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 Post subject: Triangularization of underdetermined systemsPosted: Thu Aug 11, 2005 5:32 pm
Is there a Singular library analogous to "triang.lib" which triangularizes
underdetermined systems (ideals with dim > 0)?

For example,

> LIB "triang.lib";
...
> ring r=0,(x,y,z,w),lp;
> ideal i = x^2*y+z^3+w*x*y, x*w+y^2*x,x+x^2*y+z*w;
> ideal j = groebner(i);
> j;
j[1]=z11w-4z9w2-2z8w2+6z7w3+2z6w3-4z5w4+z5w3+z5+2z4w4+z3w5-2z2w5+zw8+zw5
j[2]=yzw18-4yzw15+6yzw12-4yzw9-yzw8+yzw6-2yzw5-yzw2+z10w7-2z10w4+z10w+z9w4+
z9w+z8w11-6z8w8+9z8w5-4z8w2-z7w8+z7w5-6z7w2+z6w15-6z6w12+15z6w9-16z6w6-2z6w5+
6z6w3-2z6w2-z5w12+3z5w9-4z5w6+8z5w3-2z4w16+9z4w13-16z4w10+13z4w7+z4w6-4z4w4+
2z4w3+z4-z3w16+3z3w13-2z3w10+z3w6-2z3w4+2z3w3+z3+z2w17-4z2w14+6z2w11-z2w10-
4z2w8+z2w5+z2w4-zw17+4zw14-6zw11+4zw8-zw5
j[3]=4yz2w2+yzw15-5yzw12+7yzw9-3yzw6-yzw5-yzw2+z10w4-3z10w+z9w+z8w8-7z8w5+
12z8w2-z7w5+2z7w2+z6w12-7z6w9+18z6w6-18z6w3-2z6w2-z5w9+4z5w6-4z5w3-2z4w13+
11z4w10-19z4w7+12z4w4+z4w3-3z4-z3w13+4z3w10-2z3w7-2z3w4+z3w3+z3+z2w14-5z2w11+
7z2w8-z2w7-3z2w5+z2w4-zw14+5zw11-7zw8+3zw5
j[4]=yz3-yzw4-z6w+2z4w2+z3w2-z2w3+zw3
j[5]=y2zw+zw2
j[6]=xw3+x-yz3w+yzw2-z3+zw
j[7]=xz3-xzw-y2z3-yzw3-z6+2z4w-z2w2
j[8]=xy+xw2-yz3+yzw
j[9]=x2-xz3+xzw+xw2
> triangL(j);
? Error: ideal is not zero-dimensional.
> dim(j);
2

To paraphrase my question, how can I reproduce the output given by Maple's
gsolve?

> ideal := [x^2*y+z^3+w*x*y, x*w+y^2*x,x+x^2*y+z*w]:
> vars := [x,y,z,w]:
> Groebner[gsolve](ideal,vars);
2
{[[z, y - y + 1, x - y + 1, w + y - 1], plex(w, x, y, z), {y - 1, y, y + 1}],

[[x, z], plex(z, x, y, w), {}], [[

2 3 3 5 3 2 8 5 5 2 5 2 3
-z - 2 z y - z y - y z + y - y z + y , -z + y + x - z y - z

4 2 6 3 3 5 6 7 2 9 6 2 4 7
- z y + z y - z y - z y + z x - 2 z y - z - z y + z y

5 4 2 6 3 3 4 5 2 7 4 2
- z y , -y z + y + x y - z y - y z + z x - 2 z y - z - z y

2 7 3 4
+ z y - z y ,

2 2 7 7 2 4 4 2 2 3 2 2 6 5
x + z x + z y + y - z y - 2 z y - 3 y z - z y - z - z ,

2
w + y ], plex(w, x, y, z), {x, z}],

[[z, y + 1, x - 1, w + 1], plex(w, x, y, z), {}]}

email: jtshorwo@math.uwaterloo.ca
Posted in old Singular Forum on: 2003-05-27 14:32:34+02

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