Tue Feb 10 12:28:06 2004 run on Linux
*** ^ redefined
% Test file for XIDEAL package (Groebner bases for exterior algebra)
% Declare EXCALC variables
pform {x,y,z,t}=0,f(i)=1,{u,u(i),u(i,j)}=0;
% Reductions with xmodideal (all should be zero)
d x^d y xmodideal {d x - d y};
0
d x^d y^d z xmodideal {d x^d y - d z^d t};
0
d x^d z^d t xmodideal {d x^d y - d z^d t};
0
f(2)^d x^d y xmodideal {d t^f(1) - f(2)^f(3),
f(3)^f(1) - d x^d y};
0
d t^f(1)^d z xmodideal {d t^f(1) - f(2)^f(3),
f(1)^d z - d x^d y,
d t^d y - d x^f(2)};
0
f(3)^f(4)^f(5)^f(6)
xmodideal {f(1)^f(2) + f(3)^f(4) + f(5)^f(6)};
0
f(1)^f(4)^f(5)^f(6)
xmodideal {f(1)^f(2) + f(2)^f(3) + f(3)^f(4)
+ f(4)^f(5) + f(5)^f(6)};
0
d x^d y^d z xmodideal {x**2+y**2+z**2-1,x*d x+y*d y+z*d z};
0
% Changing the division between exterior variables and parameters
xideal {a*d x+y*d y};
d x*a + d y*y
{---------------}
a
xvars {a};
xideal {a*d x+y*d y};
{d x*a + d y*y,d x^d y}
xideal({a*d x+y*d y},{a,y});
{d x*a + d y*y,
d x^d y*y}
xvars {};
% all 0-forms are coefficients
excoeffs(d u - (a*p - q)*d y);
{1, - a*p + q}
exvars(d u - (a*p - q)*d y);
{d u,d y}
xvars {p,q};
% p,q are no longer coefficients
excoeffs(d u - (a*p - q)*d y);
{ - a,1,1}
exvars(d u - (a*p - q)*d y);
{d y*p,d y*q,d u}
xvars nil;
% Exterior system for heat equation on 1st jet bundle
S := {d u - u(-t)*d t - u(-x)*d x,
d u(-t)^d t + d u(-x)^d x,
d u(-x)^d t - u(-t)*d x^d t};
s := { - d t*u + d u - d x*u ,
t x
- (d t^d u + d x^d u ),
t x
u *d t^d x - d t^d u }
t x
% Check that it's closed.
dS := d S xmodideal S;
ds := {}
% Exterior system for a Monge-Ampere equation
korder d u(-y,-y),d u(-x,-y),d u(-x,-x),d u(-y),d u(-x),d u;
M := {u(-x,-x)*u(-y,-y) - u(-x,-y)**2,
d u - u(-x)*d x - u(-y)*d y,
d u(-x) - u(-x,-x)*d x - u(-x,-y)*d y,
d u(-y) - u(-x,-y)*d x - u(-y,-y)*d y}$
% Get the full Groebner basis
gbdeg := xideal M;
2
gbdeg := {u *u - (u ) ,
x x y y x y
d u - d x*u - d y*u ,
x y
d u - d x*u - d y*u ,
x x x x y
d u - d x*u - d y*u }
y x y y y
% Changing the term ordering can be dramatic
xorder gradlex;
gradlex
gbgrad := xideal M;
2
gbgrad := {u *u - (u ) ,
x x y y x y
- d u + d x*u + d y*u ,
x y
- d u + d x*u + d y*u ,
y x y y y
- d u + d x*u + d y*u ,
x x x x y
d u ^d x + d u ^d y,
x y
- d u *u + d u *u ,
x y y y x y
- d u *u + d u *u ,
x x y y x x
d u ^d u ,
y x
d u *u - d u*u + d y*u *u - d y*u *u ,
y x x y x y y x y y
d u *u - d u*u + d y*u *u - d y*u *u ,
x x x x x x y x y x
u *d x^d y + d u^d x,
y
u *d x^d y + d u ^d x,
y y y
d u^d x^d y,
- u *d u^d y + u *d u ^d y - d u ^d u,
x y y x x
- u *d u^d y + u *d u ^d y,
x x x x
u *d u^d y + u *d u ^d x + d u ^d u,
y y y x y
d u ^d x^d y,
x
d u ^d u^d y,
x
d u ^d u^d x,
x
- u *d u^d x + u *d u ^d x,
y y y y
d u ^d u^d x}
y
% But the bases are equivalent
gbdeg xmod gbgrad;
{}
xorder deglex;
deglex
gbgrad xmod gbdeg;
{}
% Some Groebner bases
gb := xideal {f(1)^f(2) + f(3)^f(4)};
1 2 3 4
gb := {f ^f + f ^f ,
2 3 4
f ^f ^f ,
1 3 4
f ^f ^f }
gb := xideal {f(1)^f(2), f(1)^f(3)+f(2)^f(4)+f(5)^f(6)};
1 3 2 4 5 6
gb := {f ^f + f ^f + f ^f ,
1 2
f ^f ,
2 5 6
f ^f ^f ,
2 3 4 3 5 6
f ^f ^f - f ^f ^f ,
1 5 6
f ^f ^f ,
3 4 5 6
f ^f ^f ^f }
% Non-graded ideals
% Left and right ideals are not the same
d t^(d z+d x^d y) xmodideal {d z+d x^d y};
0
(d z+d x^d y)^d t xmodideal {d z+d x^d y};
- 2*d t^d z
% Higher order forms can now reduce lower order ones
d x xmodideal {d y^d z + d x,d x^d y + d z};
0
% Anything whose even part is a parameter generates the trivial ideal!!
gb := xideal({x + d y},{});
gb := {1}
gb := xideal {1 + f(1) + f(1)^f(2) + f(2)^f(3)^f(4) + f(3)^f(4)^f(5)^f(6)};
gb := {1}
xvars nil;
% Tracing Groebner basis calculations
on trxideal;
gb := xideal {x-y+y*d x-x*d y};
Input Basis
xpoly(1)= - x^d y + d x^y + x - y
New Basis
xpoly(1)=x^d y - d x^y - x + y
wedge_pair{d y,1} -> xpoly(2)=d x^y^d y - x^d y + y^d y
spoly_pair{2,1} -> xpoly(3)=x^x - 2*x^y + y^y
spoly_pair{1,3} -> xpoly(4)=x^d x^y - 2*x^y^d y + y^y^d y + x^x - x^y
spoly_pair{4,3} -> 0
spoly_pair{4,1} -> 0
spoly_pair{2,4} -> criterion 1 hit
wedge_pair{d x,4} -> 0
wedge_pair{d x,2} -> xpoly(5)=x^d x - x^d y - d x^y + y^d y
New Basis
xpoly(1)=x^d y - d x^y - x + y
xpoly(2)=d x^y^d y - x^d y + y^d y
xpoly(3)=x^x - 2*x^y + y^y
xpoly(4)=x^d x - x^d y - d x^y + y^d y
spoly_pair{4,3} -> 0
spoly_pair{4,1} -> 0
spoly_pair{2,4} -> criterion 1 hit
wedge_pair{d x,4} -> 0
2 2
gb := {x - 2*x*y + y ,
- d x*y + d y*x - x + y,
d x*x - 2*d x*y + d y*y - x + y,
- d x*y + d y*y + d x^d y*y - x + y}
off trxideal;
% Same thing in lexicographic order, without full reduction
xorder lex;
lex
off xfullreduce;
gblex := xideal {x-y+y*d x-x*d y};
gblex := {d x*y - d y*y - d x^d y*y + x - y,
d x*y - d y*x + x - y}
% Manual autoreduction
gblex := xauto gblex;
gblex := {d x*y - d y*y - d x^d y*y + x - y}
% Tracing reduction
on trxmod;
first gb xmod gblex;
x^x - 2*x^y + y^y =
x^(x - d x^y^d y + d x^y - y^d y - y) +
(d x^y^d y)^(x - d x^y^d y + d x^y - y^d y - y) +
( - d x^y)^(x - d x^y^d y + d x^y - y^d y - y) +
(y^d y)^(x - d x^y^d y + d x^y - y^d y - y) +
( - y)^(x - d x^y^d y + d x^y - y^d y - y) +
0
0
% Restore defaults
on xfullreduce;
off trxideal,trxmod;
xvars nil;
xorder deglex;
deglex
end;
Time for test: 60 ms, plus GC time: 10 ms