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{SECT 0 {EXCHG {PARA 201 "" 0 "" {TEXT 216 51 "Introduction to the MR \+
(Macaulay Resultant) Package" }}{PARA 202 "" 0 "" {TEXT 217 25 "(C) 20
04 Manfred Minimair" }}{PARA 203 "" 0 "" }{PARA 203 "" 0 "" {TEXT 218 
11 "Description" }}{PARA 203 "" 0 "" {TEXT 219 226 "This package provi
des the function MResultant which computes the projective (dense, Maca
ulay) resultant of a list of multivariate polynomials. The function im
plements Macaulay's algorithm (see D. Cox, J. Little and D. O'Shea. " 
}{TEXT 220 24 "Using Algebraic Geometry" }{TEXT 219 55 ". Springer Ver
lag, New York, Berlin, Heidelberg, 1998)." }}{PARA 203 "" 0 "" }{PARA 
203 "" 0 "" {TEXT 218 16 "Calling Sequence" }}{PARA 204 "" 0 "" {TEXT 
221 41 "MR:-MResultant(polylist, varlist, optseq)" }}{PARA 204 "" 0 ""
 {TEXT 221 41 "MR[MResultant](polylist, varlist, optseq)" }}{PARA 204 
"" 0 "" {TEXT 221 36 "MResultant(polylist, varlist, optseq" }{TEXT 
221 1 ")" }}{PARA 203 "" 0 "" }{PARA 203 "" 0 "" {TEXT 218 10 "Paramet
ers" }}{PARA 204 "" 0 "" {TEXT 221 78 "polylist     - list of polynomi
als (multivariate, not necessarily homogeneous)" }}{PARA 204 "" 0 "" 
{TEXT 221 64 "varlist       - list of variables of the polynomials in \+
polylist" }}{PARA 204 "" 0 "" {TEXT 221 87 "optseq      - sequence of \+
optional arguments: degrees=D, method=charpoly[lambda], out=L" }}
{PARA 203 "" 0 "" }{PARA 203 "" 0 "" {TEXT 218 37 "Description of the \+
optional arguments" }}{PARA 205 "" 0 "" {TEXT 222 172 "D in degrees=D \+
stands for a list of the total degrees of the polynomials in plist. If
 this optional argument is not given then the degrees will be computed
 by the function." }}{PARA 205 "" 0 "" {TEXT 222 202 "lambda in method
=charpoly[lambda] stands for any variable symbol. If this optional arg
ument is given then the function will compute the generalized characte
ristic polynomial even if it is not necessary." }}{PARA 205 "" 0 "" 
{TEXT 222 667 "L in out=L is a list of 'resultant', 'charpoly'. An out
put sequence will be returned in the same order as the list. 'resultan
t' causes the function to return the resultant. If the function comput
es the generalized characteristic polynomial 'charpoly' causes the fun
ction to return a triple: numerator, denominator, remainder. The gener
alized characteristic polynomial of the input polynomials is numerator
/denominator. If the input polynomials are given exactly then remainde
r equals 0. If the function does not compute the generalized polynomia
l, it will return the tripel FAIL, FAIL, FAIL. If this optional argume
nt is not given then only the resultant is returned." }}{PARA 205 "" 0
 "" {TEXT 222 89 "There are other optional arguments (of more technica
l interest) described in the package." }}{PARA 203 "" 0 "" }{PARA 203 
"" 0 "" {TEXT 218 10 "Infolevels" }}{PARA 203 "" 0 "" {TEXT 219 29 "in
folevel[MR:-MResultant]:=k;" }}{PARA 205 "" 0 "" {TEXT 222 36 "k >= 2:
 print steps in the algorithm" }}{PARA 205 "" 0 "" {TEXT 222 28 "k >= \+
3: print overall timing" }}{PARA 205 "" 0 "" {TEXT 222 51 "k >= 4: pri
nt timings of all steps in the algorithm" }}{PARA 203 "" 0 "" }{PARA 
203 "" 0 "" {TEXT 218 8 "Examples" }}}{EXCHG {PARA 203 "> " 0 "" 
{MPLTEXT 1 223 9 "with(MR);" }}{PARA 206 "" 1 "" {XPPMATH 20 "6#7#%+MR
esultantG" }{TEXT 224 1 " " }}}{EXCHG {PARA 203 "> " 0 "" {MPLTEXT 1 
223 45 "f1 := randpoly([x1,x2,p1,p2],dense,degree=2);" }}{PARA 207 "" 
1 "" {XPPMATH 20 "6#>%#f1G,@\"#\\!\"\"*&\"#^\"\"\")%#x1G\"\"#F*F'*(\"#
tF*F,F*%#x2GF*F'*(F/F*F,F*%#p1GF*F'*(\"#\"*F*F,F*%#p2GF*F'F,F**&\"\"&F
*)F0F-F*F**(\"#')F*F0F*F2F*F'*(\"#VF*F0F*F5F*F**&\"\"%F*F0F*F'*&\"#]F*
)F2F-F*F'*(F@F*F2F*F5F*F**&\"#nF*F2F*F**&\"#RF*)F5F-F*F'*&\"\")F*F5F*F
*" }{TEXT 225 1 " " }}}{EXCHG {PARA 203 "> " 0 "" {MPLTEXT 1 223 45 "f
2 := randpoly([x1,x2,p1,p2],dense,degree=2);" }}{PARA 207 "" 1 "" 
{XPPMATH 20 "6#>%#f2G,@\"#()!\"\"*&\"#6\"\"\")%#x1G\"\"#F*F**(\"#$*F*F
,F*%#x2GF*F**(\"#9F*F,F*%#p1GF*F'*(\"#**F*F,F*%#p2GF*F'*&\"#nF*F,F*F'*
&\"#oF*)F0F-F*F**(\"#XF*F0F*F3F*F**(\"#wF*F0F*F6F*F**&\"\"'F*F0F*F**&
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F6F*F*" }{TEXT 225 1 " " }}}{EXCHG {PARA 203 "> " 0 "" {MPLTEXT 1 223 
45 "f3 := randpoly([x1,x2,p1,p2],dense,degree=2);" }}{PARA 207 "" 1 ""
 {XPPMATH 20 "6#>%#f3G,@\"#f\"\"\"*&\"#sF')%#x1G\"\"#F'F'*(\"#YF'F+F'%
#x2GF'!\"\"*(\"#oF'F+F'%#p1GF'F0*(\"#UF'F+F'%#p2GF'F0*&\"#ZF'F+F'F0*&
\"#KF')F/F,F'F0*(\"#PF'F/F'F3F'F'*(\"#$*F'F/F'F6F'F0*&\"#eF'F/F'F0*&\"
#!*F')F3F,F'F0*(\"#`F'F3F'F6F'F0*&\"#pF'F3F'F0*&\"#%)F')F6F,F'F0*&F.F'
F6F'F'" }{TEXT 225 1 " " }}}{EXCHG {PARA 203 "> " 0 "" {MPLTEXT 1 223 
35 "MR:-MResultant([f1,f2,f3],[x1,x2]);" }}{PARA 207 "" 1 "" {XPPMATH 
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z&3Xi&F&*&\":88?@&G`r*4#HIEF&F7F&F**&\"9q37zo\")>%*RFOnF&F/F&F**&\":!
\\\"pHD6R^30%e<F&)F/F2F&F**&\"9e\"\\Byz\"*R8=PN)F&FjnF&F&*&\":M!Qp9/to
'3%e&y$F&FCF&F**&\":H'>^8w9x$Q@u_#F&F[pF&F*F&F(F&F&*&\"98%)4vTyVAJ#zn
\"F&F7F&F&*&\"9o)GAH]8w5<6,#F&)F/F)F&F&*&\"94=%Q(36-FW-C^F&FjnF&F**&\"
9Xp1')y&=yP[!oVF&F[pF&F**&\"9a=J[X%QBL$GEHF&F^qF&F&*&\"8ajZ!R&e4(z`jsF
&FCF&F&\"8D\"Qwe<[S]')\\fF&*&\"9`Y;3We`!)z3eiF&FSF&F**&\"9vt\")y@&4RVF
/P\"F&F/F&F&" }{TEXT 225 1 " " }}}{EXCHG {PARA 203 "> " 0 "" {MPLTEXT 
1 223 31 "infolevel[MR:-MResultant] := 2;" }}{PARA 206 "" 1 "" 
{XPPMATH 20 "6#>&%*infolevelG6#%+MResultantG\"\"#" }{TEXT 224 1 " " }}
}{EXCHG {PARA 203 "> " 0 "" {MPLTEXT 1 223 35 "MR:-MResultant([f1,f2,f
3],[x1,x2]);" }}{PARA 208 "" 1 "" {TEXT 226 37 "MResultantGeneric:   m
ethod: standard" }}{PARA 208 "" 1 "" {TEXT 226 42 "MResultantGeneric: \+
  STEP: NumeratorMatrix" }}{PARA 208 "" 1 "" {TEXT 226 44 "MResultantG
eneric:   STEP: DenominatorMatrix" }}{PARA 208 "" 1 "" {TEXT 226 41 "M
ResultantGeneric:   STEP: DenominatorDet" }}{PARA 208 "" 1 "" {TEXT 
226 39 "MResultantGeneric:   STEP: NumeratorDet" }}{PARA 208 "" 1 "" 
{TEXT 226 38 "MResultantGeneric:   STEP: DetQuotient" }}{PARA 207 "" 
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