F*F*%\"xGF*F**$)F.\"\"#F*F*" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "g:=add(b[i]*x^i,i=0..2)+x^3; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gG,*&%\"bG6#\"\"!\"\"\"*&&F'6# F*F*%\"xGF*F**&&F'6#\"\"#F*)F.F2F*F**$)F.\"\"$F*F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 39 "Now lets consider the following matrix." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(linalg):" }}{PARA 7 "" 1 "" {TEXT -1 80 "Warning, the protected names norm and trace have bee n redefined and unprotected\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "M:=sylvester(f,g,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"MGK% 'matrixG6#7'7'\"\"\"&%\"aG6#F*&F,6#\"\"!F0F07'F0F*F+F.F07'F0F0F*F+F.7' F*&%\"bG6#\"\"#&F5F-&F5F/F07'F0F*F4F8F9Q(pprint06\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "d:=det(M);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"dG,<*$)&%\"bG6#\"\"!\"\"#\"\"\"F-*(&F)6#F-F-&%\"aGF0F-F(F-! \"\"*&&F2F*F-)F/F,F-F-**F,F-&F)6#F,F-F5F-F(F-F3*(F8F-)F1F,F-F(F-F-**F8 F-F1F-F5F-F/F-F3*&)F8F,F-)F5F,F-F-**\"\"$F-F1F-F5F-F(F-F-*(F,F-F/F-F?F -F3*&)F1FAF-F(F-F3*(F;F-F5F-F/F-F-*(F1F-F8F-F?F-F3*$)F5FAF-F-" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 77 "The matrix formed in this way from the coefficients of f and g is called the " }{TEXT 259 16 "Sylvester \+ matrix" }{TEXT -1 48 ". The determinant of that matrix is called the: \+ " }{TEXT 260 10 "resultant." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "R:=resultant(f,g,x); # same as d" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"RG,<*$)&%\"bG6#\"\"!\"\"#\"\"\"F-*(&F)6#F-F-&%\"aGF0F-F(F-!\"\" *&&F2F*F-)F/F,F-F-**F,F-&F)6#F,F-F5F-F(F-F3*(F8F-)F1F,F-F(F-F-**F8F-F1 F-F5F-F/F-F3*&)F8F,F-)F5F,F-F-**\"\"$F-F1F-F5F-F(F-F-*(F,F-F/F-F?F-F3* &)F1FAF-F(F-F3*(F;F-F5F-F/F-F-*(F1F-F8F-F?F-F3*$)F5FAF-F-" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 100 "We claim that the resultant equals zero \+ if and only if f and g have a root in common. In other words" }}{PARA 0 "" 0 "" {TEXT -1 63 " resultant(f,g,x) = 0 if and only if gcd(f,g) is not trivial." }}{PARA 0 "" 0 "" {TEXT -1 11 "Lets check:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "b[0]:=1;b[1]:=3;a[0]:=2;a[1] :=5;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"bG6#\"\"!\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"bG6#\"\"\"\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"aG6#\"\"!\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >&%\"aG6#\"\"\"\"\"&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "R;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(\"#V\"\"\"*&\"#HF%&%\"bG6#\"\"#F%! \"\"*&\"\"%F%)F(F+F%F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "so lve(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$,&#\"#H\"\")\"\"\"*(\"\"$F' F&!\"\"\"#<#F'\"\"#F',&F$F'*(F)F'F&F*F+F,F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "b[2]:=%[1];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&% \"bG6#\"\"#,&#\"#H\"\")\"\"\"*(\"\"$F,F+!\"\"\"#<#F,F'F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "expand(R);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "gcd (f,g); # non-trivial." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(%\"xG\"\"\" #\"\"&\"\"#F%*&F(!\"\"\"#<#F%F(F%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 16 "Another example." }}{PARA 0 "" 0 "" {TEXT -1 100 "Suppose that thi s time we know the roots alpha.i of f and beta.j of g. Then we can wri te f and g as:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "f:=mul(x- alpha[i],i=1..2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG*&,&%\"xG\" \"\"&%&alphaG6#F(!\"\"F(,&F'F(&F*6#\"\"#F,F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "g:=mul(x-beta[j],j=1..3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gG*(,&%\"xG\"\"\"&%%betaG6#F(!\"\"F(,&F'F(&F*6#\"\" #F,F(,&F'F(&F*6#\"\"$F,F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 161 "Now f and g have a non-trivial gcd if and only if there exists an i and a j such that alpha.i=beta.j, which happens if and only if the followin g product equals 0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "P:=mu l(mul(alpha[i]-beta[j],i=1..2),j=1..3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"PG*.,&&%&alphaG6#\"\"\"F*&%%betaGF)!\"\"F*,&&F(6#\"\"#F*F+F- F*,&F'F*&F,F0F-F*,&F/F*F3F-F*,&F'F*&F,6#\"\"$F-F*,&F/F*F6F-F*" }}} {EXCHG {PARA 12 "" 1 "" {TEXT -1 16 "And we see that:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "R:=resultant(f,g,x); # is the same ." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"RG*.,&&%&alphaG6#\"\"\"F*&%%b etaGF)!\"\"F*,&&F(6#\"\"#F*F+F-F*,&F'F*&F,F0F-F*,&F/F*F3F-F*,&F'F*&F,6 #\"\"$F-F*,&F/F*F6F-F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 54 "We see \+ that the polynomial P equals the resultant. So:" }}{PARA 0 "" 0 "" {TEXT 258 93 "resultant(f,g,x) is an expression which is zero if and o nly if f and g have a root in common." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT 257 8 "Theorem:" }}{PARA 0 "" 0 "" {TEXT -1 43 "Let f = the product of x-alpha.i for i=1..n" }}{PARA 0 "" 0 "" {TEXT -1 41 "Let g= the product of x-beta.j for j=1..m" }}{PARA 0 "" 0 "" {TEXT -1 123 "Then resultant(f,g,x), which is defined as det(sylv ester(f,g,x)), equals the product of all alpha.i-beta.j, i=1..n, j=1.. m." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 58 "Her e is an example where the leading coefficient is not 1." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "f:=3*x^5+2*x^4+9*x^2+a*x-3;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG,,*&\"\"$\"\"\")%\"xG\"\"&F(F(*& \"\"#F()F*\"\"%F(F(*&\"\"*F()F*F-F(F(*&%\"aGF(F*F(F(F'!\"\"" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "g:=-3*x^2+5*x-8;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gG,(*&\"\"$\"\"\")%\"xG\"\"#F(!\"\"*&\" \"&F(F*F(F(\"\")F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "sylve ster(f,g,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7)7)\"\"$ \"\"#\"\"!\"\"*%\"aG!\"$F*7)F*F(F)F*F+F,F-7)F-\"\"&!\")F*F*F*F*7)F*F-F 0F1F*F*F*7)F*F*F-F0F1F*F*7)F*F*F*F-F0F1F*7)F*F*F*F*F-F0F1Q(pprint16\" " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "d:=det(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"dG,(\"'(yG\"!\"\"*&\"&Fx\"\"\"\"%\"aGF*F**& \"$['F*)F+\"\"#F*F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "fact or(d);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(\"'(yG\"!\"\"*&\"&Fx\"\"\" \"%\"aGF(F(*&\"$['F()F)\"\"#F(F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "a1,a2:=solve(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> 6$%#a1G%#a2G6$,&#\"%4f\"$K%\"\"\"*&^##!$v\"F+F,\"#r#F,\"\"#F,,&F)F,*&^ ##\"$v\"F+F,F1F2F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "gcd(f, g);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "op(subs(a=9,[f,g]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$,,*&\"\"$\"\"\")%\"xG\"\"&F&F&*&\"\"#F&)F(\"\"%F&F&*&\" \"*F&)F(F+F&F&*&F/F&F(F&F&F%!\"\",(*&F%F&F0F&F2*&F)F&F(F&F&\"\")F2" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "gcd(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "op (subs(a=a1,[f,g]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$,,*&\"\"$\"\"\" )%\"xG\"\"&F&F&*&\"\"#F&)F(\"\"%F&F&*&\"\"*F&)F(F+F&F&*&,&#\"%4f\"$K%F &*&^##!$v\"F5F&\"#r#F&F+F&F&F(F&F&F%!\"\",(*&F%F&F0F&F<*&F)F&F(F&F&\" \")F<" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "gcd(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(%\"xG\"\"\"#\"\"&\"\"'!\"\"*&^##F%F(F%\"#r #F%\"\"#F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "gcd(f,g);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 129 "We see that only if we substitute a number for a such th at the resultant becomes zero, only then will the gcd become non-trivi al." }}}}{MARK "0 0 1" 14 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 } {PAGENUMBERS 0 1 2 33 1 1 }