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-1 -1 "" 1 12 0 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 "" 0 258 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 
0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 259 1 {CSTYLE "
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{SECT 0 {EXCHG {PARA 258 "" 0 "" {TEXT -1 0 "" }{TEXT 256 63 "Example \+
Maple Worksheet for PolynomialToPDE and PDEToPolynomial" }}{PARA 261 "
" 0 "" {TEXT -1 0 "" }}{PARA 259 "" 0 "" {TEXT 261 29 "Ilias Kotsireas
 and Greg Reid" }{TEXT -1 0 "" }}{PARA 260 "" 0 "" {TEXT -1 55 "Ontari
o Research Centre for Computer Algebra, May, 2001" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 0 "" }{TEXT 259 8 "Summary:" }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 260 0 "" }{TEXT -1 81 "Exampl
es are given in which Grobner bases for polynomial systems are compute
d by:" }}{PARA 0 "" 0 "" {TEXT -1 86 "* mapping them to associated sys
tems of linear homogeneous PDEs using PolynomialToPDE;" }}{PARA 0 "" 
0 "" {TEXT -1 111 "* using the differential elimination package DEtool
s[rifsimp], to reduce them to a differential Groebner Basis;" }}{PARA 
0 "" 0 "" {TEXT -1 101 "* mapping back to a Groebner basis for the ori
ginal polynomial system, using the map PDEToPolynomial;" }}{PARA 0 "" 
0 "" {TEXT -1 94 "* the polynomial solving is then done using some alt
ernative methods, including the functions " }}{PARA 0 "" 0 "" {TEXT 
-1 80 "    Groebner[MulMatrix] and Groebner[NormalSet] implemented in \+
Maple by Corless." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 35 "Particular problems considered are:" }}{PARA 0 "" 0 "" 
{TEXT -1 48 "*  Planar 4-body problem with equal masses (S4);" }}
{PARA 0 "" 0 "" {TEXT -1 50 "*  Spatial  5-body problem with equal mas
ses (S5);" }}{PARA 0 "" 0 "" {TEXT -1 53 "*  Robotics example, chapter
 6, Cox, Little, O'Shea, " }{TEXT 262 31 "Ideals Varieties and Algorit
hms" }{TEXT -1 1 ";" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 69 "The associated Technical Document is ORCCA Technical Re
port TR-01-03." }}{PARA 0 "" 0 "" {TEXT -1 60 "The associated code is \+
part of the PolynomialTools package, " }}{PARA 0 "" 0 "" {TEXT -1 59 "
 which is distributed with the upcoming release of Maple 7." }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 1 " " }
{TEXT 257 72 "Solve the system S4 (newtonian planar 4-body problem wit
h equal masses) " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 90 "The equations of central configurations in the newto
nian 4-body problem with equal masses." }}{PARA 0 "" 0 "" {TEXT -1 
157 "Reference: Ilias S. Kotsireas, \"Central Configurations in the Ne
wtonian N-body Problem \nof Celestial Mechanics\", To appear in Contem
porary Mathematics, 2001." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
70 "e1:=-2*p**3+2*p**3*phi**3-4*phi**3*s*p**2+5*phi**3*s**3*p-phi**3*s
**5;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#e1G,,*$)%\"pG\"\"$\"\"\"!\"
#*(\"\"#F*F'F*)%$phiGF)F*F***\"\"%F*)F(F-F*%\"sGF*F.F*!\"\"**\"\"&F*F.
F*)F3F)F*F(F*F**&F.F*)F3F6F*F4" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 67 "e2:=-2*s*p**3-2*phi**3*s**2+phi**3*s**4-3*phi**3*s**2*p+2*phi*
*3*p;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#e2G,,*&)%\"pG\"\"$\"\"\"%
\"sGF*!\"#*(\"\"#F*)F+F.F*)%$phiGF)F*!\"\"*&F0F*)F+\"\"%F*F***F)F*F(F*
F/F*F0F*F2*(F.F*F(F*F0F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
40 "e3:=-2*s**2+s**4-4*s**2*p+phi**2+1+4*p;\n" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#e3G,.*$)%\"sG\"\"#\"\"\"!\"#*$)F(\"\"%F*F**(F.F*%\"p
GF*F'F*!\"\"*$)%$phiGF)F*F*F*F**&F.F*F0F*F*" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 18 "S4:= [e1, e2, e3]:" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 72 "The plex  basis of [e1,e2,e3] cannot  be computed with Ma
ple's Groebner." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 87 "The tdeg basis
 of  [e1,e2,e3] can be computed with  either Maple's Groebner  or  usi
ng " }}{PARA 0 "" 0 "" {TEXT -1 39 "PolynomialToPDE and then using rif
simp." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 97 "The univariate polynomia
l in phi, that results from the elimination called P37, is of degree 3
7.." }}{PARA 0 "" 0 "" {TEXT -1 51 "P37 a part of the plex Grobner bas
is of [e1,e2,e3]." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 50 "P37 cannot b
e computed with Groebner at this time." }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 46 "We show two alternative ways of computing P37." }}}{SECT 
1 {PARA 4 "" 0 "" {TEXT -1 24 " using Maple's Groebner " }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "degOrd:= tdeg(p,s,phi):" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "st:=time(): degBasisS4:=Groebner[gb
asis](S4,degOrd): time()-st;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"\"!
F$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 97 "The calculation was complet
ed in the time given above, and resulted in 11 equations in the basis:
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "nops(degBasisS4);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#6" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 65 "nsS4, rvS4 := Groebner[SetBasis](degBasisS4, degOrd):
\nnops(nsS4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#**" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 90 "st:=time():\nMphiS4 := Groebner[Mul
Matrix](phi, nsS4, rvS4, degBasisS4, degOrd);\ntime()-st;" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#>%'MphiS4G-%'RTABLEG6+\"*G!f%Q\"%)anythingG%'Ma
trixG%'sparseG%.Fortran_orderG7\"\"\"#;\"\"\"\"#**F/" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#$\"%\")>!\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 92 "st:=time():\nsort(factor(LinearAlgebra[CharacteristicPolynomia
l](MphiS4,lambda)));\ntime()-st;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#**
,&*$)%'lambdaG\"\"#\"\"\"F)\"\"$!\"\"F),bo*$)F'\"#PF)F)*&\"#hF))F'\"#M
F)F+*&\"$O$F))F'\"#LF)F)*&\"$S#F))F'\"#KF)F+*&\"%_?F))F'\"#JF)F)*&\"&?
@\"F))F'\"#IF)F+*&\"%+%)F))F'\"#HF)F)*&\"&c/$F))F'\"#GF)F+*&\"'8^<F))F
'\"#FF)F)*&\"&[&))F))F'\"#EF)F+*&\"'S5CF))F'\"#DF)F)*&\"(&Qk8F))F'\"#C
F)F+*&\"'%**Q$F))F'\"#BF)F)*&\"(%)>3\"F))F'\"#AF)F+*&\"(1:C'F))F'\"#@F
)F)*&\"'i@kF))F'\"#?F)F)*&\"(2&>BF))F'\"#>F)F)*&\")y-z:F))F'\"#=F)F+*&
\")wtG7F))F'\"#<F)F+*&\"(4pQ\"F))F'\"#;F)F)*&\")#*H@6F))F'\"#:F)F)*&\"
)OX*e&F))F'\"#9F)F)*&\")'\\*))>F))F'\"#8F)F+*&\")k*QP&F))F'\"#7F)F)*&
\"*HLNG\"F))F'\"#6F)F+*&\")3`@WF))F'\"#5F)F)*&\"*SAXs\"F))F'\"\"*F)F+*
&\"*ts\"4;F))F'\"\")F)F)*&\"))fkF%F))F'\"\"(F)F+*&\"*[_h<#F))F'\"\"'F)
F)*&\"*&zSa6F))F'\"\"&F)F+*&\")5U7<F))F'\"\"%F)F)*&\"*&Rg!R\"F))F'F*F)
F+*&\")v!e)RF)F&F)F)FjsF)F)),&F%F)F)F)F]sF))F'\"#[F)" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#$\"&4r#!\"$" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 
35 " using  rifsimp and PolynomialTools" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 62 "desysS4 := PolynomialTools[PolynomialToPDE](S4,[s,p,p
hi],[u]);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%(desysS4G7%,,-%%diffG6$
-%\"uG6%%\"sG%\"pG%$phiG-%\"$G6$F.\"\"$!\"#*&\"\"#\"\"\"-F(6%F*F0-F16$
F/F3F7F7*&\"\"%F7-F(6&F*-F16$F.F6F:F-F7!\"\"*&\"\"&F7-F(6&F*F.F:-F16$F
-F3F7F7-F(6%F*F:-F16$F-FDFB,,-F(6%F*F0F-F4*&F6F7-F(6%F*F:-F16$F-F6F7FB
-F(6%F*F:-F16$F-F=F7*&F3F7-F(6&F*F.F:FSF7FB*&F6F7-F(6%F*F.F:F7F7,.-F(6
$F*FSF4-F(6$F*FWF7*&F=F7-F(6%F*F.FSF7FB-F(6$F*-F16$F/F6F7F*F7*&F=F7-F(
6$F*F.F7F7" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "infolevel[rif
simp] := 0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%*infolevelG6#%(rifsi
mpG\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "st:=time(): \nt
degrsysS4 := DEtools[rifsimp](desysS4): \ntime()-st;" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#$\"&?>#!\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 81 "degBasisS4 := PolynomialTools[PDEToPolynomial](tdegrsysS4[Solved
],[s,p,phi],[u]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "nops(d
egBasisS4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#>" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 72 "nsS4, rvS4 := Groebner[SetBasis](degBasis
S4, degOrd):\nnsS4: nops(nsS4);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
\"#**" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 90 "st:=time():\nMphiS
4 := Groebner[MulMatrix](phi, nsS4, rvS4, degBasisS4, degOrd);\ntime()
-st;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'MphiS4G-%'RTABLEG6+\"*![b%Q
\"%)anythingG%'MatrixG%'sparseG%.Fortran_orderG7\"\"\"#;\"\"\"\"#**F/
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"%!)Q!\"$" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 92 "st:=time():\nsort(factor(LinearAlgebra[Charact
eristicPolynomial](MphiS4,lambda)));\ntime()-st;" }}{PARA 12 "" 1 "" 
{XPPMATH 20 "6#**,&*$)%'lambdaG\"\"#\"\"\"F)\"\"$!\"\"F),bo*$)F'\"#PF)
F)*&\"#hF))F'\"#MF)F+*&\"$O$F))F'\"#LF)F)*&\"$S#F))F'\"#KF)F+*&\"%_?F)
)F'\"#JF)F)*&\"&?@\"F))F'\"#IF)F+*&\"%+%)F))F'\"#HF)F)*&\"&c/$F))F'\"#
GF)F+*&\"'8^<F))F'\"#FF)F)*&\"&[&))F))F'\"#EF)F+*&\"'S5CF))F'\"#DF)F)*
&\"(&Qk8F))F'\"#CF)F+*&\"'%**Q$F))F'\"#BF)F)*&\"(%)>3\"F))F'\"#AF)F+*&
\"(1:C'F))F'\"#@F)F)*&\"'i@kF))F'\"#?F)F)*&\"(2&>BF))F'\"#>F)F)*&\")y-
z:F))F'\"#=F)F+*&\")wtG7F))F'\"#<F)F+*&\"(4pQ\"F))F'\"#;F)F)*&\")#*H@6
F))F'\"#:F)F)*&\")OX*e&F))F'\"#9F)F)*&\")'\\*))>F))F'\"#8F)F+*&\")k*QP
&F))F'\"#7F)F)*&\"*HLNG\"F))F'\"#6F)F+*&\")3`@WF))F'\"#5F)F)*&\"*SAXs
\"F))F'\"\"*F)F+*&\"*ts\"4;F))F'\"\")F)F)*&\"))fkF%F))F'\"\"(F)F+*&\"*
[_h<#F))F'\"\"'F)F)*&\"*&zSa6F))F'\"\"&F)F+*&\")5U7<F))F'\"\"%F)F)*&\"
*&Rg!R\"F))F'F*F)F+*&\")v!e)RF)F&F)F)FjsF)F)),&F%F)F)F)F]sF))F'\"#[F)
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"&@3$!\"$" }}}}}{SECT 1 {PARA 
256 "" 0 "" {TEXT -1 75 " Solve the system S5 (newtonian spatial  5-bo
dy problem with equal masses) " }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 90 "
The equations of central configurations in the newtonian 5-body proble
m with equal masses." }}{PARA 0 "" 0 "" {TEXT -1 157 "Reference: Ilias
 S. Kotsireas, \"Central Configurations in the Newtonian N-body Proble
m \nof Celestial Mechanics\", To appear in Contemporary Mathematics, 2
001." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 99 "e1:=-6*p**3+4*p**3*phi**3+15*phi**3*s**3*p-3*phi
**3*s**5-12*phi**3*s*p**2-3*phi**3*s*p+phi**3*s**3;" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%#e1G,0*$)%\"pG\"\"$\"\"\"!\"'*(\"\"%F*F'F*)%$phiGF)
F*F***\"#:F*F.F*)%\"sGF)F*F(F*F**(F)F*F.F*)F3\"\"&F*!\"\"**\"#7F*)F(\"
\"#F*F3F*F.F*F7**F)F*F(F*F3F*F.F*F7*&F2F*F.F*F*" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 69 "e2:=-9*phi**3*s**2*p-5*phi**3*s**2-6*s*p**3+3*
phi**3*s**4+5*phi**3*p;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#e2G,,*(%
\"pG\"\"\")%\"sG\"\"#F()%$phiG\"\"$F(!\"**(\"\"&F(F)F(F,F(!\"\"*(\"\"'
F()F'F.F(F*F(F2*(F.F(F,F()F*\"\"%F(F(*(F1F(F'F(F,F(F(" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "e3:=-12*s**2*p-6*s**2+3*s**4+4*phi*
*2+3+12*p;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#e3G,.*&%\"pG\"\"\")%
\"sG\"\"#F(!#7*&\"\"'F(F)F(!\"\"*&\"\"$F()F*\"\"%F(F(*&F3F()%$phiGF+F(
F(F1F(*&\"#7F(F'F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "S5:
= [e1, e2, e3]:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 72 "The plex  basi
s of [e1,e2,e3] cannot  be computed with Maple's Groebner." }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 119 "The tdeg basis of  [e1,e2,e3] can be com
puted with   either Maple's Groebner  or  PolynomialToPDE together wit
h rifsimp" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 114 "The univariate poly
nomial in phi, called P43, is of degree 43.\nP43 a part of the plex Gr
obner basis of [e1,e2,e3]." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 37 "P43
 cannot be computed with Groebner." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 46 "We show two alternative ways of computing P43." }}}{SECT 1 
{PARA 4 "" 0 "" {TEXT -1 23 "Using Maple's Groebner " }}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "deg
Ord:= tdeg(p,s,phi):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "st:
=time(): degBasisS5:=Groebner[gbasis](S5,degOrd): \ntime()-st;" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"%J7!\"$" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 17 "nops(degBasisS5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#\"#6" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "nsS5, rvS5 := Gro
ebner[SetBasis](degBasisS5, degOrd):\nnops(nsS5);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#\"#**" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 90 "st:
=time():\nMphiS5 := Groebner[MulMatrix](phi, nsS5, rvS5, degBasisS5, d
egOrd);\ntime()-st;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'MphiS5G-%'RT
ABLEG6+\"*/plN\"%)anythingG%'MatrixG%'sparseG%.Fortran_orderG7\"\"\"#;
\"\"\"\"#**F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"%!e#!\"$" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "st:=time():\nsort(factor(Lin
earAlgebra[CharacteristicPolynomial](MphiS5,lambda)));\ntime()-st;" }}
{PARA 12 "" 1 "" {XPPMATH 20 "6#,$**,&*$)%'lambdaG\"\"#\"\"\"\"\"$\"\"
)!\"\"F*,`p*$)F(\"#VF*\"&/6$*&\"(y'QAF*)F(\"#TF*F-*&\"(3E<$F*)F(\"#SF*
F-*&\"('p*H%F*)F(\"#RF*F**&\")d5S#)F*)F(\"#QF*F**&\"*o#ys9F*)F(\"#PF*F
**&\"*3'>*4$F*)F(\"#OF*F-*&\"*wkFl)F*)F(\"#NF*F-*&\"+5-&3!QF*)F(\"#MF*
F-*&\"+ydJErF*)F(\"#LF*F**&\"*9^mL(F*)F(\"#KF*F**&\",SxnL.&F*)F(\"#JF*
F**&\",vhROX)F*)F(\"#IF*F-*&\",Q'*ohZ&F*)F(\"#HF*F**&\"-?@p&=2%F*)F(\"
#GF*F-*&\"-g0f**HjF*)F(\"#FF*F**&\"->b(fk)eF*)F(\"#EF*F-*&\".OmHdON#F*
)F(\"#DF*F**&\".EvqjFC$F*)F(\"#CF*F-*&\".cH`*fHNF*)F(\"#BF*F**&\".I*)>
k[$)*F*)F(\"#AF*F-*&\"/1y#Q=u3\"F*)F(\"#@F*F**&\"/KXH1*HH\"F*)F(\"#?F*
F-*&\"/%)erw\"oa#F*)F(\"#>F*F**&\"/vrV+!*z?F*)F(\"#=F*F-*&\"/3%QcFik#F
*)F(\"#<F*F**&\"/7`Q&f/C$F*)F(\"#;F*F-*&\"/%Qh,*[#z\"F*)F(\"#:F*F**&\"
/sK^ZsECF*)F(\"#9F*F-*&\"/[?/w\"G6\"F*)F(\"#8F*F**&\".[+sh#fGF*)F(\"#7
F*F-*&\".oZG_jc\"F*)F(\"#6F*F**&\".3M;UF\"))F*)F(\"#5F*F**&\".w\\&Qf)*
*)F*)F(\"\"*F*F-*&\".;'z77l)*F*)F(F,F*F**&\".+g$f,.**F*)F(\"\"(F*F-*&
\".3I+,az)F*)F(\"\"'F*F**&\".kCa7M([F*)F(\"\"&F*F-*&\".!3ycw\"4'F*)F(
\"\"%F*F**&\"-oZe*R\\*F*)F(F+F*F-*&\".ce-kVs#F*F'F*F*\"-#z8*Q`\\F*F*),
&F&FcuF+F*F+F*)F(\"#[F*#F*\"(o>(f" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$
\"&\\$f!\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT 
-1 34 "Using  rifsimp and PolynomialTools" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 62 "desysS5 := PolynomialTools[PolynomialToPDE](S5,[p,s,p
hi],[u]);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%(desysS5G7%,0-%%diffG6$
-%\"uG6%%\"pG%\"sG%$phiG-%\"$G6$F-\"\"$!\"'*&\"\"%\"\"\"-F(6%F*F0-F16$
F/F3F7F7*&\"#:F7-F(6&F*F-F:-F16$F.F3F7F7*&F3F7-F(6%F*F:-F16$F.\"\"&F7!
\"\"*&\"#7F7-F(6&F*-F16$F-\"\"#F:F.F7FH*&F3F7-F(6&F*F-F:F.F7FH-F(6%F*F
:F@F7,,-F(6&F*F-F:-F16$F.FO!\"**&FGF7-F(6%F*F:FXF7FH*&\"\"'F7-F(6%F*F0
F.F7FH*&F3F7-F(6%F*F:-F16$F.F6F7F7*&FGF7-F(6%F*F-F:F7F7,.-F(6%F*F-FX!#
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3F7F*F7F7*&FJF7-F(6$F*F-F7F7" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 24 "infolevel[rifsimp] := 0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%*
infolevelG6#%(rifsimpG\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
65 "st:=time(): \ntdegrsysS5 := DEtools[rifsimp](desysS5): \ntime()-st
;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"%H(*!\"$" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 81 "degBasisS5 := PolynomialTools[PDEToPolynomial
](tdegrsysS5[Solved],[p,s,phi],[u]):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 17 "nops(degBasisS5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
\"#6" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "nsS5, rvS5 := Groeb
ner[SetBasis](degBasisS5, degOrd):\nnsS5: nops(nsS5);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#\"#**" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
90 "st:=time():\nMphiS5 := Groebner[MulMatrix](phi, nsS5, rvS5, degBas
isS5, degOrd);\ntime()-st;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'MphiS
5G-%'RTABLEG6+\"*/p%y8%)anythingG%'MatrixG%'sparseG%.Fortran_orderG7\"
\"\"#;\"\"\"\"#**F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"%qG!\"$" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "st:=time():\nsort(factor(Lin
earAlgebra[CharacteristicPolynomial](MphiS5,lambda)));\ntime()-st;" }}
{PARA 12 "" 1 "" {XPPMATH 20 "6#,$**,&*$)%'lambdaG\"\"#\"\"\"\"\"$\"\"
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\"&5+'!\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}{SECT 
1 {PARA 257 "" 0 "" {TEXT -1 117 "Inverse Kinematic Problem and Robot \+
Motion Planning, \nChapter 6, Cox, Little, O'Shea, Ideals Varieties an
d Algorithms" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "eq1 := expan
d(l3*(c1*c2-s1*s2)+l2*c1-a);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eq1
G,**(%#l3G\"\"\"%#c1GF(%#c2GF(F(*(F'F(%#s1GF(%#s2GF(!\"\"*&%#l2GF(F)F(
F(%\"aGF." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "eq2 := expand(
l3*(c1*s2+c2*s1)+l2*s1-b);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eq2G,
**(%#l3G\"\"\"%#c1GF(%#s2GF(F(*(F'F(%#c2GF(%#s1GF(F(*&%#l2GF(F-F(F(%\"
bG!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "eq3 := c1^2+s1^2
-1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eq3G,(*$)%#c1G\"\"#\"\"\"F**
$)%#s1GF)F*F*F*!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "eq4
 := c2^2+s2^2-1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eq4G,(*$)%#c2G
\"\"#\"\"\"F**$)%#s2GF)F*F*F*!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 21 "G:=[eq1,eq2,eq3,eq4];" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%\"GG7&,**(%#l3G\"\"\"%#c1GF)%#c2GF)F)*(F(F)%#s1GF)%#s2GF)!\"\"
*&%#l2GF)F*F)F)%\"aGF/,**(F(F)F*F)F.F)F)*(F(F)F+F)F-F)F)*&F1F)F-F)F)%
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{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "desys := PolynomialTools[Pol
ynomialToPDE](G,[c2,s2,c1,s1],[v]);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6
#>%&desysG7&,**&%#l3G\"\"\"-%%diffG6%-%\"vG6&%#c2G%#s2G%#c1G%#s1GF2F0F
)F)*&F(F)-F+6%F-F3F1F)!\"\"*&%#l2GF)-F+6$F-F2F)F)*&%\"aGF)F-F)F7,**&F(
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F7,(-F+6$F--%\"$G6$F2\"\"#F)-F+6$F--FN6$F3FPF)F-F7,(-F+6$F--FN6$F0FPF)
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nfolevel[rifsimp]:=0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%*infolevel
G6#%(rifsimpG\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 126 "st:=
time(): \ntdegrsys:=DEtools[rifsimp](desys,ranking=[[1,0,0,0,0],[0,1,0
,0,0],[0,0,1,0,0],[0,0,0,1,0]],casesplit);\ntime()-st;" }}{PARA 12 "" 
1 "" {XPPMATH 20 "6#>%)tdegrsysG-%&TABLEG6#70/\"\"\"-F&6#7%/%'SolvedG7
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 69" "Curve 70" "Curve 71" "Curve 72" "Curve 73" "Curve 74" "Curve 75
" "Curve 76" "Curve 77" "Curve 78" }}}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "for i f
rom 1 to 13 do \nex||i:=map(z->expand(lhs(z)-rhs(z)),copy(tdegrsys[i])
[Solved]); \nod;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$ex1G7&,,-%%diff
G6$-%\"vG6&%#c2G%#s2G%#c1G%#s1GF-\"\"\"**#F1\"\"#F1%#l2GF1F*F1%#l3G!\"
\"F1**F3F1F*F1F6F1F5F7F1*&#F1F4F1**F*F1F6F7F5F7%\"aGF4F1F7*&#F1F4F1**F
*F1F6F7F5F7%\"bGF4F1F7,0-F(6$F*F.F1*.F3F1F<F7F5F7F6F1F@F1F*F1F1**F<F7F
6F7F@F4-F(6$F*F0F1F1*&#F1F4F1*,F<F7F5F1F6F7F@F1F*F1F1F7*&#F1F4F1*,F<F1
F5F7F6F7F@F1F*F1F1F7*&#F1F4F1*,F<F7F5F7F6F7F@\"\"$F*F1F1F7*(F<F1F6F7FF
F1F1,.-F(6$F*F/F1*,F3F1F<F7F5F7F6F4F*F1F1*(F<F7F@F1FFF1F1*&#F1F4F1*(F<
F7F5F1F*F1F1F7*&#F1F4F1*(F<F1F*F1F5F7F1F7*&#F1F4F1**F<F7F5F7F@F4F*F1F1
F7,@-F(6$F*-%\"$G6$F0F4F1*(,&*&)F<F4F1)F5F4F1\"\"%*(FfoF1)F@F4F1FeoF1F
1F7F<FfoF*F1F1*,F4F1FboF7F5F4F<F4F*F1F7*.FfoF1FboF7F5F1F@F1F6F4FFF1F1*
,F4F1FboF7F6F4F*F1F5F4F7*,FfoF1FboF7F@F1F5FQFFF1F7*.FfoF1FboF7F5F1F@F1
F<F4FFF1F7*,FfoF1FboF7F5F1F@FQFFF1F7*(FboF7F5FfoF*F1F1*,F4F1FboF7F6F4F
*F1F<F4F7*,F4F1FboF7F6F4F*F1F@F4F7*(FboF7F6FfoF*F1F1*(FboF7F@FfoF*F1F1
*,F4F1FboF7F5F4F*F1F@F4F1*,F4F1FboF7F<F4F*F1F@F4F1" }}{PARA 12 "" 1 "
" {XPPMATH 20 "6#>%$ex2G7&,(-%%diffG6$-%\"vG6&%#c2G%#s2G%#c1G%#s1GF-\"
\"\"**#F1\"\"#F1%#l2GF1F*F1%#l3G!\"\"F1**F3F1F*F1F6F1F5F7F1,(-F(6$F*F.
F1*&#F1F4F1*,%\"aGF7F5F1F6F7%\"bGF1F*F1F1F7*.F3F1F?F7F5F7F6F1F@F1F*F1F
1,,-F(6$F*F/F1*,F5F1,&*$)F5F4F1F7*$)F6F4F1F1F7F?F7F*F1F@F4F7*,#F1\"\"%
F1F5\"\"$FFF7F?F7F*F1F1*&#F1F4F1*,F5F1FFF7F?F7F6F4F*F1F1F7*.FLF1F5F7FF
F7F?F7F6FMF*F1F1,,-F(6$F*F0F1**F@F1F5F1FFF7F*F1F1*,FLF1F@F7F5FNFFF7F*F
1F1*&#F1F4F1*,F@F7F5F1FFF7F6F4F*F1F1F7*.FLF1F@F7F5F7FFF7F6FMF*F1F1" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ex3G7#-%\"vG6&%#c2G%#s2G%#c1G%#s1G
" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$ex4G7',*-%%diffG6$-%\"vG6&%#c2G
%#s2G%#c1G%#s1GF-\"\"\"**#F1\"\"#F1%#l2GF1F*F1%#l3G!\"\"F1**F3F1F*F1F6
F1F5F7F1*&#F1F4F1**F*F1F6F7F5F7%\"bGF4F1F7,&-F(6$F*F.F1*(F<F1-F(6$F*F/
F1F6F7F7,0-F(6$F*-%\"$G6$F/F4F1*,#F1\"\"%F1F*F1F<!\"#F5FLF6FKF1*&#F1F4
F1*(F*F1F5FLF6F4F1F7*&#F1F4F1*(F*F1F<FLF6F4F1F7**FJF1F*F1F<F4F5FLF1*&#
F1F4F1F*F1F7**FJF1F*F1F<FLF5F4F1,*-F(6$F*F0F1*,F3F1F<F7F5F7F6F4F*F1F1*
&#F1F4F1*(F<F1F*F1F5F7F1F7*&#F1F4F1*(F<F7F5F1F*F1F1F7%\"aG" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%$ex5G7%-%\"vG6&%#c2G%#s2G%#c1G%#s1G%\"aG%
\"bG" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$ex6G7',&-%%diffG6$-%\"vG6&%
#c2G%#s2G%#c1G%#s1GF-\"\"\"*(%#l2GF1F*F1%#l3G!\"\"F1-F(6$F*F.,(-F(6$F*
-%\"$G6$F/\"\"#F1-F(6$F*-F<6$F0F>F1F*F5%\"aG%\"bG" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%$ex7G7$-%\"vG6&%#c2G%#s2G%#c1G%#s1G%#l2G" }}{PARA 12 
"" 1 "" {XPPMATH 20 "6#>%$ex8G7&,(-%%diffG6$-%\"vG6&%#c2G%#s2G%#c1G%#s
1GF-\"\"\"*(%#l3G!\"\"%\"aGF1-F(6$F*F/F1F4*(F3F4%\"bGF1-F(6$F*F0F1F4,(
-F(6$F*F.F1*(F9F1F6F1F3F4F4*(F5F1F3F4F:F1F1,(-F(6$F*-%\"$G6$F/\"\"#F1-
F(6$F*-FE6$F0FGF1F*F4%#l2G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ex9G7
$-%\"vG6&%#c2G%#s2G%#c1G%#s1G%#l3G" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#
>%%ex10G7&,(-%%diffG6$-%\"vG6&%#c2G%#s2G%#c1G%#s1G-%\"$G6$F-\"\"#\"\"
\"-F(6$F*-F26$F.F4F5F*!\"\",&-F(6$F*F/F5*(%\"aGF5F*F5%#l2GF:F:,&-F(6$F
*F0F5*(%\"bGF5F*F5F@F:F:%#l3G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%ex
11G7%-%\"vG6&%#c2G%#s2G%#c1G%#s1G%#l2G%#l3G" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%%ex12G7&-%\"vG6&%#c2G%#s2G%#c1G%#s1G%\"aG%#l2G%#l3G" 
}}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%%ex13G7(,(-%%diffG6$-%\"vG6&%#c2G%
#s2G%#c1G%#s1G-%\"$G6$F-\"\"#\"\"\"-F(6$F*-F26$F.F4F5F*!\"\",(-F(6$F*-
F26$F/F4F5-F(6$F*-F26$F0F4F5F*F:%\"aG%\"bG%#l2G%#l3G" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 93 "copy(tdegrsys[1])[Case];\nmap(factor,Poly
nomialTools[PDEToPolynomial](ex1,[c2,s2,c1,s1],[v]));" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#7%7$0%#l3G\"\"!-%%diffG6%-%\"vG6&%#c2G%#s2G%#c1G%#
s1GF0F.7$0*&%\"aG\"\"\"%#l2GF6F'-F)6$F+F07$0,&*$)F5\"\"#F6F6*$)%\"bGF?
F6F6F'-F)6$F+-%\"$G6$F1F?" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7&,$*(,,*
(%#c2G\"\"\"%#l3GF)%#l2GF)!\"#*$)F+\"\"#F)!\"\"*$)F*F/F)F0*$)%\"aGF/F)
F)*$)%\"bGF/F)F)F)F*F0F+F0#F0F/,$**,0**%#s2GF)F5F)F+F)F*F)F/*&F2F)F8F)
F)**F/F)F7F)%#s1GF)F+F)F)*&F.F)F8F)F0*&F4F)F8F)F0*$)F8\"\"$F)F0**F/F)F
4F)FAF)F+F)F)F)F5F0F+F0F*F0#F)F/,$*(,.*(%#c1GF)F5F)F+F)F,F1F0**F/F)F8F
)FAF)F+F)F0F-F)F3F)F6F)F)F5F0F+F0F9,$*(,B*()FAF/F)F.F)F4F)\"\"%**FTF)F
SF)F.F)F7F)F)*$)F5FTF)F)*(F/F)F4F)F.F)F0*,FTF)F8F)F2F)FAF)F+F)F)*(F/F)
F2F)F.F)F0**FTF))F+FFF)F8F)FAF)F0*,FTF)F8F)F4F)FAF)F+F)F0**FTF)FEF)FAF
)F+F)F0*$)F+FTF)F)*(F/F)F2F)F4F)F0*(F/F)F2F)F7F)F0*$)F*FTF)F)*$)F8FTF)
F)*(F/F)F7F)F.F)F)*(F/F)F4F)F7F)F)F)F+F,,&F3F)F6F)F0#F)FT" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "for i from 1 to 13 do copy(tdegrsys
[i])[Case]; od;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%7$0%#l3G\"\"!-%%d
iffG6%-%\"vG6&%#c2G%#s2G%#c1G%#s1GF0F.7$0*&%\"aG\"\"\"%#l2GF6F'-F)6$F+
F07$0,&*$)F5\"\"#F6F6*$)%\"bGF?F6F6F'-F)6$F+-%\"$G6$F1F?" }}{PARA 12 "
" 1 "" {XPPMATH 20 "6#7&7$0%#l3G\"\"!-%%diffG6%-%\"vG6&%#c2G%#s2G%#c1G
%#s1GF0F.7$0*&%\"aG\"\"\"%#l2GF6F'-F)6$F+F07$/,&*$)F5\"\"#F6F6*$)%\"bG
F?F6F6F'-F)6$F+-%\"$G6$F1F?7$0*(FBF6,&F&F6F7!\"\"F6,&F&F6F7F6F6F'-F)6$
F+F1" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7(7$0%#l3G\"\"!-%%diffG6%-%\"v
G6&%#c2G%#s2G%#c1G%#s1GF0F.7$0*&%\"aG\"\"\"%#l2GF6F'-F)6$F+F07$/,&*$)F
5\"\"#F6F6*$)%\"bGF?F6F6F'-F)6$F+-%\"$G6$F1F?7$/*(FBF6,&F&F6F7!\"\"F6,
&F&F6F7F6F6F'-F)6$F+F17%/*&,&F7F6F&FLF6FMF6F'FBQ,false~split6\"7%0FBF'
F+FT" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&7$0%#l3G\"\"!-%%diffG6%-%\"v
G6&%#c2G%#s2G%#c1G%#s1GF0F.7$/*&%\"aG\"\"\"%#l2GF6F'-F)6$F+F07$0F7F'F5
7$0%\"bGF'-F)6$F+F1" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7'7$0%#l3G\"\"!
-%%diffG6%-%\"vG6&%#c2G%#s2G%#c1G%#s1GF0F.7$/*&%\"aG\"\"\"%#l2GF6F'-F)
6$F+F07$0F7F'F57$/%\"bGF'-F)6$F+F17$0*&,&F7F6F&!\"\"F6,&F&F6F7F6F6F'F+
" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7'7$0%#l3G\"\"!-%%diffG6%-%\"vG6&%
#c2G%#s2G%#c1G%#s1GF0F.7$/*&%\"aG\"\"\"%#l2GF6F'-F)6$F+F07$0F7F'F57$/%
\"bGF'-F)6$F+F17$/*&,&F7F6F&!\"\"F6,&F&F6F7F6F6F'F+" }}{PARA 12 "" 1 "
" {XPPMATH 20 "6#7&7$0%#l3G\"\"!-%%diffG6%-%\"vG6&%#c2G%#s2G%#c1G%#s1G
F0F.7$/*&%\"aG\"\"\"%#l2GF6F'-F)6$F+F07$/F7F'F57$0,(*$)F&\"\"#F6!\"\"*
$)%\"bGFAF6F6*$)F5FAF6F6F'F+" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7&7$0%
#l3G\"\"!-%%diffG6%-%\"vG6&%#c2G%#s2G%#c1G%#s1GF0F.7$/*&%\"aG\"\"\"%#l
2GF6F'-F)6$F+F07$/F7F'F57$/,(*$)F&\"\"#F6!\"\"*$)%\"bGFAF6F6*$)F5FAF6F
6F'F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%7$/%#l3G\"\"!-%%diffG6%-%\"
vG6&%#c2G%#s2G%#c1G%#s1GF0F.7$0%#l2GF'-F)6$F+F07$0,(*$)F4\"\"#\"\"\"F=
*$)%\"aGF<F=!\"\"*$)%\"bGF<F=FAF'F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#7%7$/%#l3G\"\"!-%%diffG6%-%\"vG6&%#c2G%#s2G%#c1G%#s1GF0F.7$0%#l2GF'-F
)6$F+F07$/,(*$)F4\"\"#\"\"\"F=*$)%\"aGF<F=!\"\"*$)%\"bGF<F=FAF'F+" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#7%7$/%#l3G\"\"!-%%diffG6%-%\"vG6&%#c2G
%#s2G%#c1G%#s1GF0F.7$/%#l2GF'-F)6$F+F07$0%\"aGF'F+" }}{PARA 12 "" 1 "
" {XPPMATH 20 "6#7&7$/%#l3G\"\"!-%%diffG6%-%\"vG6&%#c2G%#s2G%#c1G%#s1G
F0F.7$/%#l2GF'-F)6$F+F07$/%\"aGF'F+7$0%\"bGF'F+" }}{PARA 12 "" 1 "" 
{XPPMATH 20 "6#7&7$/%#l3G\"\"!-%%diffG6%-%\"vG6&%#c2G%#s2G%#c1G%#s1GF0
F.7$/%#l2GF'-F)6$F+F07$/%\"aGF'F+7$/%\"bGF'F+" }}}}{SECT 1 {PARA 3 "" 
0 "" {TEXT 263 43 "A system from the Maple 6 help file on FGLM" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "G:=[x^31-x^6-x-y,x^8-z,x^10-
t,x^2+y^2+z^2-t^2];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"GG7&,**$)%
\"xG\"#J\"\"\"F+*$)F)\"\"'F+!\"\"F)F/%\"yGF/,&*$)F)\"\")F+F+%\"zGF/,&*
$)F)\"#5F+F+%\"tGF/,**$)F)\"\"#F+F+*$)F0F>F+F+*$)F5F>F+F+*$)F:F>F+F/" 
}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 82 " The degree basis of G can be c
omputed with Groebner or (rifsimp+PolynomialTools) " }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 58 " The plex basis of G cannot be computed with Gr
oebner-FGLM" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 58 "We show how to com
pute the univariate polynomial in say x," }}{PARA 0 "" 0 "" {TEXT -1 
35 "which is a part of the plex baisis." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 71 "st:=time(): \ndegBasis := Groebner[gbasis](G,tdeg(t,z
,y,x)): \ntime()-st;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"%,H!\"$" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "st:=time():\nns, rv := Groe
bner[SetBasis](degBasis, tdeg(t,z,y,x)):\nns; nops(ns);\ntime()-st;" }
}{PARA 12 "" 1 "" {XPPMATH 20 "6#7jn\"\"\"%\"xG%\"yG%\"zG%\"tG*$)F%\"
\"#F$*&F&F$F%F$*$)F&F+F$*&F'F$F%F$*&F'F$F&F$*$)F'F+F$*&F(F$F%F$*&F(F$F
&F$*&F(F$F'F$*$)F%\"\"$F$*&F&F$F*F$*&F.F$F%F$*$)F&F8F$*(F'F$F&F$F%F$*&
F'F$F.F$*&F2F$F%F$*&F2F$F&F$*$)F'F8F$*&F(F$F*F$*(F(F$F&F$F%F$*&F(F$F.F
$*(F(F$F'F$F%F$*(F(F$F'F$F&F$*&F(F$F2F$*$)F%\"\"%F$*&F&F$F7F$*&F.F$F*F
$*&F<F$F%F$*$)F&FKF$*(F'F$F.F$F%F$*&F'F$F<F$*(F2F$F&F$F%F$*&F2F$F.F$*&
FBF$F%F$*&FBF$F&F$*$)F'FKF$*&F(F$F7F$*(F(F$F&F$F*F$*(F(F$F.F$F%F$*&F(F
$F<F$**F(F$F'F$F&F$F%F$*(F(F$F'F$F.F$*(F(F$F2F$F%F$*(F(F$F2F$F&F$*$)F%
\"\"&F$*&F&F$FJF$*&F.F$F7F$*&F<F$F*F$*&FPF$F%F$*$)F&F]oF$*(F'F$F<F$F%F
$*(FBF$F&F$F%F$*&F(F$FJF$*(F(F$F&F$F7F$*(F(F$F<F$F%F$*&F<F$F7F$*(F(F$F
&F$FJF$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#i" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#$\"#!*!\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
85 "st:=time():\nMt := Groebner[MulMatrix](x, ns, rv, degBasis, tdeg(t
,z,y,x));\ntime()-st;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#MtG-%'RTAB
LEG6+\"*wa*z8%)anythingG%'MatrixG%'sparseG%.Fortran_orderG7\"\"\"#;\"
\"\"\"#iF/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"%*z\"!\"$" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "st:=time():\nsort(factor(LinearAlge
bra[CharacteristicPolynomial](Mt,x)));\ntime()-st;" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#*&,2*$)%\"xG\"#g\"\"\"F)*&\"\"#F))F'\"#NF)!\"\"*&F+F)
)F'\"#IF)F.*$)F'\"#=F)F.*$)F'\"#9F)F)*$)F'\"#5F)F)*&F+F))F'\"\"&F)F)F+
F)F))F'F+F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"%^\\!\"$" }}}}{SECT 
1 {PARA 3 "" 0 "" {TEXT -1 1 " " }{TEXT 258 70 "A System from a CASC'2
000 paper: compute the whole plex Groebner basis" }}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 46 "G:=[x^8*y^6+6*x^4*y^6+y^3+6,4*x^4+x^2*y^9-10];
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"GG7$,**&)%\"xG\"\")\"\"\")%\"y
G\"\"'F+F+*(F.F+)F)\"\"%F+F,F+F+*$)F-\"\"$F+F+F.F+,(*$F0F+F1*&)F)\"\"#
F+)F-\"\"*F+F+\"#5!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 64 "the pl
ex Grobner basis computation for G does not work in Maple." }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 78 "we show how to compute the plex Grobner b
asis with rifsimp and PolynomialTools" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 55 "desys := PolynomialTools[PolynomialToPDE](G,[x,y],[u]
);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%&desysG7$,*-%%diffG6%-%\"uG6$%
\"xG%\"yG-%\"$G6$F-\"\")-F06$F.\"\"'\"\"\"*&F5F6-F(6%F*-F06$F-\"\"%F3F
6F6-F(6$F*-F06$F.\"\"$F6*&F5F6F*F6F6,(-F(6$F*F:F<-F(6%F*-F06$F-\"\"#-F
06$F.\"\"*F6*&\"#5F6F*F6!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 23 "infolevel[rifsimp]:= 0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%*i
nfolevelG6#%(rifsimpG\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 62 "  t
he following ranking corresponds to a plex order with x > y" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "st:=time(): \nPLEXsys:= DEto
ols[rifsimp](desys, ranking=[[1,0,0]]); time()-st; " }}{PARA 12 "" 1 "
" {XPPMATH 20 "6#>%(PLEXsysG-%&TABLEG6#7#/%'SolvedG7$/-%%diffG6$-%\"uG
6$%\"xG%\"yG-%\"$G6$F3\"\"#,@-F.6$F0-F66$F4\"#U#!/j4Eh/Yd\"2CMIr:kB)>*
&#\"0B7q@'eZ<\"/%)zk;g!H\"\"\"\"F0FF!\"\"*&#\"2^)R%pLL5]\"\"03ex*>s[:F
F-F.6$F0-F66$F4\"\"$FFFG*&#\"10*o*H!*oq)*FKFF-F.6$F0-F66$F4\"\"'FFFG*&
#\"-:Mh$z#=\"-k)QNLf\"FF-F.6$F0-F66$F4\"#:FFFF*&#\")+pA>\",P&Q]()HFF-F
.6$F0-F66$F4\"#7FFFF*&#\"3\"45Z<A&pdT\"1kk?)fx*Q7FF-F.6$F0-F66$F4\"\"*
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