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{SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "# This is the way to
 work with Grobner Basis. First - ordering. For this we need those two
 lines." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "with(Ore_algebra
);\nwith(Groebner);A:=poly_algebra(x,y,z,t);" }}{PARA 12 "" 1 "" 
{XPPMATH 20 "6#79%-OA_InternalsG%-Ore_to_DESolG%-Ore_to_RESolG%,Ore_to
_diffG%-Ore_to_shiftG%-annihilatorsG%)applyoprG%-diff_algebraG%-dual_a
lgebraG%0dual_polynomialG%-poly_algebraG%/qshift_algebraG%/rand_skew_p
olyG%0reverse_algebraG%3reverse_polynomialG%.shift_algebraG%-skew_alge
braG%*skew_elimG%+skew_gcdexG%*skew_pdivG%+skew_powerG%*skew_premG%-sk
ew_productG" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#78%*MulMatrixG%)SetBasi
sG%0ToricIdealBasisG%*fglm_algoG%'gbasisG%'gsolveG%+hilbertdimG%,hilbe
rtpolyG%.hilbertseriesG%-inter_reduceG%*is_finiteG%,is_solvableG%*lead
coeffG%(leadmonG%)leadtermG%(normalfG%/pretend_gbasisG%'reduceG%&spoly
G%*termorderG%*testorderG%)univpolyG" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#>%\"AG%,Ore_algebraG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "f
:=x^3*y*z^2+x^4-2*x^2*y^2*z^2-3;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%
\"fG,**()%\"xG\"\"$\"\"\"%\"yGF*)%\"zG\"\"#F*F**$)F(\"\"%F*F***F.F*)F(
F.F*)F+F.F*F,F*!\"\"F)F5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 98 
"# Now we see differnt leading terms depending of orderin. Note that l
eadmonom gives both lc and lm" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 108 "leadterm(f,plex(x,y,z));leadterm(f,tdeg(x,y,z));leadterm(f,plex
(y,x,z));l:=leadmon(f,plex(y,x,z));l[1];l[2];" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#*$)%\"xG\"\"%\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
*()%\"xG\"\"$\"\"\"%\"yGF')%\"zG\"\"#F'" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#*()%\"xG\"\"#\"\"\")%\"yGF&F')%\"zGF&F'" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"lG6$!\"#*()%\"xG\"\"#\"\"\")%\"yGF*F+)%\"zGF*F+" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#*()%\"xG\"\"#\"\"\")%\"yGF&F')%\"zGF&F'" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 73 "#Now let us calculate a small GB. Unfortunately the
 result is not sorted." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "S
:=\{f, 3*x*y-2\}; G1:=gbasis(S,plex(x,y,z));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"SG<$,**()%\"xG\"\"$\"\"\"%\"yGF+)%\"zG\"\"#F+F+*$)F
)\"\"%F+F+**F/F+)F)F/F+)F,F/F+F-F+!\"\"F*F6,&*(F*F+F)F+F,F+F+F/F6" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#G1G7$,*\"#;!\"\"*(\"#s\"\"\")%\"yG
\"\"%F+)%\"zG\"\"#F+F+*&\"$V#F+F,F+F+*(\"#CF+)F-F1F+F/F+F(,**&\"\")F+%
\"xGF+F+*(F5F+)F-\"\"$F+F/F+F(*&\"#\")F+F<F+F(*(F9F+F-F+F/F+F+" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "#We can create a correspondi
ng rewriting rules" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "rule:
= proc (f,T) printf (\"%+18a -> %a\",leadterm (f,T),leadterm(f,T)-f/le
adcoeff(f,T)) end:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "ruleG
B:=proc (L,T) local f;for f in L do rule(f,T);print(); od end;" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'ruleGBGj+6$%\"LG%\"TG6#%\"fG6\"F+?&
8$9$%%trueGC$-%%ruleG6$F-9%-%&printGF+F+F+F+6$\"\"!F8" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "ruleGB(G1,plex(x,y,z));" }}{PARA 6 
"" 1 "" {TEXT -1 46 "           y^4*z^2 -> 2/9-27/8*y^4+1/3*y^2*z^2" }
}{PARA 6 "" 1 "" {TEXT -1 46 "                 x -> 3*y^3*z^2+81/8*y^3
-y*z^2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "#But RHS is still
 not sorted. So we write a short procedure to print sorted GB. " }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 294 "printsorteds:=proc(f,T)loca
l l;if f=0 then printf(\";\");print(); elif type(f,rational)then print
f(\"%a\",f);printf(\";\");print(); else l:=leadmon(f,T): if abs(l[1]) \+
<> 1 then printf(\"%+a%a\",l[1],l[2]) elif l[1]=1 then printf(\"+%a\",
l[2]) else printf(\"-%a\",l[2])fi;printsorteds(f-l[1]*l[2],T) fi;end:
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 232 "printsorted:=proc(f,T) local l
; if f=0 then print(0);return; else l:=leadmon(f,T); if l[1]<0 then pr
intsorteds(f,T); return; else if l[1]=1 then printf(%a,l[2]) else prin
tf(\"%a%a\",l[1],l[2]) fi;fi;printsorteds(f-l[1]*l[2],T) fi;end:" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 56 "f;printsorted(f,plex(x,z,y));printsorted(f,plex(y,z,x
));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,**()%\"xG\"\"$\"\"\"%\"yGF()%
\"zG\"\"#F(F(*$)F&\"\"%F(F(**F,F()F&F,F()F)F,F(F*F(!\"\"F'F3" }}{PARA 
6 "" 1 "" {TEXT -1 29 "x^4+x^3*y*z^2-2x^2*y^2*z^2-3;" }}{PARA 6 "" 1 "
" {TEXT -1 30 "-2x^2*y^2*z^2+x^3*y*z^2+x^4-3;" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 36 "#Now we are ready to print sorted GB" }}{PARA 0 
"> " 0 "" {MPLTEXT 1 0 69 "printGB:=proc(GB,T) local f; for f in GB do
 printsorted(f,T); od end:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
25 "printGB (G1,plex(x,z,y));" }}{PARA 6 "" 1 "" {TEXT -1 30 "72y^4*z^
2-24y^2*z^2+243y^4-16;" }}{PARA 6 "" 1 "" {TEXT -1 26 "8x-24y^3*z^2+8y
*z^2-81y^3;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "#now we chec
k normal forms: we see that f is in the ideal, x^5 is not. compare red
uce and normalf" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "normalf(f, G1,pl
ex(x,y,z));normalf (x^5, G1,plex(x,y,z));reduce (x^5,G1,plex(x,y,z));
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 
1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,0*&#\"\"%\"\"
$\"\"\"*&)%\"yGF'F()%\"zG\"\"'F(F(F(*&#\"#VF.F(*&F*F()F-F&F(F(F(*&#F&
\"\"*F(*&F,F(F+F(F(!\"\"*(\"#=F(F*F()F-\"\"#F(F(*&#\"#;F6F(*&F3F(F+F(F
(F8*&#\"$V#\"\")F(*$F*F(F(F(*(F.F(F+F(F;F(F8" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#,0*(\"#'*\"\"\")%\"yG\"\"$F&)%\"zG\"\"'F&F&*(\"$;&F&F'F
&)F+\"\"%F&F&*(\"#KF&F*F&F(F&!\"\"*(\"%'H\"F&F'F&)F+\"\"#F&F&*(\"$G\"F
&F/F&F(F&F3*&\"%(=#F&F'F&F&*(\"$K%F&F(F&F6F&F3" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 68 "#Let us check intersection of two ideals: Elem
ents are GB themselves" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "I
1:=\{x^2-y^2\}; I2:=\{x^3-y^3\};" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%
#I1G<#,&*$)%\"xG\"\"#\"\"\"F+*$)%\"yGF*F+!\"\"" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#I2G<#,&*$)%\"xG\"\"$\"\"\"F+*$)%\"yGF*F+!\"\"" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "IJ:=\{expand(t*I1[1]),expand
((t-1)*I2[1])\};" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#IJG<$,**&%\"tG
\"\"\")%\"xG\"\"$F)F)*&F(F))%\"yGF,F)!\"\"*$F*F)F0*$F.F)F),&*&F(F))F+
\"\"#F)F)*&F(F))F/F6F)F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 
"G2:=gbasis(IJ, plex(t,x,y,z));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#
G2G7%,**$)%\"xG\"\"%\"\"\"F+*&F)F+)%\"yG\"\"$F+!\"\"*&)F)F/F+F.F+F+*$)
F.F*F+F0,**&%\"tGF+F-F+F0*$F2F+F0*$F-F+F+*(F)F+F7F+)F.\"\"#F+F+,&*&F7F
+)F)F<F+F+*&F7F+F;F+F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 100 "
#We see that the first element generates GB for the intersection. Chec
k that it lies in both ideals." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 60 "g:=G2[1]; reduce(g,I1,plex(x,y,z));reduce(g,I2,plex(x,y,z));" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gG,**$)%\"xG\"\"%\"\"\"F**&F(F*)
%\"yG\"\"$F*!\"\"*&)F(F.F*F-F*F**$)F-F)F*F/" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "#now check  if we get  a fin
te set of solutions. Note that the ordering is not used." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "is_finite (G1);is_finite(\{x^2-y^2,
x*y\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%&falseG" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#%%trueG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 
"#Analyzing the parametrization" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 54 "I3:=\{x-t^2,y-t^3,z-t^5\};G3:=gbasis(I3, plex(t,x,y,z));" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#I3G<%,&%\"xG\"\"\"*$)%\"tG\"\"#F(!
\"\",&%\"yGF(*$)F+\"\"$F(F-,&%\"zGF(*$)F+\"\"&F(F-" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%#G3G7+,&*$)%\"zG\"\"$\"\"\"!\"\"*$)%\"yG\"\"&F+F+,&
*&%\"xGF+)F)\"\"#F+F+*$)F/\"\"%F+F,,&F)F,*&F3F+F/F+F+,&*$)F/F*F+F,*&)F
3F5F+F)F+F+,&*$)F/F5F+F,*$)F3F*F+F+,&*&%\"tGF+F)F+F+FAF,,&*$F?F+F,*&FG
F+F/F+F+,&*&FGF+F3F+F+F/F,,&F3F,*$)FGF5F+F+" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 65 "#we see that z can be eliminated, so let us try an
other ordering:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "G4:=gbas
is(I3, plex(t,z,y,x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#G4G7',&*$
)%\"yG\"\"#\"\"\"F+*$)%\"xG\"\"$F+!\"\",&%\"zGF+*&F.F+F)F+F0,&*&%\"tGF
+F.F+F+F)F0,&*$)F.F*F+F0*&F6F+F)F+F+,&F.F0*$)F6F*F+F+" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "#and the result looks much better n
ow: the first two equations is what we need." }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 91 "#At last some computations with the radical. Let
 us chech if x, x+y belongs to radical of J" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 26 "J:=\{x^3+x^2*y,y^3+y^2*x\}; " }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"JG<$,&*$)%\"xG\"\"$\"\"\"F+*&)F)\"\"#F+%\"yGF+F+,&*
$)F/F*F+F+*&)F/F.F+F)F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
105 "J1:= J union\{x*t-1\};G5:=gbasis (J1, tdeg(x,y,z,t));J2:=J union
\{x*t+y*t-1\};G6:=gbasis (J2, tdeg(x,y,z,t));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#J1G<%,&*&%\"tG\"\"\"%\"xGF)F)F)!\"\",&*$)F*\"\"$F)F)
*&)F*\"\"#F)%\"yGF)F),&*$)F3F/F)F)*&)F3F2F)F*F)F)" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#G5G7$,&%\"xG\"\"\"%\"yGF(,&F(F(*&%\"tGF(F)F(F(" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#J2G<%,(*&%\"tG\"\"\"%\"xGF)F)*&F(F)
%\"yGF)F)F)!\"\",&*$)F*\"\"$F)F)*&)F*\"\"#F)F,F)F),&*$)F,F1F)F)*&)F,F4
F)F*F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#G6G7#\"\"\"" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "# We see that x+y is in the radical
, but x is not." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}
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