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{SECT 0 {EXCHG {PARA 208 "" 0 "" {TEXT 209 1 " " }}}{EXCHG {PARA 0 "J
C > " 0 "" {MPLTEXT 1 0 24 "restart;with(Groebner):
" }{MPLTEXT 1 0 
77 "with(DifferentialGeometry): 
with(JetCalculus):
with(Tools):
with(
ListTools):" }{MPLTEXT 1 0 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 215 
67 "Cross-section algorithm  for generating sets of rational invariant
s" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "rgi:=proc(act,var,G:=[]
,K:=[])\n" }{MPLTEXT 1 0 21 "description \"Input: \n" }{MPLTEXT 1 0 
54 "   act= the list of polynomials defining the action; \n" }{MPLTEXT
 1 0 205 "   var=[[group-par],[target],[source]] are indeterminants (i
f the action is rational, clear the denominators and introduce an equa
tion with a dummy variable to ensure that denominators     are non-zer
o), \n" }{MPLTEXT 1 0 45 "   G= the ideal of the group (can be empty).
\n" }{MPLTEXT 1 0 53 "   K= the ideal of the cross-section (can be emp
ty).\n" }{MPLTEXT 1 0 138 "  Output: The reduced Crobner basis got the
 graph-section ideal and a(non-minimal)set of              generators \+
of rational invariants.\";" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 40 "local
 all,gb_all,idealI,gb_I,Ie,R,gs,i;\n" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 
0 28 "all:=[op(act),op(G),op(K)]:\n" }{MPLTEXT 1 0 37 "gb_all:=Basis(a
ll, lexdeg(op(var)));\n" }{MPLTEXT 1 0 4 "   (" }{MPLTEXT 1 0 1 "*" }
{MPLTEXT 1 0 40 "Elimination ideal and its reduced GB:*)\n" }{MPLTEXT 
1 0 6 "idealI" }{MPLTEXT 1 0 52 ":=map(collect, remove(has, gb_all,var
[1]), var[2]);\n" }{MPLTEXT 1 0 39 "gb_I:=Basis(idealI, tdeg(op(var[2]
)));\n" }{MPLTEXT 1 0 3 "   " }{MPLTEXT 1 0 29 "(*print(evaln(gb_I)=gb
_I);*)\n" }{MPLTEXT 1 0 1 "I" }{MPLTEXT 1 0 80 "e:=map(p-> collect( p/
 LeadingCoefficient(p, tdeg(op(var[2]))), var[2]), gb_I);\n" }{MPLTEXT
 1 0 3 "   " }{MPLTEXT 1 0 40 "(*Extracting the set of coefficients of
 " }{MPLTEXT 1 0 14 "reduced GB of " }{MPLTEXT 1 0 5 "Ie:*)" }{MPLTEXT
 1 0 0 "" }{MPLTEXT 1 0 66 "R:= map(simplify,MakeUnique(map(coeffs,map
(expand,Ie), var[2])));\n" }{MPLTEXT 1 0 3 "   " }{MPLTEXT 1 0 35 "(*R
emoving constant coefficients:*)" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 8 "
gs:=\{\};\n" }{MPLTEXT 1 0 27 "for i from 1 to nops(R) do\n" }{MPLTEXT
 1 0 26 " if indets(R[i])<>\{\} then\n" }{MPLTEXT 1 0 21 "  gs:=gs uni
on\{R[i]\}\n" }{MPLTEXT 1 0 5 " fi;\n" }{MPLTEXT 1 0 10 "od;return(" }
{MPLTEXT 1 0 1 "I" }{MPLTEXT 1 0 7 "e,gs);\n" }{MPLTEXT 1 0 9 "end pro
c:" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT 215 7 "scaling" }}{EXCHG {PARA 0
 "> " 0 "" {MPLTEXT 1 0 28 "G:=[a*b-1];A:=[X-a*x,Y-a*y];" }}{PARA 11 "
" 1 "" {XPPMATH 20 ">I\"GG6\"7#,&*&I\"aGF$\"\"\"I\"bGF$F)F)!\"\"F)" }}
{PARA 11 "" 1 "" {XPPMATH 20 ">I\"AG6\"7$,&*&I\"aGF$\"\"\"I\"xGF$F)!\"
\"I\"XGF$F),&*&F(F)I\"yGF$F)F+I\"YGF$F)" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 27 "K1:=[X-1]; K2:=[X^2+Y^2-1];" }}{PARA 11 "" 1 "" 
{XPPMATH 20 ">I#K1G6\"7#,&I\"XGF$\"\"\"!\"\"F(" }}{PARA 11 "" 1 "" 
{XPPMATH 20 ">I#K2G6\"7#,(*$I\"XGF$\"\"#\"\"\"*$I\"YGF$F)F*!\"\"F*" }}
}{EXCHG {PARA 0 "" 0 "" {TEXT 209 23 "without a cross-section" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "undebug(rgi); (*change to " 
}{MPLTEXT 1 0 29 "debug(rgi) to see all steps*)" }{MPLTEXT 1 0 30 " rg
i(A,[[a,b],[X,Y],[x,y]],G);" }}{PARA 11 "" 1 "" {XPPMATH 20 "I$rgiG6\"
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$7#,&I\"XG6\"\"\"\"*(I\"yGF&!\"\"I
\"YGF&F'I\"xGF&F'F*<#,$*&F)F*F,F'F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
209 25 "cross-section of degree 1" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 45 "undebug(rgi);rgi(A,[[a,b],[X,Y],[x,y]],G,K1);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "I$rgiG6\"" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6$7$,&I\"YG6\"\"\"\"*&I\"yGF&F'I\"xGF&!\"\"F+,&I\"XGF&F'F'F+<#,$F(
F+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 209 25 "cross-section of degree 2"
 }}{PARA 0 "" 0 "" {TEXT 209 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT
 1 0 51 "undebug(rgi);Ie,R:=rgi(A,[[a,b],[X,Y],[x,y]],G,K2);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "I$rgiG6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 ">6$
I#IeG6\"I\"RGF%6$7$,&I\"XGF%\"\"\"*(I\"yGF%!\"\"I\"YGF%F+I\"xGF%F+F.,&
*$)F/\"\"#F+F+*&,&*$)F0F4F+F+*$)F-F4F+F+F.F:F+F.<$,$*&F-F.F0F+F.,$F5F.
" }}}{EXCHG {PARA 0 "JC > " 0 "" {MPLTEXT 1 0 16 "solve(Ie,[X,Y]);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "7#7$/I\"XG6\"*&-I'RootOfG6$%*protectedGI
(_syslibG6\"6#,&*&,&*$)I\"xG6\"\"\"#\"\"\"\"\"\"*$)I\"yG6\"\"\"#\"\"\"
\"\"\"\"\"\")I#_ZG6$%*protectedGI(_syslibG6\"\"\"#\"\"\"\"\"\"\"\"\"!
\"\"\"\"\"I\"xG6\"\"\"\"/I\"YG6\"*&-I'RootOfG6$%*protectedGI(_syslibG6
\"6#,&*&,&*$)I\"xG6\"\"\"#\"\"\"\"\"\"*$)I\"yG6\"\"\"#\"\"\"\"\"\"\"\"
\")I#_ZG6$%*protectedGI(_syslibG6\"\"\"#\"\"\"\"\"\"\"\"\"!\"\"\"\"\"I
\"yG6\"\"\"\"" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT 215 18 "\"positive\" \+
scaling" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "G:=[a*b-1];A:=[X-
a^2*x,Y-a^2*y];" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I\"GG6\"7#,&*&I\"aGF
$\"\"\"I\"bGF$F)F)F)!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I\"AG6\"7$
,&*&)I\"aGF$\"\"#\"\"\"I\"xGF$F+!\"\"I\"XGF$F+,&*&F(F+I\"yGF$F+F-I\"YG
F$F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "K1:=[X-1]; K2:=[X^2
+Y^2-1];" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I#K1G6\"7#,&I\"XGF$\"\"\"F(
!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I#K2G6\"7#,(*$)I\"XGF$\"\"#\"
\"\"F+*$)I\"YGF$F*F+F+F+!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 209 23 
"without a cross-section" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 
"undebug(rgi);rgi(A,[[a,b],[X,Y],[x,y]],G);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "I$rgiG6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$7#,&I\"XG6\"
\"\"\"*(I\"yGF&!\"\"I\"YGF&F'I\"xGF&F'F*<#,$*&F)F*F,F'F*" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT 209 25 "cross-section of degree 1" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "undebug(rgi);rgi(A,[[a,b],[X,Y],[x,
y]],G,K1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "I$rgiG6\"" }}{PARA 11 "" 1
 "" {XPPMATH 20 "6$7$,&I\"YG6\"\"\"\"*&I\"yGF&F'I\"xGF&!\"\"F+,&I\"XGF
&F'F'F+<#,$F(F+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 209 25 "cross-section
 of degree 2" }}{PARA 0 "" 0 "" {TEXT 209 0 "" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 51 "undebug(rgi);Ie,R:=rgi(A,[[a,b],[X,Y],[x,y]],G,K
2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "I$rgiG6\"" }}{PARA 11 "" 1 "" 
{XPPMATH 20 ">6$I#IeG6\"I\"RGF%6$7$,&I\"XGF%\"\"\"*(I\"yGF%!\"\"I\"YGF
%F+I\"xGF%F+F.,&*$)F/\"\"#F+F+*&,&*$)F0F4F+F+*$)F-F4F+F+F.F:F+F.<$,$*&
F-F.F0F+F.,$F5F." }}}{EXCHG {PARA 0 "JC > " 0 "" {MPLTEXT 1 0 16 "solv
e(Ie,[X,Y]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "7#7$/I\"XG6\"*&-I'RootOf
G6$%*protectedGI(_syslibG6\"6#,&*&,&*$)I\"xG6\"\"\"#\"\"\"\"\"\"*$)I\"
yG6\"\"\"#\"\"\"\"\"\"\"\"\")I#_ZG6$%*protectedGI(_syslibG6\"\"\"#\"\"
\"\"\"\"\"\"\"!\"\"\"\"\"I\"xG6\"\"\"\"/I\"YG6\"*&-I'RootOfG6$%*protec
tedGI(_syslibG6\"6#,&*&,&*$)I\"xG6\"\"\"#\"\"\"\"\"\"*$)I\"yG6\"\"\"#
\"\"\"\"\"\"\"\"\")I#_ZG6$%*protectedGI(_syslibG6\"\"\"#\"\"\"\"\"\"\"
\"\"!\"\"\"\"\"I\"yG6\"\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 0 "" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT 215 8 "rotation" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "G:=[c^2+s^2-1];A:=[X-c*x+s*y,Y-s*x-
c*y];" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I\"GG6\"7#,(*$)I\"cGF$\"\"#\"
\"\"F+*$)I\"sGF$F*F+F+F+!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I\"AG6
\"7$,(*&I\"cGF$\"\"\"I\"xGF$F)!\"\"*&I\"sGF$F)I\"yGF$F)F)I\"XGF$F),(*&
F(F)F.F)F+*&F-F)F*F)F+I\"YGF$F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 38 "K1:=[X]; K2:=[X-Y^2]; nK:=[X^2+Y^2-1];" }}{PARA 11 "" 1 "" 
{XPPMATH 20 ">I#K1G6\"7#I\"XGF$" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I#K2
G6\"7#,&*$)I\"YGF$\"\"#\"\"\"!\"\"I\"XGF$F+" }}{PARA 11 "" 1 "" 
{XPPMATH 20 ">I#nKG6\"7#,(*$)I\"XGF$\"\"#\"\"\"F+*$)I\"YGF$F*F+F+F+!\"
\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 209 23 "without a cross-section" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "undebug(rgi);rgi(A,[[c,s],
[X,Y],[x,y]],G);" }}{PARA 11 "" 1 "" {XPPMATH 20 "I$rgiG6\"" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6$7#,**$)I\"XG6\"\"\"#\"\"\"F**$)I\"YGF(F)F*F*
*$)I\"xGF(F)F*!\"\"*$)I\"yGF(F)F*F1<#,&F.F1F2F1" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT 209 25 "cross-section of degree 2" }}{PARA 0 "" 0 "" {TEXT 
209 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "undebug(rgi);\n"
 }{MPLTEXT 1 0 40 "Ie,R:=rgi(A,[[c,s],[X,Y],[x,y]],G,K1); \n" }
{MPLTEXT 1 0 16 "solve(Ie,[X,Y]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "I$r
giG6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 ">6$I#IeG6\"I\"RGF%6$7$I\"XGF%,
(*$)I\"YGF%\"\"#\"\"\"F/*$)I\"xGF%F.F/!\"\"*$)I\"yGF%F.F/F3<#,&F0F3F4F
3" }}{PARA 11 "" 1 "" {XPPMATH 20 "7#7$/I\"XG6\"\"\"!/I\"YG6\"-I'RootO
fG6$%*protectedGI(_syslibG6\"6#,(*$)I#_ZG6$%*protectedGI(_syslibG6\"\"
\"#\"\"\"\"\"\"*$)I\"xG6\"\"\"#\"\"\"!\"\"*$)I\"yG6\"\"\"#\"\"\"!\"\""
 }}}{EXCHG {PARA 0 "" 0 "" {TEXT 209 25 "cross-section of degree 4" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "undebug(rgi);\n" }{MPLTEXT 
1 0 39 "Ie,R:=rgi(A,[[c,s],[X,Y],[x,y]],G,K2);\n" }{MPLTEXT 1 0 16 "so
lve(Ie,[X,Y]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "I$rgiG6\"" }}{PARA 11 
"" 1 "" {XPPMATH 20 ">6$I#IeG6\"I\"RGF%6$7$,&*$)I\"YGF%\"\"#\"\"\"F.I
\"XGF%!\"\",**$)F/F-F.F.*$)I\"xGF%F-F.F0*$)I\"yGF%F-F.F0F/F.<#,&F4F0F7
F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "7#7$/I\"XG6\"-I'RootOfG6$%*protect
edGI(_syslibG6\"6#,**$)I#_ZG6$%*protectedGI(_syslibG6\"\"\"#\"\"\"\"\"
\"*$)I\"xG6\"\"\"#\"\"\"!\"\"*$)I\"yG6\"\"\"#\"\"\"!\"\"I#_ZG6$%*prote
ctedGI(_syslibG6\"\"\"\"/I\"YG6\"-I'RootOfG6$%*protectedGI(_syslibG6\"
6#,&*$)I#_ZG6$%*protectedGI(_syslibG6\"\"\"#\"\"\"\"\"\"-I'RootOfG6$%*
protectedGI(_syslibG6\"6#,**$)I#_ZG6$%*protectedGI(_syslibG6\"\"\"#\"
\"\"\"\"\"*$)I\"xG6\"\"\"#\"\"\"!\"\"*$)I\"yG6\"\"\"#\"\"\"!\"\"I#_ZG6
$%*protectedGI(_syslibG6\"\"\"\"!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT
 209 25 "not a valid cross-section" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 45 "undebug(rgi);rgi(A,[[c,s],[X,Y],[x,y]],G,nK);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "I$rgiG6\"" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6$7#\"\"\"<\"" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT 215 30 "actions w
ith horizontal orbits" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "A1:
=[X-x-t,Y-y]; #translation\n" }{MPLTEXT 1 0 43 "A2:=[X-x-t*y,Y-y]; #tr
anslation with boost\n" }{MPLTEXT 1 0 60 "A3:=[X-x-t,Y-a*y]; G3:=[a^2-
1]; #translation and reflection\n" }{MPLTEXT 1 0 1 "\n" }}{PARA 11 "" 
1 "" {XPPMATH 20 ">I#A1G6\"7$,(I\"XGF$\"\"\"I\"xGF$!\"\"I\"tGF$F*,&I\"
YGF$F(I\"yGF$F*" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I#A2G6\"7$,(*&I\"tGF
$\"\"\"I\"yGF$F)!\"\"I\"XGF$F)I\"xGF$F+,&I\"YGF$F)F*F+" }}{PARA 11 "" 
1 "" {XPPMATH 20 ">I#A3G6\"7$,(I\"XGF$\"\"\"I\"xGF$!\"\"I\"tGF$F*,&*&I
\"aGF$F(I\"yGF$F(F*I\"YGF$F(" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I#G3G6
\"7#,&*$)I\"aGF$\"\"#\"\"\"F+F+!\"\"" }}}{EXCHG {PARA 0 "JC > " 0 "" 
{MPLTEXT 1 0 21 "K1:=[X]; K2:=[Y-X^2];" }}{PARA 11 "" 1 "" {XPPMATH 20
 ">I#K1G6\"7#I\"XGF$" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I#K2G6\"7#,&*$)
I\"XGF$\"\"#\"\"\"!\"\"I\"YGF$F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT
 1 0 42 "undebug(rgi);rgi(A1,[[t],[X,Y],[x,y]],K2);" }}{PARA 11 "" 1 "
" {XPPMATH 20 "I$rgiG6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$7$,&I\"YG6
\"\"\"\"I\"yGF&!\"\",&*$)I\"XGF&\"\"#F'F'F(F)<#,$F(F)" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "undebug(rgi);rgi(A2,[[t],[X,Y],[x,y
]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "I$rgiG6\"" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6$7#,&I\"YG6\"\"\"\"I\"yGF&!\"\"<#,$F(F)" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "undebug(rgi);rgi(A3,[[t,a],[X,Y],[x
,y]],G3,K2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "I$rgiG6\"" }}{PARA 11 ""
 1 "" {XPPMATH 20 "6$7$,&*$)I\"YG6\"\"\"#\"\"\"F**$)I\"yGF(F)F*!\"\",&
*$)I\"XGF(F)F*F*F'F.<#,$F+F." }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT 215 
40 "Differential invariants of planar curves" }}{EXCHG {PARA 0 "" 0 ""
 {TEXT 209 227 "This procedure prolongs the action of G from R^2 to th
e jet space of a given order. It outputs the equations of the prolonge
d action in the polynomial form and the list of variables,  in the for
mat suitable for the \"rgi\" input." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 31 "pr:=proc(action, order, G:=[])\n" }{MPLTEXT 1 0 75 "l
ocal k, lact, act,act_eq, pr_act,TJ,delta,jv, z,Z,i,eq,J,Q,A,Lambda, v
ar;\n" }{MPLTEXT 1 0 10 "k:=order;\n" }{MPLTEXT 1 0 6 "   (*R" }
{MPLTEXT 1 0 57 "ewrite the action in DiffGeom notation and prolong it
:*)\n" }{MPLTEXT 1 0 35 "lact:=subs(y=y[],map(rhs,action));\n" }
{MPLTEXT 1 0 33 "act_eq:=zip(`=`, [x,y[]], lact);\n" }{MPLTEXT 1 0 23 
"DGsetup([x],[y],JC,k);\n" }{MPLTEXT 1 0 35 "act:=Transformation(JC,JC
,act_eq);\n" }{MPLTEXT 1 0 24 "TJ:=TotalJacobian(act);\n" }{MPLTEXT 1 
0 39 "J := LinearAlgebra[MatrixInverse](TJ);\n" }{MPLTEXT 1 0 42 "pr_a
ct:= map(simplify,Prolong(act, k,J));\n" }{MPLTEXT 1 0 7 "    (* " }
{MPLTEXT 1 0 34 "print(pr_act,k); optional print*)\n" }{MPLTEXT 1 0 
39 "delta:=LinearAlgebra[Determinant](TJ);\n" }{MPLTEXT 1 0 37 "jv:=DG
info(JC, \"FrameJetVariables\");\n" }{MPLTEXT 1 0 5 "   (*" }{MPLTEXT 
1 0 114 "change the notation back and rewrite the equations and variab
les in the format appropriate for the \"rgi\" input:*)\n" }{MPLTEXT 1 
0 18 "z:=jet_not(jv,k); " }{MPLTEXT 1 0 25 "delta:=jet_not(delta,k);\n
" }{MPLTEXT 1 0 21 "Z:=subs(x=X,y=Y,z); \n" }{MPLTEXT 1 0 32 "eq:=map2
(Pullback, pr_act, jv);\n" }{MPLTEXT 1 0 20 "Q:=zip(`-`, Z, eq);\n" }
{MPLTEXT 1 0 66 "A:=map(collect,map(simplify,[op(map(numer,Q)),dummy*d
elta-1]),Z);\n" }{MPLTEXT 1 0 31 "A:=map(simplify,jet_not(A,k));\n" }
{MPLTEXT 1 0 70 "Lambda:=convert((indets(A)minus \{op(z),op(Z)\})union
 indets(G), list);\n" }{MPLTEXT 1 0 19 "var:=[Lambda,Z,z];\n" }
{MPLTEXT 1 0 15 "return(A,var);\n" }{MPLTEXT 1 0 9 "end proc:" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT 209 24 "Notation simplification:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "jet_not:=proc(object, order)
\n" }{MPLTEXT 1 0 11 "local k,i;\n" }{MPLTEXT 1 0 10 "k:=order;\n" }
{MPLTEXT 1 0 53 "subs ([y[]=y,seq(y[seq(1$i)]=y[i],i=1..k)],object); \+
\n" }{MPLTEXT 1 0 9 "end proc:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT 209 0 "" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT 216 41 "(Specia
l) Euclidean group - rigid motions" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 40 "E:=[X=c*x-s*y+a,Y=lambda*(s*x+c*y+b)]; \n" }{MPLTEXT 
1 0 15 "K:=[X,Y,Y[1]];\n" }{MPLTEXT 1 0 50 "G:=[s^2+c^2-1,lambda^2-1];
(*the Euclidean group*)\n" }{MPLTEXT 1 0 26 "SG:=[s^2+c^2-1,lambda-1];
 " }{MPLTEXT 1 0 6 "(*the " }{MPLTEXT 1 0 26 "special Euclidean group*
)\n" }{MPLTEXT 1 0 14 "undebug(pr); \n" }}{PARA 11 "" 1 "" {XPPMATH 20
 ">I\"EG6\"7$/I\"XGF$,(*&I\"cGF$\"\"\"I\"xG%*protectedGF+F+*&I\"sGF$F+
I\"yGF$F+!\"\"I\"aGF$F+/I\"YGF$*&I'lambdaGF$F+,(*&F*F+F0F+F+*&F/F+F,F+
F+I\"bGF$F+F+" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I\"KG6\"7%I\"XGF$I\"YG
F$&F'6#\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I\"GG6\"7$,(*$)I\"cGF$
\"\"#\"\"\"F+*$)I\"sGF$F*F+F+F+!\"\",&*$)I'lambdaGF$F*F+F+F+F/" }}
{PARA 11 "" 1 "" {XPPMATH 20 ">I#SGG6\"7$,(*$)I\"cGF$\"\"#\"\"\"F+*$)I
\"sGF$F*F+F+F+!\"\",&I'lambdaGF$F+F+F/" }}{PARA 11 "" 1 "" {XPPMATH 20
 "I#prG6\"" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT 217 7 "on J^2 " }}{EXCHG 
{PARA 0 "JC > " 0 "" {MPLTEXT 1 0 14 "T := time(): \n" }{MPLTEXT 1 0 
19 "A,var:=pr(E,2,G): \n" }{MPLTEXT 1 0 21 "RunTime := time()-T; " }
{MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I(RunTimeG6\"$\"#8!
\"$" }}}{SECT 0 {PARA 0 "" 0 "" {TEXT 209 4 "SE_2" }{TEXT 214 0 "" }}
{EXCHG {PARA 0 "" 0 "" {TEXT 209 6 "with K" }}}{EXCHG {PARA 0 "JC > " 
0 "" {MPLTEXT 1 0 57 "T := time(): Ie,R:=rgi(A,var,SG,K); RunTime := t
ime()-T; " }}{PARA 11 "" 1 "" {XPPMATH 20 ">6$I#IeG6\"I\"RGF%6$7&&I\"Y
GF%6#\"\"\"F*I\"XGF%,&*$)&F*6#\"\"#F3F,F,*&,**$)&I\"yGF%F+\"\"'F,F,*&
\"\"$F,)F8\"\"%F,F,*&F<F,)F8F3F,F,F,F,!\"\")&F9F2F3F,FA<#,$*&FBF,),&*$
F@F,F,F,F,F<FAFA" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I(RunTimeG6\"$\"\"*
!\"$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 209 5 "w/o K" }}}{EXCHG {PARA 0 
"JC > " 0 "" {MPLTEXT 1 0 48 "T := time(): rgi(A,var,SG); RunTime := t
ime()-T;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$7#,,*$)&I\"YG6\"6#\"\"\"\"
\"'F+F+*&\"\"$F+)F'\"\"%F+F+*&F.F+)F'\"\"#F+F+F+F+*(,**$)&I\"yGF)F*F,F
+!\"\"*&F.F+)F8F0F+F:*&F.F+)F8F3F+F:F+F:F+)&F96#F3F3F:)&F(FAF3F+F+<#,$
*&),&*$F>F+F+F+F+F.F+F?F:F:" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I(RunTim
eG6\"$\"%s;!\"$" }}}{EXCHG {PARA 0 "JC > " 0 "" {MPLTEXT 1 0 0 "" }}}}
{SECT 0 {PARA 0 "" 0 "" {TEXT 209 3 "E_2" }}{EXCHG {PARA 0 "" 0 "" 
{TEXT 209 6 "with K" }}}{EXCHG {PARA 0 "JC > " 0 "" {MPLTEXT 1 0 56 "T
 := time(): Ie,R:=rgi(A,var,G,K); RunTime := time()-T; " }}{PARA 11 ""
 1 "" {XPPMATH 20 ">6$I#IeG6\"I\"RGF%6$7&&I\"YGF%6#\"\"\"F*I\"XGF%,&*$
)&F*6#\"\"#F3F,F,*&,**$)&I\"yGF%F+\"\"'F,F,*&\"\"$F,)F8\"\"%F,F,*&F<F,
)F8F3F,F,F,F,!\"\")&F9F2F3F,FA<#,$*&FBF,),&*$F@F,F,F,F,F<FAFA" }}
{PARA 11 "" 1 "" {XPPMATH 20 ">I(RunTimeG6\"$\"&pu%!\"$" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT 209 5 "w/o K" }}}{EXCHG {PARA 0 "JC > " 0 "" 
{MPLTEXT 1 0 47 "T := time(): rgi(A,var,G); RunTime := time()-T;" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6$7#,,*$)&I\"YG6\"6#\"\"\"\"\"'F+F+*&\"
\"$F+)F'\"\"%F+F+*&F.F+)F'\"\"#F+F+F+F+*(,**$)&I\"yGF)F*F,F+!\"\"*&F.F
+)F8F0F+F:*&F.F+)F8F3F+F:F+F:F+)&F96#F3F3F:)&F(FAF3F+F+<#,$*&),&*$F>F+
F+F+F+F.F+F?F:F:" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I(RunTimeG6\"$\"'r
\\;!\"$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 209 0 "" }}}}}{SECT 0 {PARA 5
 "" 0 "" {TEXT 217 6 "on J^3" }}{EXCHG {PARA 0 "JC > " 0 "" {MPLTEXT 
1 0 14 "T := time(): \n" }{MPLTEXT 1 0 19 "A,var:=pr(E,3,G): \n" }
{MPLTEXT 1 0 20 "RunTime := time()-T;" }{MPLTEXT 1 0 0 "" }}{PARA 11 "
" 1 "" {XPPMATH 20 ">I(RunTimeG6\"$\"#T!\"$" }}}{SECT 0 {PARA 20 "" 0 
"" {TEXT 218 4 "SE_2" }}{EXCHG {PARA 0 "" 0 "" {TEXT 209 6 "with K" }}
}{EXCHG {PARA 0 "JC > " 0 "" {MPLTEXT 1 0 48 "T := time(): rgi(A,var,S
G); RunTime := time()-T;" }{MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6$7%,**(#\"\"\"\"\"$F')&I\"YG6\"6#F'\"\"#F'&F+6#F(F'!\"\"
*&F*F')&F+6#F.F.F'F'**F&F',(*&)&I\"yGF,F-F.F'&F;F0F'F'*(F(F'F:F')&F;F5
F.F'F1F<F'F'F>F1F3F'F'*&F&F'F/F'F1,0**F/F',4**F(F')F:\"\"%F')F?FFF')F<
FFF'F1**\"#OF')F:F(F')F?\"\"'F')F<F(F'F'**\"$i\"F'F9F')F?\"\")F')F<F.F
'F1**\"$C$F'F:F')F?\"#5F'F<F'F'*&\"$V#F')F?\"#7F'F1**FMF'F9F'FGF'FHF'F
1**FJF'F:F'FLF'FNF'F'*(\"#aF'FQF'FSF'F1*(F(F'FGF'FHF'F1F',&*(\"$H(F'F9
F'FVF'F'*&F]oF'FVF'F'F1)F*F(F'F'*&,&*(,&*(\"#\")F'F9F'FVF'F1*&FeoF'FVF
'F1F'F[oF1)F/F.F'F'*(,@*()F:FMF'F>F')F<\"\"&F'F'**\"#:F')F:F]pF'FGF'FH
F'F1**\"#!*F'FEF'FLF'FNF'F'**\"$q#F'FKF'FQF'FSF'F1**\"$0%F'F9F'FVF'F<F
'F'*(FYF'F:F'FZF'F1**F(F'FEF'F>F'F\\pF'F'**\"#IF'FKF'FGF'FHF'F1**\"$3
\"F'F9F'FLF'FNF'F'**FPF'F:F'FQF'FSF'F1*(FeoF'FVF'F<F'F'**F(F'F9F'F>F'F
\\pF'F'**F_pF'F:F'FGF'FHF'F1*(\"#=F'FLF'FNF'F'*&F>F'F\\pF'F'F'F[oF1F/F
'F'F'F)F'F'*&,&*(,,**FinF'FEF'FQF'F<F'F1*(FPF'FKF'FVF'F'**F\\qF'F9F'FQ
F'F<F'F1*(FPF'F:F'FVF'F'*(FinF'FQF'F<F'F1F'F[oF1FgoF'F'*(F/F'FCF'F[oF1
F'F'F*F'F'*$)F4FFF'F'*&,&*(,,**FUF'FEF'FQF'F<F'F'*(\"$s*F'FKF'FVF'F1**
\"$['F'F9F'FQF'F<F'F'*(FfrF'F:F'FVF'F1*(FUF'FQF'F<F'F'F'F[oF1F/F'F'*&,
P*&)F:FRF')F<FMF'F1**FbqF')F:\"\"(F'F>F'F\\pF'F'**\"$N\"F'F[pF'FGF'FHF
'F1**\"$S&F'F`pF'FLF'FNF'F'**\"%:7F'FEF'FQF'FSF'F1**\"%e9F'FKF'FVF'F<F
'F'*(F]oF'F9F'FZF'F1*(FFF'F[pF'F_sF'F1**FinF'F`pF'F>F'F\\pF'F'**\"$(HF
'FEF'FGF'FHF'F1**\"$k)F'FKF'FLF'FNF'F'**FjsF'F9F'FQF'FSF'F1**FjsF'F:F'
FVF'F<F'F'*&F]oF'FZF'F1*(FMF'FEF'F_sF'F1**FinF'FKF'F>F'F\\pF'F'**\"$*=
F'F9F'FGF'FHF'F1**FUF'F:F'FLF'FNF'F'*(FYF'FQF'FSF'F1*(FFF'F9F'F_sF'F1*
*FbqF'F:F'F>F'F\\pF'F'*(\"#FF'FGF'FHF'F1*$F_sF'F1F'F[oF1F'F'F3F'F'*(,6
**\"\"*F'F[pF'FLF'FSF'F'**FinF'F`pF'FQF'F<F'F1*(FeoF'FEF'FVF'F'**F^uF'
FEF'FLF'FSF'F'**F\\qF'FKF'FQF'F<F'F1FdoF1**F^uF'F9F'FLF'FSF'F'**FinF'F
:F'FQF'F<F'F1*&FPF'FVF'F1*(FcuF'FLF'FSF'F'F'F[oF1FgoF'F'FhoF',.*$)F*FF
F'F'*(,**(F.F'F9F'FSF'F'**FenF'F:F'F>F'F<F'F1*&FbqF'FGF'F'*&F.F'FSF'F'
F',**&F9F'FSF'F'**FMF'F:F'F>F'F<F'F1*&FcuF'FGF'F'*$FSF'F'F1F)F'F'**,(*
(F(F'FEF'F>F'F'*(FMF'F9F'F>F'F'*&F(F'F>F'F'F'F/F'FevF1F*F'F'*(,(*(FcuF
'FEF'F>F'F1*(FbqF'F9F'F>F'F1*&FcuF'F>F'F1F'FevF1F3F'F'F'F'*(,0*&F[pF'F
<F'F1*(F(F'F`pF'F>F'F'*(F(F'FEF'F<F'F1*(FMF'FKF'F>F'F'*(F(F'F9F'F<F'F1
F=F'F<F1F'FevF1F/F'F'<,,$*(#F'F^uF',(*(F.F'F9F'F<F'F1*(FMF'F:F'F>F'F'*
&F.F'F<F'F1F'F>F1F',$*(F&F'F7F'F>F1F',$*(#FFFcuF'F7F'F>F1F',$*(#F'F]oF
')FevF(F'FVF1F1,$*(#F'FeoF',0*&FEF'FSF'F'**FMF'FKF'F>F'F<F'F1*(FcuF'F9
F'FGF'F'FavF'FgvF1FcvF1FivF'F'FGF1F',$**FcuF'),&*$F9F'F'F'F'F.F'F>F'Fe
vF1F1,$**F(F'FeyF'F>F'FevF1F',$*(FeyF'F7F'FevF1F1,$**#F'FYF')FevF.F'Ff
yF1FLF1F1,$*,FjxF'F_zF'F7F'FQF1FfyF1F'" }}{PARA 11 "" 1 "" {XPPMATH 20
 ">I(RunTimeG6\"$\"'24:!\"$" }}}}}}}{SECT 0 {PARA 3 "" 0 "" {TEXT 215 
41 "Invariants and covariants of binary forms" }}{EXCHG {PARA 0 "" 0 "
" {TEXT 209 44 "\"fm\" outputs generic binary form of degree m" }}}
{EXCHG {PARA 0 "JC > " 0 "" {MPLTEXT 1 0 24 "fm:=proc(m) local i, f;\n
" }{MPLTEXT 1 0 6 "f:=0;\n" }{MPLTEXT 1 0 21 "for i from 0 to m do\n" 
}{MPLTEXT 1 0 40 " f:=f+c[m-i]*binomial(m,i)*x^(m-i)*y^i;\n" }{MPLTEXT
 1 0 4 "od;\n" }{MPLTEXT 1 0 3 "f;\n" }{MPLTEXT 1 0 9 "end proc:" }}
{PARA 0 "JC > " 0 "" {MPLTEXT 1 0 6 "fm(2);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 ",(*&I\"xG6\"\"\"#&I\"cGF%6#F&\"\"\"F**(F$F*I\"yGF%F*&F(6#
F*F*F&*&F,F&&F(6#\"\"!F*F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 209 23 "Li
near action on (x,y):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 209 0 "" }}}
{EXCHG {PARA 0 "JC > " 0 "" {MPLTEXT 1 0 55 "act_var:=[X-(a[1,1]*x+a[1
,2]*y),Y-(a[2,1]*x+a[2,2]*y)];" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I(act
_varG6\"7$,(*&I\"xGF$\"\"\"&I\"aGF$6$F)F)F)!\"\"*&I\"yGF$F)&F+6$F)\"\"
#F)F-I\"XGF$F),(*&F(F)&F+6$F2F)F)F-*&F/F)&F+6$F2F2F)F-I\"YGF$F)" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT 209 122 "\"act_coeff\" outputs polynomial
 equations of the actions on the coefficients of the binary form and t
he list of variables  " }}}{EXCHG {PARA 0 "JC > " 0 "" {MPLTEXT 1 0 
48 "act_coeff:=proc(m) local f, bf, i,A, delta,var;\n" }{MPLTEXT 1 0 
11 " f:=fm(m):\n" }{MPLTEXT 1 0 82 " bf:=collect(simplify(subs(\{x=a[2
,2]*x-a[1,2]*y,y=-a[2,1]*x+a[1,1]*y\},f)),[x,y]);\n" }{MPLTEXT 1 0 37 
"delta:= a[1,1]*a[2,2]-a[1,2]*a[2,1];\n" }{MPLTEXT 1 0 150 " A:=[delta
^m*C[0]-1/(m!)*diff(bf,y$m),seq(delta^m*C[i]-1/(m!)*diff(diff(bf,x$i),
y$(m-i)),i=1..(m-1)),delta^m*C[m]-1/(m!)*diff(bf,x$m),delta*dummy-1];
\n" }{MPLTEXT 1 0 92 " var:=[[a[1,1],a[1,2],a[2,1],a[2,2], dummy],[X,Y
,seq(C[i],i=0..m)],[x,y,seq(c[i],i=0..m)]];\n" }{MPLTEXT 1 0 8 "A, var
;\n" }{MPLTEXT 1 0 9 "end proc:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 
"" 0 "" {TEXT 209 0 "" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT 216 4 "SL_2" }
}{EXCHG {PARA 0 "JC > " 0 "" {MPLTEXT 1 0 14 "SL:=[dummy-1];" }}{PARA 
11 "" 1 "" {XPPMATH 20 ">I#SLG6\"7#,&I&dummyGF$\"\"\"!\"\"F(" }}}
{SECT 0 {PARA 5 "" 0 "" {TEXT 217 9 "quadratic" }}{EXCHG {PARA 0 "JC >
 " 0 "" {MPLTEXT 1 0 20 "A,var:=act_coeff(2);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 ">6$I\"AG6\"I$varGF%6$7&,**&),&*&&I\"aGF%6$\"\"\"F1F1&F/6$
\"\"#F4F1F1*&&F/6$F1F4F1&F/6$F4F1F1!\"\"F4F1&I\"CGF%6#\"\"!F1F1*&)F.F4
F1&I\"cGF%F=F1F:**F4F1F.F1F6F1&FB6#F1F1F1*&)F6F4F1&FB6#F4F1F:,(*&F+F1&
F<FEF1F1*&,&*&F8F1FAF1F1*&F2F1FDF1F:F1F.F1F1*&F6F1,&*&F8F1FDF1F1*&F2F1
FHF1F:F1F:,**&F+F1&F<FIF1F1*&)F8F4F1FAF1F:**F4F1F8F1F2F1FDF1F1*&)F2F4F
1FHF1F:,&*&F,F1I&dummyGF%F1F1F1F:7%7'F.F6F8F2Fin7'I\"XGF%I\"YGF%F;FLFW
7'I\"xG%*protectedGI\"yGF%FAFDFH" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 209 
10 "invariants" }}}{EXCHG {PARA 0 "JC > " 0 "" {MPLTEXT 1 0 39 "K:=[C[
1]];undebug(rgi);rgi(A,var,SL,K);" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I
\"KG6\"7#&I\"CGF$6#\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "I$rgiG6\"" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6$7$&I\"CG6\"6#\"\"\",(*&&F%6#\"\"!F(&
F%6#\"\"#F(F(*&&I\"cGF&F,F(&F3F/F(!\"\"*$)&F3F'F0F(F(<#,&F1F5F6F(" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT 209 10 "covariants" }}}{EXCHG {PARA 0 "JC
 > " 0 "" {MPLTEXT 1 0 27 "Acov:=[op(A),op(act_var)]; " }}{PARA 11 "" 
1 "" {XPPMATH 20 ">I%AcovG6\"7(,**&),&*&&I\"aGF$6$\"\"\"F.F.&F,6$\"\"#
F1F.F.*&&F,6$F.F1F.&F,6$F1F.F.!\"\"F1F.&I\"CGF$6#\"\"!F.F.*&)F+F1F.&I
\"cGF$F:F.F7**F1F.F+F.F3F.&F?6#F.F.F.*&)F3F1F.&F?6#F1F.F7,(*&F(F.&F9FB
F.F.*&,&*&F5F.F>F.F.*&F/F.FAF.F7F.F+F.F.*&F3F.,&*&F5F.FAF.F.*&F/F.FEF.
F7F.F7,**&F(F.&F9FFF.F.*&)F5F1F.F>F.F7**F1F.F5F.F/F.FAF.F.*&)F/F1F.FEF
.F7,&*&F)F.I&dummyGF$F.F.F.F7,(*&I\"xG%*protectedGF.F+F.F7*&I\"yGF$F.F
3F.F7I\"XGF$F.,(*&FinF.F5F.F7*&F\\oF.F/F.F7I\"YGF$F." }}}{EXCHG {PARA 
0 "JC > " 0 "" {MPLTEXT 1 0 32 "undebug(rgi);rgi(Acov,var,SL,K);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "I$rgiG6\"" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6$7&&I\"CG6\"6#\"\"\",(*&&F%6#\"\"!F(&F%6#\"\"#F(F(*&&I\"cGF&F,F(&
F3F/F(!\"\"*$)&F3F'F0F(F(,,*&)I\"XGF&F0F(F.F(F(*&)I\"YGF&F0F(F+F(F(*&)
I\"xG%*protectedGF0F(F4F(F5**F0F(FBF(I\"yGF&F(F8F(F5*&)FEF0F(F2F(F5,(*
&F;F()F.F0F(F(*&,&F1F(F6F5F(F>F(F(*&,(F@F5FDF5FFF5F(F.F(F(<%,&F1F5F6F(
FLFN" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT 217 5 "cubic" }}{EXCHG {PARA 0
 "JC > " 0 "" {MPLTEXT 1 0 31 "A,var:=act_coeff(3); K:=[C[1]];" }}
{PARA 11 "" 1 "" {XPPMATH 20 ">6$I\"AG6\"I$varGF%6$7',,*&),&*&&I\"aGF%
6$\"\"\"F1F1&F/6$\"\"#F4F1F1*&&F/6$F1F4F1&F/6$F4F1F1!\"\"\"\"$F1&I\"CG
F%6#\"\"!F1F1*&)F.F;F1&I\"cGF%F>F1F:**F;F1)F.F4F1F6F1&FC6#F1F1F1**F;F1
F.F1)F6F4F1&FC6#F4F1F:*&)F6F;F1&FC6#F;F1F1,**&F+F1&F=FGF1F1*&,&*&F8F1F
BF1F:*&F2F1FFF1F1F1FEF1F:**F4F1F6F1,&*&F8F1FFF1F:*&F2F1FJF1F1F1F.F1F1*
&FIF1,&*&F8F1FJF1F:*&F2F1FNF1F1F1F:,(*&F+F1&F=FKF1F1*&,(*&)F8F4F1FBF1F
1**F4F1F8F1F2F1FFF1F:*&)F2F4F1FJF1F1F1F.F1F:*&F6F1,(*&F_oF1FFF1F1**F4F
1F8F1F2F1FJF1F:*&FboF1FNF1F1F1F1,,*&F+F1&F=FOF1F1*&)F8F;F1FBF1F1**F;F1
F_oF1F2F1FFF1F:**F;F1FboF1FJF1F8F1F1*&)F2F;F1FNF1F:,&*&F,F1I&dummyGF%F
1F1F1F:7%7'F.F6F8F2Fcp7(I\"XGF%I\"YGF%F<FRF[oFjo7(I\"xG%*protectedGI\"
yGF%FBFFFJFN" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I\"KG6\"7#&I\"CGF$6#\"
\"\"" }}}{EXCHG {PARA 209 "" 0 "" {TEXT 209 10 "invariants" }}}{EXCHG 
{PARA 0 "JC > " 0 "" {MPLTEXT 1 0 36 "undebug(rgi);Ie,gs:=rgi(A,var,SL
,K);" }}{PARA 11 "" 1 "" {XPPMATH 20 "I$rgiG6\"" }}{PARA 11 "" 1 "" 
{XPPMATH 20 ">6$I#IeG6\"I#gsGF%6$7$&I\"CGF%6#\"\"\",0*(#F,\"\"%F,)&F*6
#\"\"!\"\"#F,)&F*6#\"\"$F5F,F,*&F2F,)&F*6#F5F9F,F,*(F/F,)&I\"cGF%F3F5F
,)&FAF8F5F,!\"\"*,#F9F5F,F@F,&FAF+F,&FAF=F,FCF,F,*&F@F,)FHF9F,FD*&)FGF
9F,FCF,FD*(#F9F0F,)FGF5F,)FHF5F,F,<#,*F>FD*(F/F,,&**\"\"'F,FGF,FHF,FCF
,F,*&F0F,FJF,FDF,F@F,F,FKFDFMF," }}}{EXCHG {PARA 0 "" 0 "" {TEXT 209 
58 "The  computed invariant is a multiple of the discriminant." }}}
{EXCHG {PARA 0 "JC > " 0 "" {MPLTEXT 1 0 75 "Delta[3]:=(c[0]*c[3]-c[1]
*c[2])^2-4*(c[1]*c[3]-c[2]^2)*(c[0]*c[2]-c[1]^2);\n" }}{PARA 11 "" 1 "
" {XPPMATH 20 ">&I&DeltaG6\"6#\"\"$,&*$),&*&&I\"cGF%6#\"\"!\"\"\"&F.F&
F1F1*&&F.6#F1F1&F.6#\"\"#F1!\"\"F8F1F1*(\"\"%F1,&*&F4F1F2F1F1*$)F6F8F1
F9F1,&*&F-F1F6F1F1*$)F4F8F1F9F1F9" }}}{EXCHG {PARA 0 "JC > " 0 "" 
{MPLTEXT 1 0 28 "simplify(Delta[3]+4*op(gs));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "\"\"!" }}}{EXCHG {PARA 210 "" 0 "" {TEXT 209 10 "covarian
ts" }}}{EXCHG {PARA 0 "JC > " 0 "" {MPLTEXT 1 0 44 "Acov:=[op(A),op(ac
t_var)];  K:=[X-1,Y,C[1]];" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I%AcovG6
\"7),,*&),&*&&I\"aGF$6$\"\"\"F.F.&F,6$\"\"#F1F.F.*&&F,6$F.F1F.&F,6$F1F
.F.!\"\"\"\"$F.&I\"CGF$6#\"\"!F.F.*&)F+F8F.&I\"cGF$F;F.F7**F8F.)F+F1F.
F3F.&F@6#F.F.F.**F8F.F+F.)F3F1F.&F@6#F1F.F7*&)F3F8F.&F@6#F8F.F.,**&F(F
.&F:FDF.F.*&,&*&F5F.F?F.F7*&F/F.FCF.F.F.FBF.F7**F1F.F3F.,&*&F5F.FCF.F7
*&F/F.FGF.F.F.F+F.F.*&FFF.,&*&F5F.FGF.F7*&F/F.FKF.F.F.F7,(*&F(F.&F:FHF
.F.*&,(*&)F5F1F.F?F.F.**F1F.F5F.F/F.FCF.F7*&)F/F1F.FGF.F.F.F+F.F7*&F3F
.,(*&F\\oF.FCF.F.**F1F.F5F.F/F.FGF.F7*&F_oF.FKF.F.F.F.,,*&F(F.&F:FLF.F
.*&)F5F8F.F?F.F.**F8F.F\\oF.F/F.FCF.F7**F8F.F_oF.FGF.F5F.F.*&)F/F8F.FK
F.F7,&*&F)F.I&dummyGF$F.F.F.F7,(*&I\"xG%*protectedGF.F+F.F7*&I\"yGF$F.
F3F.F7I\"XGF$F.,(*&FcpF.F5F.F7*&FfpF.F/F.F7I\"YGF$F." }}{PARA 11 "" 1 
"" {XPPMATH 20 ">I\"KG6\"7%,&I\"XGF$\"\"\"F(!\"\"I\"YGF$&I\"CGF$6#F(" 
}}}{EXCHG {PARA 0 "JC > " 0 "" {MPLTEXT 1 0 74 "T := time():undebug(rg
i); Ie,gs:=rgi(Acov,var,SL,K); RunTime := time()-T; " }}{PARA 11 "" 1 
"" {XPPMATH 20 "I$rgiG6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 ">6$I#IeG6\"
I#gsGF%6$7(,,*&)I\"xG%*protectedG\"\"$\"\"\"&I\"cGF%6#F.F/!\"\"**F.F/)
F,\"\"#F/I\"yGF%F/&F16#F6F/F3**F.F/F,F/)F7F6F/&F16#F/F/F3*&)F7F.F/&F16
#\"\"!F/F3&I\"CGF%F2F/&FDF=,(&FDFAF/*(,.**F6F/F5F/F<F/F0F/F3*(F6F/F5F/
)F8F6F/F/*,F6F/F,F/F7F/F@F/F0F/F3*,F6F/F,F/F7F/F<F/F8F/F/**F6F/F;F/F@F
/F8F/F3*(F6F/F;F/)F<F6F/F/F/,6*&)F,\"\"'F/)F0F6F/F/*,FUF/)F,\"\"&F/F7F
/F8F/F0F/F/*,FUF/)F,\"\"%F/F;F/F<F/F0F/F/**\"\"*F/FenF/F;F/FLF/F/*,F6F
/F+F/F?F/F@F/F0F/F/*,\"#=F/F+F/F?F/F<F/F8F/F/*,FUF/F5F/)F7FfnF/F@F/F8F
/F/**FhnF/F5F/F]oF/FQF/F/*,FUF/F,F/)F7FYF/F@F/F<F/F/*&)F7FUF/)F@F6F/F/
F3&FDF9F/F/*&,:*(F+F/F@F/FVF/F3*,F.F/F+F/F<F/F8F/F0F/F/*(F6F/F+F/)F8F.
F/F3*.F.F/F5F/F7F/F@F/F8F/F0F/F3*,FUF/F5F/F7F/FQF/F0F/F/*,F.F/F5F/F7F/
F<F/FLF/F3*.F.F/F,F/F;F/F@F/F<F/F0F/F/*,FUF/F,F/F;F/F@F/FLF/F3*,F.F/F,
F/F;F/FQF/F8F/F/*(F?F/FcoF/F0F/F/*,F.F/F?F/F@F/F<F/F8F/F3*(F6F/F?F/)F<
F.F/F/F/FRF3F/I\"YGF%,&I\"XGF%F/F/F3,0*(F5F/F<F/F0F/F/*&F5F/FLF/F3**F,
F/F7F/F@F/F0F/F/**F,F/F7F/F<F/F8F/F3*(F;F/F@F/F8F/F/*&F;F/FQF/F3*$)Fdo
F6F/F/<&*&,(*&,&*(F6F/F<F/F0F/F3*&F6F/FLF/F/F/F5F/F/**F6F/F7F/,&*&F@F/
F0F/F/*&F<F/F8F/F3F/F,F/F3*(F6F/,&*&F@F/F8F/F/*$FQF/F3F/F;F/F3F/),*F*F
/F4F/F:F/F>F/F6F3*&,**&,(*&F@F/FVF/F3**F.F/F<F/F8F/F0F/F/*&F6F/FjoF/F3
F/F+F/F/**F.F/F7F/,(*(F@F/F8F/F0F/F/*(F6F/FQF/F0F/F3*&F<F/FLF/F/F/F5F/
F3**F.F/F;F/,(*(F@F/F<F/F0F/F/*(F6F/F@F/FLF/F3*&FQF/F8F/F/F/F,F/F/*&F?
F/,(*&FcoF/F0F/F/**F.F/F@F/F<F/F8F/F3*&F6F/FdpF/F/F/F/F/F`rF3,(*&F5F/,
&*&F<F/F0F/F/*$FLF/F3F/F/*(F7F/FiqF/F,F/F/*&F]rF/F;F/F/,*F*F3F4F3F:F3F
>F3" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I(RunTimeG6\"$\"#Y!\"$" }}}
{SECT 1 {PARA 211 "" 0 "" {TEXT 209 41 "The meaning of the generating \+
covariants:" }}{EXCHG {PARA 0 "JC > " 0 "" {MPLTEXT 1 0 9 "nops(gs);" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT
 209 4 "form" }}}{EXCHG {PARA 0 "JC > " 0 "" {MPLTEXT 1 0 12 "f[3]:=fm
(3);" }}{PARA 11 "" 1 "" {XPPMATH 20 ">&I\"fG6\"6#\"\"$,**&)I\"xG%*pro
tectedGF'\"\"\"&I\"cGF%F&F-F-**F'F-)F+\"\"#F-I\"yGF%F-&F/6#F2F-F-**F'F
-F+F-)F3F2F-&F/6#F-F-F-*&)F3F'F-&F/6#\"\"!F-F-" }}}{EXCHG {PARA 0 "" 0
 "" {TEXT 209 7 "Hessian" }}}{EXCHG {PARA 0 "JC > " 0 "" {MPLTEXT 1 0 
75 "H[3]:=1/36*LinearAlgebra[Determinant](VectorCalculus[Hessian](f[3]
,[x,y]));" }}{PARA 11 "" 1 "" {XPPMATH 20 ">&I\"HG6\"6#\"\"$,.*()I\"xG
%*protectedG\"\"#\"\"\"&I\"cGF%6#F.F.&F0F&F.F.*&F*F.)&F06#F-F-F.!\"\"*
*F+F.I\"yGF%F.&F06#\"\"!F.F2F.F.**F+F.F9F.F/F.F5F.F7*()F9F-F.F:F.F5F.F
.*&F?F.)F/F-F.F7" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 209 36 "Jacobian of \+
the form and the Hessian" }}}{EXCHG {PARA 0 "JC > " 0 "" {MPLTEXT 1 0 
91 "JHf:=simplify(1/3*LinearAlgebra[Determinant](VectorCalculus[Jacobi
an]([f[3],H[3]],[x,y])));" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I$JHfG6\",
**&,(*&&I\"cGF$6#\"\"!\"\"\")&F*6#\"\"$\"\"#F-F-**F1F-&F*6#F-F-&F*6#F2
F-F/F-!\"\"*&F2F-)F6F1F-F-F-)I\"xG%*protectedGF1F-F-**F1F-I\"yGF$F-,(*
(F)F-F6F-F/F-F-*(F2F-)F4F2F-F/F-F8*&F4F-)F6F2F-F-F-)F<F2F-F-**F1F-)F?F
2F-,(*(F)F-F4F-F/F-F-*(F2F-F)F-FEF-F8*&FCF-F6F-F-F-F<F-F8*&)F?F1F-,(*&
)F)F2F-F/F-F-**F1F-F)F-F4F-F6F-F8*&F2F-)F4F1F-F-F-F8" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT 209 57 "Expressing the generators in terms of f[3], H[
3] and FHA." }}}{EXCHG {PARA 0 "JC > " 0 "" {MPLTEXT 1 0 30 "simplify(
gs[1]+2*H[3]/f[3]^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}}
{EXCHG {PARA 0 "JC > " 0 "" {MPLTEXT 1 0 27 "simplify(gs[2]+JHf/f[3]^2
);" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}}{EXCHG {PARA 0 "JC > " 0
 "" {MPLTEXT 1 0 21 "simplify(gs[3]-H[3]);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "\"\"!" }}}{EXCHG {PARA 0 "JC > " 0 "" {MPLTEXT 1 0 11 "gs
[4]+f[3];" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}}}}}{SECT 0 {PARA 
4 "" 0 "" {TEXT 216 2 "GL" }}{EXCHG {PARA 0 "JC > " 0 "" {MPLTEXT 1 0 
6 "GL:=[]" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I#GLG6\"7\"" }}}{SECT 0 
{PARA 5 "" 0 "" {TEXT 217 9 "quadratic" }}{EXCHG {PARA 0 "JC > " 0 "" 
{MPLTEXT 1 0 21 "A,var:=act_coeff(2); " }{MPLTEXT 1 0 10 "K:=[C[1]];" 
}}{PARA 11 "" 1 "" {XPPMATH 20 ">6$I\"AG6\"I$varGF%6$7&,**&,&*&&I\"aGF
%6$\"\"\"F0F0&F.6$\"\"#F3F0F0*&&F.6$F0F3F0&F.6$F3F0F0!\"\"F3&I\"CGF%6#
\"\"!F0F0*&F-F3&I\"cGF%F<F0F9*(F-F0F5F0&F@6#F0F0F3*&F5F3&F@6#F3F0F9,(*
&F+F3&F;FCF0F0*&,&*&F7F0F?F0F0*&F1F0FBF0F9F0F-F0F0*&F5F0,&*&F7F0FBF0F0
*&F1F0FEF0F9F0F9,**&F+F3&F;FFF0F0*&F7F3F?F0F9*(F7F0F1F0FBF0F3*&F1F3FEF
0F9,&*&F+F0I&dummyGF%F0F0F9F07%7'F-F5F7F1FZ7'I\"XGF%I\"YGF%F:FIFT7'I\"
xGF%I\"yGF%F?FBFE" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I\"KG6\"7#&I\"CGF$
6#\"\"\"" }}}{EXCHG {PARA 212 "" 0 "" {TEXT 209 26 "no non-constant in
variants" }}}{EXCHG {PARA 0 "JC > " 0 "" {MPLTEXT 1 0 27 "undebug(rgi)
;rgi(A,var,GL);" }}{PARA 11 "" 1 "" {XPPMATH 20 "I$rgiG6\"" }}{PARA 11
 "" 1 "" {XPPMATH 20 "6$7\"<\"" }}}{EXCHG {PARA 213 "" 0 "" {TEXT 209 
10 "covariants" }}}{EXCHG {PARA 0 "JC > " 0 "" {MPLTEXT 1 0 27 "Acov:=
[op(A),op(act_var)]; " }}{PARA 11 "" 1 "" {XPPMATH 20 ">I%AcovG6\"7(,*
*&,&*&&I\"aGF$6$\"\"\"F-F-&F+6$\"\"#F0F-F-*&&F+6$F-F0F-&F+6$F0F-F-!\"
\"F0&I\"CGF$6#\"\"!F-F-*&F*F0&I\"cGF$F9F-F6*(F*F-F2F-&F=6#F-F-F0*&F2F0
&F=6#F0F-F6,(*&F(F0&F8F@F-F-*&,&*&F4F-F<F-F-*&F.F-F?F-F6F-F*F-F-*&F2F-
,&*&F4F-F?F-F-*&F.F-FBF-F6F-F6,**&F(F0&F8FCF-F-*&F4F0F<F-F6*(F4F-F.F-F
?F-F0*&F.F0FBF-F6,&*&F(F-I&dummyGF$F-F-F6F-,(*&I\"xGF$F-F*F-F6*&I\"yGF
$F-F2F-F6I\"XGF$F-,(*&FZF-F4F-F6*&FfnF-F.F-F6I\"YGF$F-" }}}{EXCHG 
{PARA 0 "JC > " 0 "" {MPLTEXT 1 0 24 "K:=[X-1,Y, C[0]-1,C[1]];" }}
{PARA 11 "" 1 "" {XPPMATH 20 ">I\"KG6\"7&,&I\"XGF$\"\"\"!\"\"F(I\"YGF$
,&&I\"CGF$6#\"\"!F(F)F(&F-6#F(" }}}{EXCHG {PARA 0 "JC > " 0 "" 
{MPLTEXT 1 0 19 "rgi(Acov,var,GL,K);" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6$7',**&I\"xG6\"\"\"#&I\"cGF'6#F(\"\"\"!\"\"*(F&F,I\"yGF'F,&F*6#F,F,!
\"#*&F/F(&F*6#\"\"!F,F-&I\"CGF'F+F,&F8F1,&&F8F5F,F-F,I\"YGF',&I\"XGF'F
,F-F,<#,(F%F-F.F2F3F-" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT 217 5 "cubic"
 }}{EXCHG {PARA 0 "JC > " 0 "" {MPLTEXT 1 0 51 "A,var:=act_coeff(3);K \+
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