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{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 27 "f:=add(a.i*x^i,i=0..1)+x^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>
%\"fG,(*&%#a1G\"\"\"%\"xGF(F(%#a0GF(*$)F)\"\"#\"\"\"F(" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "g:=add(b.i*x^i,i=0..2)+x^3;" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gG,*%#b0G\"\"\"*&%#b1GF'%\"xGF'F'*
&%#b2GF')F*\"\"#\"\"\"F'*$)F*\"\"$F/F'" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 39 "Now lets consider the following matrix." }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 13 "with(linalg):" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 20 "M:=sylvester(f,g,x);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%\"MG-%'matrixG6#7'7'\"\"\"%#a1G%#a0G\"\"!F-7'F-F*F+F,F-7'F-F-F
*F+F,7'F*%#b2G%#b1G%#b0GF-7'F-F*F1F2F3" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 10 "d:=det(M);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"dG,
<*$)%#b0G\"\"#\"\"\"\"\"\"*(%#a1GF+%#b1GF+F(F+!\"\"*&%#a0GF+)F.F)F*F+*
(F1F*%#b2GF+F(F*!\"#*(F4F*)F-F)F*F(F*F+**F-F*F.F*F1F*F4F*F/*&)F1F)F*)F
4F)F*F+*(F1F*F-F*F(F*\"\"$*&F:F*F.F*F5*&)F-F=F*F(F*F/*(F7F*F.F*F1F*F+*
(F:F*F4F*F-F*F/*$)F1F=F*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 265 16 "Sylvester matrix" }{TEXT -1 48 ". The determinant
 of that matrix is called the: " }{TEXT 266 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,<*$)%#b0G\"\"#\"\"\"\"\"\"*(%
#a1GF+%#b1GF+F(F+!\"\"*&%#a0GF+)F.F)F*F+*(F1F*%#b2GF+F(F*!\"#*(F4F*)F-
F)F*F(F*F+**F-F*F.F*F1F*F4F*F/*&)F1F)F*)F4F)F*F+*(F1F*F-F*F(F*\"\"$*&F
:F*F.F*F5*&)F-F=F*F(F*F/*(F7F*F.F*F1F*F+*(F:F*F4F*F-F*F/*$)F1F=F*F+" }
}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 100 "We claim that the resultant equ
als zero if and only if f and g have a root in common. In other words
" }}{PARA 0 "" 0 "" {TEXT -1 63 " resultant(f,g,x) = 0   if and only i
f gcd(f,g) is not trivial." }}{PARA 0 "" 0 "" {TEXT -1 11 "Lets check:
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "b0:=1;b1:=3;a0:=2;a1:=5
;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#b0G\"\"\"" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#b1G\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#a0G\"
\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#a1G\"\"&" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 2 "R;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(\"#V
\"\"\"%#b2G!#H*$)F&\"\"#\"\"\"\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 9 "solve(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$,&#\"#H\"
\")\"\"\"*$-%%sqrtG6#\"#<\"\"\"#\"\"$F&,&F$F'F(#!\"$F&" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "b2:=%[1];" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#b2G,&#\"#H\"\")\"\"\"*$-%%sqrtG6#\"#<\"\"\"#\"\"$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%*$-%%sqrtG6#\"#<\"\"\"#F%F(" }}}{EXCHG {PARA 0 
"" 0 "" {TEXT -1 16 "Another example." }}{PARA 0 "" 0 "" {TEXT -1 100 
"Suppose that this time we know the roots alpha.i of f and beta.j of g
. Then we can write f and g as:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 25 "f:=mul(x-alpha.i,i=1..2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>
%\"fG*&,&%\"xG\"\"\"%'alpha1G!\"\"F(,&F'F(%'alpha2GF*F(" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "g:=mul(x-beta.j,j=1..3);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%\"gG*(,&%\"xG\"\"\"%&beta1G!\"\"F(,&F'F(%&
beta2GF*F(,&F'F(%&beta3GF*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 an
d a j such that alpha.i=beta.j, which happens if and only if the follo
wing product equals 0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "P:
=mul(mul(alpha.i-beta.j,i=1..2),j=1..3);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%\"PG*.,&%'alpha1G\"\"\"%&beta1G!\"\"F(,&%'alpha2GF(F)F*F(,&F'F
(%&beta2GF*F(,&F,F(F.F*F(,&F'F(%&beta3GF*F(,&F,F(F1F*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*.,&%'alpha1G\"\"\"%&beta1G!\"\"F(,&%'
alpha2GF(F)F*F(,&F'F(%&beta2GF*F(,&F,F(F.F*F(,&F'F(%&beta3GF*F(,&F,F(F
1F*F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 54 "We see that the polynomi
al P equals the resultant. So:" }}{PARA 0 "" 0 "" {TEXT 264 93 "result
ant(f,g,x) is an expression which is zero if and only 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 prod
uct 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(sylvester(f,g,x)), equals th
e 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 "Here 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*\"\"#*$)F(F
/F*\"\"**&%\"aG\"\"\"F(F5F5!\"$F5" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 16 "g:=-3*x^2+5*x-8;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>
%\"gG,(*$)%\"xG\"\"#\"\"\"!\"$F(\"\"&!\")\"\"\"" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 17 "sylvester(f,g,x);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#-%'matrixG6#7)7)\"\"$\"\"#\"\"!\"\"*%\"aG!\"$F*7)F*F(F)
F*F+F,F-7)F-\"\"&!\")F*F*F*F*7)F*F-F0F1F*F*F*7)F*F*F-F0F1F*F*7)F*F*F*F
-F0F1F*7)F*F*F*F*F-F0F1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "
d:=det(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"dG,(!'(yG\"\"\"\"%\"
aG\"&Fx\"*$)F(\"\"#\"\"\"!$['" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 10 "factor(d);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(!'(yG\"\"\"\"%\"
aG\"&Fx\"*$)F&\"\"#\"\"\"!$['" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 16 "a1,a2:=solve(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>6$%#a1G%#a2
G6$,&#\"%4f\"$K%\"\"\"*&%\"IGF,-%%sqrtG6#\"#r\"\"\"#\"$v\"F+,&F)F,F-#!
$v\"F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "gcd(f,g);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 20 "op(subs(a=9,[f,g]));" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6$,,*$)%\"xG\"\"&\"\"\"\"\"$*$)F&\"\"%F(\"\"#*$)F&F-F(\"\"*F&F0!\"$\"
\"\",(F.F1F&F'!\")F2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "gcd(
%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 21 "op(subs(a=a1,[f,g]));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6$,,*$)%\"xG\"\"&\"\"\"\"\"$*$)F&\"\"%F(\"\"#*$)F&F-F(\"
\"**&,&#\"%4f\"$K%\"\"\"*&%\"IGF6-%%sqrtG6#\"#rF(#\"$v\"F5F6F&F6F6!\"$
F6,(F.F?F&F'!\")F6" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "gcd(%)
;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(%\"xG\"\"\"#!\"&\"\"'F%*&%\"IGF
%-%%sqrtG6#\"#r\"\"\"#!\"\"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 numb
er for a such that the resultant becomes zero, only then will the gcd \+
become non-trivial." }}}}{MARK "36 0 0" 129 }{VIEWOPTS 1 1 0 1 1 1803 
}