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Mon Jan 4 00:00:49 MET 1999 REDUCE 3.7, 15-Jan-99 ... 1: 1: 2: 2: 2: 2: 2: 2: 2: 2: 2: *** ^ redefined 3: 3: % 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; 4: 4: 4: 4: 4: 4: 4: 4: 4: Time for test: 460 ms 5: 5: Quitting Mon Jan 4 00:00:54 MET 1999