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REDUCE 3.6, 15-Jul-95, patched to 6 Mar 96 ... % Test file for XIDEAL package (Groebner bases for exterior algebra) % Just make sure excalc has been loaded load_package excalc$ *** ^ redefined % Declare exterior form variables pform x=0,y=0,z=0,t=0,u=1,v=1,w=1,f(i)=1,h=0,hx=0,ht=0; % Set switches for reduced Groebner bases in graded ideals on xfullreduce; % Reductions with xmodulo (all should be zero) d x^d y xmodulo {d x - d y}; 0 d x^d y^d z xmodulo {d x^d y - d z^d t}; 0 d x^d z^d t xmodulo {d x^d y - d z^d t}; 0 v^d x^d y xmodulo {d t^u - v^w, w^u - d x^d y}; 0 d t^u^d z xmodulo {d t^u - v^w, u^d z - d x^d y, d t^d y - d x^v}; 0 f(3)^f(4)^f(5)^f(6) xmodulo {f(1)^f(2) + f(3)^f(4) + f(5)^f(6)}; 0 f(1)^f(4)^f(5)^f(6) xmodulo {f(1)^f(2) + f(2)^f(3) + f(3)^f(4) + f(4)^f(5) + f(5)^f(6)}; 0 % Exterior system for heat equation on 1st jet bundle S := {d h - ht*d t - hx*d x, d ht^d t + d hx^d x, d hx^d t - ht*d x^d t}; s := {d h - d t*ht - d x*hx, d ht^d t + d hx^d x, d hx^d t + d t^d x*ht} % Check that it's closed. dS := (for each a in S collect d a) xmodulo S; ds := {} % Some Groebner bases (0-forms generate the trivial ideal) gb := xideal {x, d y}; gb := {1} gb := xideal {f(1)^f(2) + f(3)^f(4)}; 1 2 3 4 gb := {f ^f + f ^f , 1 3 4 f ^f ^f , 2 3 4 f ^f ^f } gb := xideal {f(1)^f(2), f(1)^f(3)+f(2)^f(4)+f(5)^f(6)}; 1 2 gb := {f ^f , 2 5 6 f ^f ^f , 1 5 6 f ^f ^f , 1 3 2 4 5 6 f ^f + f ^f + f ^f , 2 3 4 3 5 6 - f ^f ^f + f ^f ^f , 3 4 5 6 - f ^f ^f ^f } % The same again, but not reduced off xfullreduce; gb := xideal {x, d y}; gb := {1} gb := xideal {f(1)^f(2) + f(3)^f(4)}; 1 2 3 4 gb := {f ^f + f ^f , 1 3 4 f ^f ^f , 2 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 4 1 5 6 f ^f ^f + f ^f ^f , 2 3 4 3 5 6 - f ^f ^f + f ^f ^f , 1 2 5 6 - f ^f ^f ^f , 1 4 5 6 - f ^f ^f ^f , 2 3 5 6 f ^f ^f ^f , 3 4 5 6 - f ^f ^f ^f , 2 4 5 6 2*f ^f ^f ^f , 1 2 f ^f , 2 5 6 f ^f ^f , 1 5 6 f ^f ^f } % Reductions with a ready Groebner basis (not all zero) on xfullreduce; gb := xideal {f(1)^f(2) + f(3)^f(4) + f(5)^f(6)}; 1 2 3 4 5 6 gb := {f ^f + f ^f + f ^f , 1 3 4 1 5 6 f ^f ^f + f ^f ^f , 2 3 4 2 5 6 f ^f ^f + f ^f ^f , 1 3 5 6 - f ^f ^f ^f , 1 4 5 6 - f ^f ^f ^f , 2 3 5 6 - f ^f ^f ^f , 2 4 5 6 - f ^f ^f ^f , 3 4 5 6 2*f ^f ^f ^f } f(1)^f(3)^f(4) xmodulop gb; 1 5 6 - f ^f ^f f(3)^f(4)^f(5)^f(6) xmodulop gb; 0 % Non-graded ideals on xfullreduce; % Left and right ideals are no longer the same d t^(d z+d x^d y) xmodulo {d z+d x^d y}; 0 (d z+d x^d y)^d t xmodulo {d z+d x^d y}; - 2*d t^d z % Higher order forms can now reduce lower order ones d x xmodulo {d y^d z + d x,d x^d y + d z}; 0 % Anything with a 0-form term 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} end; (TIME: xideal 1210 1250)