Solucin algebraica

> p1:=[0,0,0]; p2:=[1,1,1]; p3:=[0,1,0]; p4:=[1,0,0]; p5:=[0,0,1]; p6:=[x,y,z];

> d:=3: n:=6: e:=12: zeros:=[0,0,0]:

> with(linalg):
R:=matrix(e,n*d,[
op(p1-p2),op(p2-p1),op(zeros),op(zeros),op(zeros),op(zeros),
op(p1-p3),op(zeros),op(p3-p1),op(zeros),op(zeros),op(zeros),
op(p1-p4),op(zeros),op(zeros),op(p4-p1),op(zeros),op(zeros),
op(p1-p5),op(zeros),op(zeros),op(zeros),op(p5-p1),op(zeros),
op(zeros),op(p2-p3),op(p3-p2),op(zeros),op(zeros),op(zeros),
op(zeros),op(p2-p5),op(zeros),op(zeros),op(p5-p2),op(zeros),
op(zeros),op(p2-p6),op(zeros),op(zeros),op(zeros),op(p6-p2),
op(zeros),op(zeros),op(p3-p4),op(p4-p3),op(zeros),op(zeros),
op(zeros),op(zeros),op(p3-p6),op(zeros),op(zeros),op(p6-p3),
op(zeros),op(zeros),op(zeros),op(p4-p5),op(p5-p4),op(zeros),
op(zeros),op(zeros),op(zeros),op(p4-p6),op(zeros),op(p6-p4),
op(zeros),op(zeros),op(zeros),op(zeros),op(p5-p6),op(p6-p5) ]);

Warning, the protected names norm and trace have been redefined and unprotected

> Rt:=transpose(R):
eqs:=geneqns(Rt,
[w12,w13,w14,w15,w23,w25,w26,w34,w36,w45,w46,w56],
vector(n*d,0));

> polys:={-w12-w14, -w12-w13, -w12-w15,
w12+w23+w25-w26*x+w26, w12+w25-w26*y+w26,
w12+w23-w26*z+w26, -w23-w34-x*w36,
w13+w34-w36*y+w36 , -w23-z*w36,
w14+w34+w45-w46*x+w46, -w34-y*w46,
-w45-z*w46, -w25-w45-x*w56 ,
-w25-y*w56, w15+w45-w56*z+w56,
-w26+w26*x+x*w36-w46+w46*x+x*w56,
-w26+w26*y-w36+w36*y+y*w46+y*w56,
-w26+w26*z+z*w36+z*w46-w56+w56*z}:
with(Groebner):
G:=gbasis(polys,lexdeg([w12,w13,w14,w15,w23,w25,
w26,w34,w36,w45,w46,w56],[x,y,z])):

> op(1,G);

> superpolys:={w56*t56-1, -w12-w14, -w12-w13,
-w12-w15, w12+w23+w25-w26*x+w26, w12+w25-w26*y+w26,
w12+w23-w26*z+w26, -w23-w34-x*w36,
w13+w34-w36*y+w36, -w23-z*w36,
w14+w34+w45-w46*x+w46, -w34-y*w46, -w45-z*w46,
-w25-w45-x*w56, -w25-y*w56, w15+w45-w56*z+w56,
-w26+w26*x+x*w36-w46+w46*x+x*w56,
-w26+w26*y-w36+w36*y+y*w46+y*w56,
-w26+w26*z+z*w36+z*w46-w56+w56*z}:
superG:=gbasis(superpolys,lexdeg([t56,w12,w13,w14,
w15,w23,w25,w26,w34,w36,w45,w46,w56],[x,y,z]));

> with(plots):implicitplot3d(y+x^2-x-y^2-z^2+z,x=-2..2,y=-2..2,z=-2..2);

Warning, the name changecoords has been redefined

>