Comparing Ideals in Polynomial Rings
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Christopher Douglas
Apr 30, 2009, 5:07:50 PM4/30/09
to Macaulay2
Dear all,
I am comparing ideals in polynomial rings, using the commands == and
isSubset, and am encountering a problem. It seems like the output I
am getting suggests a bug in these commands, but perhaps I have
misunderstood the basic syntax or some other issue. In short, (1) the
command "isSubset(A,A+B)" is returning "false", where A and B are
ideals in a polynomial ring, and (2) the command "isSubset(A,B+C)" is
returning false, even though "isSubset(A,B)" and "isSubset(A,C)" both
return "true". More specifically, the session is as follows, with ...
replacing various lists of polynomials:
-----
Macaulay 2, version 1.2
with packages: Elimination, IntegralClosure, LLLBases,
PrimaryDecomposition,
ReesAlgebra, SchurRings, TangentCone
i1 : R=ZZ[a,b,c,d];
i2 : M6=ideal "...";
o2 : Ideal of R
i3 : C1=ideal "...";
o3 : Ideal of R
i4 : C2=ideal "...";
o4 : Ideal of R
i5 : C3=ideal "...";
o5 : Ideal of R
i6 : C=C1+C2+C3;
o6 : Ideal of R
i7 : isSubset(C1,C)
o7 = true
i8 : isSubset(C2,C)
o8 = true
i9 : isSubset(C3,C)
o9 = false
i10 : isSubset(M6,C1)
o10 = true
i11 : isSubset(M6,C2)
o11 = true
i12 : isSubset(M6,C3)
o12 = true
i13 : isSubset(M6,C)
o13 = false
i14 : D=C1+C2+C3+M6;
o14 : Ideal of R
i15 : C==D
o15 = false
---------
The ideals in question are large, but not that large, and I hope to
work with even larger ideals. Below I attach the full session with
the ideals included. Does anyone know what might be going on or
whether it is possible to fix? If there are size limits on some of
the functions called (eg involving Groebner bases), I wouldn't mind
isSubset or == returning overflow errors or some such, but if it
returns 'false', then I'm uncertain if it really is false or if there
has been an internal error.
Note: I can produce similar examples that do not involve the "+"
command for ideals. These have the form "J=ideal "A,B,C,D"; M=ideal
"D"; isSubset(M,J)" returning "false". An example of this is included
at the end.
Thanks very much for any help!
Chris
-----
Example 1:
Macaulay 2, version 1.2
with packages: Elimination, IntegralClosure, LLLBases,
PrimaryDecomposition,
ReesAlgebra, SchurRings, TangentCone
i1 : R=ZZ[a,b,c,d];
i2 : M6=ideal
"(1a8b0c0d0-7a6b1c0d0+15a4b2c0d0+6a5b0c1d0-10a2b3c0d0-20a3b1c1d0+1a0b4c0d0+12a1b2c1d0-5a4b0c0d2+6a2b0c2d0+5a4b0c1d0+12a2b1c0d2-3a0b1c2d0-1a6b0c0d0-12a2b1c1d0-3a0b2c0d2+10a4b1c0d0-6a1b0c1d2+3a0b2c1d0+4a3b0c0d2-18a2b2c0d0+6a1b0c2d0-8a3b0c1d0-6a1b1c0d2+4a0b3c0d0+5a5b0c0d0+18a1b1c1d0+1a0b0c0d4-16a3b1c0d0+3a2b0c0d2+9a1b2c0d0-2a0b0c2d0-4a0b1c0d2+1a4b0c0d0+4a0b1c1d0-9a2b1c0d0-4a1b0c0d2+6a0b2c0d0+8a1b0c1d0-7a3b0c0d0+14a1b1c0d0-2a0b0c0d2+1a0b0c1d0+4a0b1c0d0+3a1b0c0d0+1a0b0c0d0)";
o2 : Ideal of R
i3 : C1=ideal
"(1a0b0c0d8-7a0b0c1d6+15a0b0c2d4-1a0b1c0d6-10a0b0c3d2+10a0b1c1d4+1a0b0c4d0-18a0b1c2d2-6a1b0c0d6-4a0b2c0d4+4a0b1c3d0+27a1b0c1d4+3a0b2c1d2-33a1b0c2d2+3a0b2c2d0+11a1b1c0d4+7a1b0c3d0+5a0b3c0d2-30a1b1c1d2-2a0b3c1d0-2a0b0c0d6+15a1b1c2d0+13a0b0c1d4+9a2b0c0d4-3a1b2c0d2-1a0b4c0d0-21a0b0c2d2-27a2b0c1d2+3a1b2c1d0-3a0b1c0d4+7a0b0c3d0+13a2b0c2d0-6a0b1c1d2-13a2b1c0d2-1a1b3c0d0+11a0b1c2d0+14a2b1c1d0+14a1b0c0d4+13a0b2c0d2-38a1b0c1d2-5a0b2c1d0-3a3b0c0d2+3a2b2c0d0+22a1b0c2d0+7a3b0c1d0-16a1b1c0d2-5a0b3c0d0+18a1b1c1d0-1a0b0c0d4+3a3b1c0d0-4a0b0c1d2-17a2b0c0d2+6a0b0c2d0+19a2b0c1d0+10a0b1c0d2-2a0b1c1d0+11a2b1c0d0-7a1b0c0d2-8a0b2c0d0+11a1b0c1d0+4a3b0c0d0+3a1b1c0d0+3a0b0c0d2-1a0b0c1d0+6a2b0c0d0-5a0b1c0d0-1a0b0c0d0),
(1a0b0c1d6-5a0b0c2d4-1a0b1c0d6+6a0b0c3d2+3a0b1c1d4-1a0b0c4d0+3a0b1c2d2+1a1b0c0d6+2a0b2c0d4-3a0b1c3d0-8a1b0c1d4-9a0b2c1d2+15a1b0c2d2-6a1b0c3d0+6a1b1c1d2+3a0b3c1d0-1a0b0c0d6-8a1b1c2d0+2a0b0c1d4-2a2b0c0d4-1a1b2c0d2+3a0b0c2d2+11a2b0c1d2+2a1b2c1d0+6a0b1c0d4-5a0b0c3d0-9a2b0c2d0-12a0b1c1d2-1a0b1c2d0-7a2b1c1d0-3a1b0c0d4-8a0b2c0d2+11a1b0c1d2+10a0b2c1d0+1a3b0c0d2-11a1b0c2d0-4a3b0c1d0+1a1b1c0d2+2a0b3c0d0-4a1b1c1d0+4a0b0c0d4-7a0b0c1d2+8a2b0c0d2+2a1b2c0d0+1a0b0c2d0-10a2b0c1d0-12a0b1c0d2+9a0b1c1d0-4a2b1c0d0-1a1b0c0d2+7a0b2c0d0-2a1b0c1d0-3a3b0c0d0+2a1b1c0d0-4a0b0c0d2+4a0b0c1d0-3a2b0c0d0+6a0b1c0d0+2a1b0c0d0+1a0b0c0d0),
(1a0b0c2d4-1a0b1c0d6-3a0b0c3d2+3a0b1c1d4+1a0b0c4d0-3a0b1c2d2+1a1b0c0d6+2a0b2c0d4+3a0b1c3d0-2a1b0c1d4-3a0b2c1d2-3a1b0c2d2+1a0b2c2d0+5a1b0c3d0-1a0b3c0d2-6a1b1c1d2-1a0b3c1d0-1a0b0c0d6+7a1b1c2d0+2a0b0c1d4-2a2b0c0d4-2a0b0c2d2+3a2b0c1d2+2a1b2c1d0+7a0b1c0d4+4a0b0c3d0+5a2b0c2d0-11a0b1c1d2+1a2b1c0d2+3a0b1c2d0+5a2b1c1d0-1a1b0c0d4-6a0b2c0d2-2a1b0c1d2+6a1b0c2d0-6a1b1c0d2+10a1b1c1d0+3a0b0c0d4-3a0b0c1d2+3a2b0c0d2+2a1b2c0d0+2a2b0c1d0-9a0b1c0d2+3a0b1c1d0+3a2b1c0d0+1a0b2c0d0+2a1b0c1d0+4a1b1c0d0-3a0b0c0d2-1a2b0c0d0+2a0b1c0d0+1a0b0c0d0),
(1a0b0c3d2-2a0b1c1d4-1a0b0c4d0+3a0b1c2d2+1a1b0c0d6+2a0b2c0d4-3a0b1c3d0-3a1b0c1d4+1a0b2c1d2+5a1b0c2d2-1a0b2c2d0-5a1b1c0d4-5a1b0c3d0-4a0b3c0d2+12a1b1c1d2+2a0b3c1d0-1a0b0c0d6-9a1b1c2d0+2a0b0c1d4-3a2b0c0d4+2a1b2c0d2+1a0b4c0d0+1a0b0c2d2+7a2b0c1d2-3a1b2c1d0+4a0b1c0d4-3a0b0c3d0-7a2b0c2d0-3a0b1c1d2+10a2b1c0d2+1a1b3c0d0-2a0b1c2d0-10a2b1c1d0+1a1b0c0d4-9a0b2c0d2+2a1b0c1d2+5a0b2c1d0+2a3b0c0d2-3a2b2c0d0-7a1b0c2d0-3a3b0c1d0+2a1b1c0d2+5a0b3c0d0-8a1b1c1d0+2a0b0c0d4-3a3b1c0d0-5a0b0c1d2+1a2b0c0d2+1a1b2c0d0+1a0b0c2d0-6a2b0c1d0-6a0b1c0d2+6a0b1c1d0-7a2b1c0d0-2a1b0c0d2+7a0b2c0d0+1a1b0c1d0-1a3b0c0d0+1a1b1c0d0-1a0b0c0d2+3a0b0c1d0+3a0b1c0d0+1a1b0c0d0),
(1a0b0c4d0-3a0b1c2d2+1a0b2c0d4+3a0b1c3d0+2a1b0c1d4-6a1b0c2d2+1a0b2c2d0-2a1b1c0d4+5a1b0c3d0-1a0b3c0d2-2a0b3c1d0-1a0b0c0d6+9a1b1c2d0+5a0b0c1d4+1a2b0c0d4+1a1b2c0d2-8a0b0c2d2-8a2b0c1d2+1a0b1c0d4+3a0b0c3d0+8a2b0c2d0-2a0b1c1d2+1a2b1c0d2+4a0b1c2d0+6a2b1c1d0+4a1b0c0d4-1a0b2c0d2-13a1b0c1d2-4a0b2c1d0-1a3b0c0d2+11a1b0c2d0+4a3b0c1d0-1a1b1c0d2-1a0b3c0d0+6a1b1c1d0+1a0b0c0d4-5a0b0c1d2-7a2b0c0d2+3a0b0c2d0+11a2b0c1d0+2a0b1c0d2-2a0b1c1d0+4a2b1c0d0-5a1b0c0d2-3a0b2c0d0+6a1b0c1d0+3a3b0c0d0+2a0b0c0d2+4a2b0c0d0-3a0b1c0d0-1a0b0c0d0),
(1a0b1c0d6-5a0b1c1d4+6a0b1c2d2-1a1b0c0d6-1a0b2c0d4-1a0b1c3d0+4a1b0c1d4+6a0b2c1d2-3a1b0c2d2-3a0b2c2d0-2a1b1c0d4-2a0b3c0d2+4a1b1c1d2-1a0b3c1d0+1a0b0c0d6-4a1b1c2d0-4a0b0c1d4+3a2b0c0d4+5a1b2c0d2+1a0b4c0d0+4a0b0c2d2-7a2b0c1d2-4a1b2c1d0-5a0b1c0d4-1a0b0c3d0+1a2b0c2d0+15a0b1c1d2+1a2b1c0d2-8a0b1c2d0-1a2b1c1d0+1a1b0c0d4+3a0b2c0d2-6a0b2c1d0-3a3b0c0d2-3a2b2c0d0-2a1b0c2d0+2a3b0c1d0+4a1b1c0d2+2a0b3c0d0-8a1b1c1d0-4a0b0c0d4+9a0b0c1d2-4a2b0c0d2-4a1b2c0d0-4a0b0c2d0+2a2b0c1d0+9a0b1c0d2+1a4b0c0d0-9a0b1c1d0-1a2b1c0d0+3a1b0c0d2-2a0b2c0d0-4a1b0c1d0+2a3b0c0d0-4a1b1c0d0+4a0b0c0d2-4a0b0c1d0-4a0b1c0d0-2a1b0c0d0-1a0b0c0d0),
(1a0b1c1d4-3a0b1c2d2-1a1b0c0d6-1a0b2c0d4+1a0b1c3d0+4a1b0c1d4+1a0b2c1d2-4a1b0c2d2+2a0b2c2d0+3a1b1c0d4+1a1b0c3d0+2a0b3c0d2-9a1b1c1d2-1a0b3c1d0+1a0b0c0d6+6a1b1c2d0-3a0b0c1d4+3a2b0c0d4-3a1b2c0d2-1a0b4c0d0+1a0b0c2d2-9a2b0c1d2+2a1b2c1d0-4a0b1c0d4+1a0b0c3d0+4a2b0c2d0+5a0b1c1d2-4a2b1c0d2+4a0b1c2d0+7a2b1c1d0+1a1b0c0d4+7a0b2c0d2-3a1b0c1d2-4a0b2c1d0-2a3b0c0d2+3a2b2c0d0+5a1b0c2d0+4a3b0c1d0-1a1b1c0d2-4a0b3c0d0+4a1b1c1d0-3a0b0c0d4+1a3b1c0d0+5a0b0c1d2-7a2b0c0d2+7a2b0c1d0+8a0b1c0d2-5a0b1c1d0+4a2b1c0d0+2a1b0c0d2-6a0b2c0d0+3a3b0c0d0-2a1b1c0d0+3a0b0c0d2-2a0b0c1d0+3a2b0c0d0-4a0b1c0d0-2a1b0c0d0-1a0b0c0d0),
(1a0b1c2d2-1a0b2c0d4-1a0b1c3d0-1a1b0c1d4+1a0b2c1d2+2a1b0c2d2-2a0b2c2d0+3a1b1c0d4-1a1b0c3d0+2a0b3c0d2-1a1b1c1d2+1a0b0c0d6-4a1b1c2d0-4a0b0c1d4-2a2b0c0d4-4a1b2c0d2+5a0b0c2d2+5a2b0c1d2-4a0b1c0d4-2a0b0c3d0-3a2b0c2d0+9a0b1c1d2-2a2b1c0d2-6a0b1c2d0-2a2b1c1d0-2a1b0c0d4+5a0b2c0d2+9a1b0c1d2-2a0b2c1d0+4a3b0c0d2+1a2b2c0d0-7a1b0c2d0-3a3b0c1d0+1a1b1c0d2-1a0b3c0d0-6a1b1c1d0-2a0b0c0d4+6a0b0c1d2+4a2b0c0d2-4a0b0c2d0-7a2b0c1d0+5a0b1c0d2-1a4b0c0d0-6a0b1c1d0-1a2b1c0d0+2a1b0c0d2-2a0b2c0d0-6a1b0c1d0-3a3b0c0d0-2a1b1c0d0+1a0b0c0d2-2a0b0c1d0-2a2b0c0d0-1a0b1c0d0),
(1a0b1c3d0-2a0b2c1d2-1a1b0c2d2+2a0b2c2d0+2a1b1c0d4+1a1b0c3d0+1a0b3c0d2-4a1b1c1d2+4a1b1c2d0+1a0b0c1d4-1a2b0c0d4-2a1b2c0d2-1a0b4c0d0-3a0b0c2d2+1a2b0c1d2+3a1b2c1d0-1a0b1c0d4+2a0b0c3d0+2a2b0c2d0-1a0b1c1d2-4a2b1c0d2+4a0b1c2d0+4a2b1c1d0+1a1b0c0d4+3a0b2c0d2-4a1b0c1d2+2a3b0c0d2+3a2b2c0d0+5a1b0c2d0-4a1b1c0d2-3a0b3c0d0+6a1b1c1d0+1a3b1c0d0-1a0b0c1d2+1a2b0c0d2+1a1b2c0d0+2a0b0c2d0+2a2b0c1d0+2a0b1c0d2-1a4b0c0d0+4a2b1c0d0-3a1b0c0d2-3a0b2c0d0+3a1b0c1d0-1a3b0c0d0+2a1b1c0d0+1a2b0c0d0-1a0b1c0d0+1a1b0c0d0),
(1a0b2c0d4-1a1b0c1d4-3a0b2c1d2+3a1b0c2d2+1a0b2c2d0-1a1b1c0d4-1a1b0c3d0-1a0b3c0d2+3a1b1c1d2+2a0b3c1d0-2a1b1c2d0+1a0b0c1d4-2a0b0c2d2+3a2b0c1d2+2a1b2c1d0+1a0b1c0d4-4a2b0c2d0-5a0b1c1d2+1a2b1c0d2+3a0b1c2d0-4a2b1c1d0-5a0b2c0d2+8a0b2c1d0-2a1b0c2d0-3a3b0c1d0+2a1b1c0d2+2a0b3c0d0+2a1b1c1d0-2a0b0c1d2+2a2b0c0d2+2a1b2c0d0+3a0b0c2d0-4a2b0c1d0-4a0b1c0d2+8a0b1c1d0-3a2b1c0d0-2a1b0c0d2+6a0b2c0d0+2a1b0c1d0-2a3b0c0d0+2a1b1c0d0+2a0b0c1d0-1a2b0c0d0+4a0b1c0d0+2a1b0c0d0),
(1a0b2c1d2-1a1b0c2d2-1a0b2c2d0-1a1b1c0d4+1a1b0c3d0-1a0b3c0d2+3a1b1c1d2-1a0b3c1d0+1a0b0c1d4+1a2b0c0d4+3a1b2c0d2+1a0b4c0d0-2a0b0c2d2-3a2b0c1d2-4a1b2c1d0+1a0b0c3d0+3a2b0c2d0-1a0b1c1d2-1a0b1c2d0-1a1b0c0d4-1a1b0c1d2-3a0b2c1d0-2a3b0c0d2-3a2b2c0d0+3a1b0c2d0+3a3b0c1d0+3a1b1c0d2+2a0b3c0d0-2a1b1c1d0-3a1b2c0d0+4a2b0c1d0+1a0b1c0d2+1a4b0c0d0-1a0b1c1d0+1a1b0c0d2-1a0b2c0d0+2a3b0c0d0-2a1b1c0d0-1a0b0c1d0-2a0b1c0d0-1a1b0c0d0),
(1a0b2c2d0-1a1b0c3d0-1a0b3c0d2+1a0b3c1d0+1a1b2c0d2+1a0b0c2d2+1a2b0c1d2+1a1b2c1d0+1a0b1c0d4-1a0b0c3d0-2a2b0c2d0-5a0b1c1d2+3a0b1c2d0-2a2b1c1d0-1a1b0c0d4-1a0b2c0d2+2a1b0c1d2+5a0b2c1d0-1a1b0c2d0-1a3b0c1d0-2a1b1c0d2+1a0b3c0d0+2a1b1c1d0+1a0b0c0d4-3a0b0c1d2+2a2b0c0d2+1a1b2c0d0+2a0b0c2d0-2a2b0c1d0-3a0b1c0d2+7a0b1c1d0-1a2b1c0d0+1a1b0c0d2+3a0b2c0d0+1a1b0c1d0-1a3b0c0d0+2a1b1c0d0-3a0b0c0d2+3a0b0c1d0-1a2b0c0d0+3a0b1c0d0+1a1b0c0d0+1a0b0c0d0),
(1a0b3c0d2-2a1b1c1d2-1a0b3c1d0+2a1b1c2d0+1a2b0c0d4-1a1b2c0d2-1a0b4c0d0+1a0b0c2d2-1a2b0c1d2+2a1b2c1d0-1a0b1c0d4-1a0b0c3d0+1a2b0c2d0+3a0b1c1d2-2a2b1c0d2-2a0b1c2d0+4a2b1c1d0-1a1b0c0d4+3a0b2c0d2-5a0b2c1d0-2a3b0c0d2+3a2b2c0d0-1a1b0c2d0+1a3b0c1d0+4a1b1c0d2-4a0b3c0d0-2a1b1c1d0+1a3b1c0d0+1a0b0c1d2-1a1b2c0d0-2a0b0c2d0+1a2b0c1d0+3a0b1c0d2-7a0b1c1d0+1a2b1c0d0+2a1b0c0d2-5a0b2c0d0-2a1b0c1d0+1a3b0c0d0-4a1b1c0d0-1a0b0c1d0-2a0b1c0d0-1a1b0c0d0),
(1a0b3c1d0-2a1b1c2d0-1a1b2c0d2+2a2b0c1d2+1a1b2c1d0+1a0b0c3d0-2a2b0c2d0+1a2b1c0d2-2a2b1c1d0-1a1b0c0d4-1a0b2c0d2+1a1b0c1d2+3a0b2c1d0-1a1b0c2d0-3a3b0c1d0+1a1b1c0d2+1a0b3c0d0+2a1b1c1d0+1a0b0c0d4-2a0b0c1d2+2a2b0c0d2+1a1b2c0d0-4a2b0c1d0-3a0b1c0d2+3a0b1c1d0-2a2b1c0d0+1a1b0c0d2+3a0b2c0d0+1a1b0c1d0-2a3b0c0d0+2a1b1c0d0-2a0b0c0d2+1a0b0c1d0-2a2b0c0d0+3a0b1c0d0+1a1b0c0d0+1a0b0c0d0),
(1a0b4c0d0-3a1b2c1d0+1a2b0c2d0+2a2b1c0d2+2a0b1c2d0-2a2b1c1d0-2a0b2c0d2-2a1b0c1d2+2a0b2c1d0-1a3b0c0d2-3a2b2c0d0+2a1b0c2d0+2a3b0c1d0+2a1b1c0d2+3a0b3c0d0+1a0b0c0d4-1a3b1c0d0-3a0b0c1d2-1a2b0c0d2+2a0b0c2d0+3a2b0c1d0-3a0b1c0d2+1a4b0c0d0+4a0b1c1d0-2a2b1c0d0-1a1b0c0d2+3a0b2c0d0+3a1b0c1d0+2a3b0c0d0-1a0b0c0d2+2a0b0c1d0+1a2b0c0d0+1a0b1c0d0),
(1a1b0c0d7-6a1b0c1d5+10a1b0c2d3-1a1b1c0d5-4a1b0c3d1+8a1b1c1d3-1a0b0c0d7-9a1b1c2d1+5a0b0c1d5-5a2b0c0d5-3a1b2c0d3-6a0b0c2d3+18a2b0c1d3+2a0b1c0d5+1a0b0c3d1-15a2b0c2d1-10a0b1c1d3+8a2b1c0d3+3a1b3c0d1+6a0b1c2d1-14a2b1c1d1+2a1b0c0d5+1a0b2c0d3-2a1b0c1d3+5a0b2c1d1+6a3b0c0d3-1a2b2c0d1-4a1b0c2d1-12a3b0c1d1-8a1b1c0d3-2a0b3c0d1+8a1b1c1d1+3a0b0c0d5-7a3b1c0d1-11a0b0c1d3+4a2b0c0d3+6a1b2c0d1+9a0b0c2d1-9a2b0c1d1-5a0b1c0d3-1a4b0c0d1+12a0b1c1d1-2a2b1c0d1-7a1b0c0d3-1a0b2c0d1+10a1b0c1d1-4a3b0c0d1+11a1b1c0d1-3a0b0c0d3+5a0b0c1d1+4a0b1c0d1+4a1b0c0d1+1a0b0c0d1),
(1a1b0c1d5-4a1b0c2d3-1a1b1c0d5+3a1b0c3d1+2a1b1c1d3-1a0b0c0d7+3a1b1c2d1+5a0b0c1d5+1a2b0c0d5+2a1b2c0d3-7a0b0c2d3-6a2b0c1d3-5a1b2c1d1+2a0b1c0d5+3a0b0c3d1+8a2b0c2d1-8a0b1c1d3-1a2b1c0d3-1a1b3c0d1+5a0b1c2d1+2a2b1c1d1+3a1b0c0d5-13a1b0c1d3+1a0b2c1d1-1a3b0c0d3+1a2b2c0d1+12a1b0c2d1+5a3b0c1d1+1a1b1c0d3+4a1b1c1d1+3a0b0c0d5+1a3b1c0d1-9a0b0c1d3-7a2b0c0d3-5a1b2c0d1+6a0b0c2d1+13a2b0c1d1-5a0b1c0d3-1a4b0c0d1+8a0b1c1d1+2a2b1c0d1-6a1b0c0d3+11a1b0c1d1+5a3b0c0d1+1a1b1c0d1-2a0b0c0d3+3a0b0c1d1+7a2b0c0d1+2a0b1c0d1+1a1b0c0d1)";
o3 : Ideal of R
i4 : C2=ideal
"(1a1b0c2d3-1a1b1c0d5-2a1b0c3d1+2a1b1c1d3-2a1b1c2d1-1a0b0c1d5+1a2b0c0d5+2a1b2c0d3+3a0b0c2d3-1a2b0c1d3+2a0b1c0d5-2a0b0c3d1-2a2b0c2d1-3a0b1c1d3-1a2b1c0d3-1a1b3c0d1-2a2b1c1d1-3a1b0c0d5-5a0b2c0d3+7a1b0c1d3+5a0b2c1d1-2a3b0c0d3-1a2b2c0d1-4a1b0c2d1+2a3b0c1d1+7a1b1c0d3+2a0b3c0d1-6a1b1c1d1+2a0b0c0d5+3a3b1c0d1-3a0b0c1d3+4a2b0c0d3-2a1b2c0d1-5a2b0c1d1-9a0b1c0d3+1a4b0c0d1+8a0b1c1d1-6a2b1c0d1+2a1b0c0d3+9a0b2c0d1-4a1b0c1d1-2a3b0c0d1-3a1b1c0d1-4a0b0c0d3+5a0b0c1d1-2a2b0c0d1+8a0b1c0d1+2a1b0c0d1+1a0b0c0d1),
(1a1b0c3d1-2a1b1c1d3+1a1b1c2d1+1a2b0c0d5+2a1b2c0d3-1a0b0c2d3-2a2b0c1d3+1a0b1c0d5+1a0b0c3d1+1a2b0c2d1-3a2b1c0d3-2a1b3c0d1+6a2b1c1d1-3a1b0c0d5-3a0b2c0d3+7a1b0c1d3-3a3b0c0d3-3a1b0c2d1+2a3b0c1d1+3a1b1c0d3+2a0b3c0d1-2a1b1c1d1+1a0b0c0d5+4a3b1c0d1-1a0b0c1d3+8a2b0c0d3-2a1b2c0d1-8a2b0c1d1-5a0b1c0d3+2a4b0c0d1+2a0b1c1d1-4a2b1c0d1+2a1b0c0d3+5a0b2c0d1-6a1b0c1d1-4a3b0c0d1+2a1b1c0d1-2a0b0c0d3+2a0b0c1d1-7a2b0c0d1+4a0b1c0d1+2a1b0c0d1+1a0b0c0d1),
(1a1b1c0d5-4a1b1c1d3+3a1b1c2d1-1a0b0c1d5-1a2b0c0d5-1a1b2c0d3+4a0b0c2d3+3a2b0c1d3+4a1b2c1d1-3a0b0c3d1-1a2b0c2d1+1a0b1c1d3-1a2b1c0d3-1a1b3c0d1-4a0b1c2d1+2a2b1c1d1+1a1b0c0d5+1a1b0c1d3+1a0b2c1d1+2a3b0c0d3+1a2b2c0d1-6a1b0c2d1-4a3b0c1d1-5a1b1c0d3+4a1b1c1d1+1a3b1c0d1+1a0b0c1d3+2a2b0c0d3+4a1b2c0d1-3a0b0c2d1-5a2b0c1d1+1a0b1c0d3-1a4b0c0d1+2a2b1c0d1-3a1b0c0d3-1a0b2c0d1+2a1b0c1d1-4a3b0c0d1+3a1b1c0d1-1a0b0c0d3+1a0b0c1d1+4a1b0c0d1+1a0b0c0d1),
(1a1b1c1d3-2a1b1c2d1-1a2b0c0d5-1a1b2c0d3-1a0b0c2d3+3a2b0c1d3+2a0b0c3d1-2a2b0c2d1+1a0b1c1d3+3a2b1c0d3+2a1b3c0d1-4a2b1c1d1+2a1b0c0d5-6a1b0c1d3-2a0b2c1d1+2a3b0c0d3-2a2b2c0d1+4a1b0c2d1-4a3b0c1d1-6a1b1c0d3+8a1b1c1d1-1a0b0c0d5-4a3b1c0d1+3a0b0c1d3-2a2b0c0d3+4a1b2c0d1-2a0b0c2d1+4a2b0c1d1+3a0b1c0d3-4a0b1c1d1+4a2b1c0d1-2a1b0c0d3-2a0b2c0d1+4a1b0c1d1+4a1b1c0d1+2a0b0c0d3-4a0b0c1d1-4a0b1c0d1),
(1a1b1c2d1-1a1b2c0d3-1a2b0c1d3-1a0b0c3d1+1a2b0c2d1+1a0b1c1d3+3a2b1c0d3+1a1b3c0d1-2a2b1c1d1+1a1b0c0d5-3a1b0c1d3-1a0b2c1d1-1a3b0c0d3-1a2b2c0d1+2a1b0c2d1+2a3b0c1d1-3a1b1c0d3-1a0b0c0d5-3a3b1c0d1+2a0b0c1d3-3a2b0c0d3+4a1b2c0d1+1a0b0c2d1+7a2b0c1d1+2a0b1c0d3+1a4b0c0d1-4a0b1c1d1+2a1b0c0d3-1a0b2c0d1+4a3b0c0d1-1a1b1c0d1+2a0b0c0d3-3a0b0c1d1-2a0b1c0d1-4a1b0c0d1-1a0b0c0d1),
(1a1b2c0d3-1a2b0c1d3-2a1b2c1d1+2a2b0c2d1-1a0b1c1d3-1a2b1c0d3-1a1b3c0d1+2a0b1c2d1+2a2b1c1d1+1a1b0c0d5-1a1b0c1d3+1a0b2c1d1+1a2b2c0d1+2a3b0c1d1-1a1b1c0d3-2a1b1c1d1-1a0b0c0d5+1a3b1c0d1+1a0b0c1d3-2a2b0c0d3-2a1b2c0d1+1a2b0c1d1+3a0b1c0d3-1a4b0c0d1+2a2b1c0d1-1a0b2c0d1+2a3b0c0d1-1a1b1c0d1+2a0b0c0d3-1a0b0c1d1+2a2b0c0d1-4a0b1c0d1-2a1b0c0d1-1a0b0c0d1),
(1a1b2c1d1-1a2b0c2d1-1a2b1c0d3-1a1b3c0d1-1a0b1c2d1+2a2b1c1d1+3a1b0c1d3+1a0b2c1d1+1a3b0c0d3+1a2b2c0d1-3a1b0c2d1-1a3b0c1d1-1a1b1c0d3-2a1b1c1d1-1a0b0c0d5+1a3b1c0d1+1a0b0c1d3-1a2b0c0d3+1a1b2c0d1-5a2b0c1d1+2a0b1c0d3-1a4b0c0d1+2a2b1c0d1+3a1b0c0d3-1a0b2c0d1-3a1b0c1d1-1a3b0c0d1-1a1b1c0d1+2a0b0c0d3-1a0b0c1d1-2a0b1c0d1-3a1b0c0d1-1a0b0c0d1),
(1a1b3c0d1-2a2b1c1d1-1a0b2c1d1+1a3b0c0d3-1a2b2c0d1+2a1b0c2d1+2a1b1c1d1-1a3b1c0d1-1a0b0c1d3-3a2b0c0d3+3a2b0c1d1+1a0b1c0d3-1a4b0c0d1+4a2b1c0d1+1a1b0c0d3-1a0b2c0d1-2a1b0c1d1+2a3b0c0d1-3a1b1c0d1+1a0b0c0d3+1a0b0c1d1+2a2b0c0d1-2a0b1c0d1-2a1b0c0d1-1a0b0c0d1),
(1a2b0c0d6-5a2b0c1d4-1a0b1c0d6+6a2b0c2d2+5a0b1c1d4-1a2b1c0d4-1a2b0c3d0-6a0b1c2d2+6a2b1c1d2-1a1b0c0d6+1a0b2c0d4+1a0b1c3d0-3a2b1c2d0+4a1b0c1d4-6a0b2c1d2-4a3b0c0d4-2a2b2c0d2-3a1b0c2d2+3a0b2c2d0+11a3b0c1d2-1a2b2c1d0+6a1b1c0d4+2a0b3c0d2-5a3b0c2d0-18a1b1c1d2+1a0b3c1d0+5a3b1c0d2+1a2b3c0d0+6a1b1c2d0+1a0b0c1d4-6a3b1c1d0+1a2b0c0d4-5a1b2c0d2-1a0b4c0d0-3a0b0c2d2+8a1b2c1d0+2a0b1c0d4+1a0b0c3d0+4a4b0c0d2-4a2b0c2d0-9a0b1c1d2-5a4b0c1d0-6a2b1c0d2+8a0b1c2d0+5a2b1c1d0+3a1b0c0d4-3a4b1c0d0-9a1b0c1d2+5a0b2c1d0+3a3b0c0d2+5a2b2c0d0+6a1b0c2d0-6a3b0c1d0-13a1b1c0d2-3a0b3c0d0-1a5b0c0d0+18a1b1c1d0-2a3b1c0d0-2a0b0c1d2-4a2b0c0d2+5a1b2c0d0+4a0b0c2d0+5a2b0c1d0-2a0b1c0d2-2a4b0c0d0+5a0b1c1d0+8a2b1c0d0-3a1b0c0d2-1a0b2c0d0+6a1b0c1d0+8a1b1c0d0+1a0b0c1d0+2a2b0c0d0+1a0b1c0d0+1a1b0c0d0),
(1a2b0c1d4-3a2b0c2d2-1a0b1c1d4-1a2b1c0d4+1a2b0c3d0+3a0b1c2d2+1a2b1c1d2-1a1b0c0d6+1a0b2c0d4-1a0b1c3d0+2a2b1c2d0+4a1b0c1d4-1a0b2c1d2+1a3b0c0d4+2a2b2c0d2-4a1b0c2d2-2a0b2c2d0-4a3b0c1d2-1a2b2c1d0+1a1b1c0d4+1a1b0c3d0-2a0b3c0d2+4a3b0c2d0-1a1b1c1d2+1a0b3c1d0+1a0b0c0d6-2a3b1c0d2-1a2b3c0d0-2a1b1c2d0-4a0b0c1d4+2a3b1c1d0+2a2b0c0d4+1a1b2c0d2+1a0b4c0d0+4a0b0c2d2-9a2b0c1d2-2a1b2c1d0-1a0b1c0d4-1a0b0c3d0-1a4b0c0d2+7a2b0c2d0+6a0b1c1d2+3a4b0c1d0+2a2b1c0d2-6a0b1c2d0-1a2b1c1d0-4a0b2c0d2+2a4b1c0d0+1a1b0c1d2+1a0b2c1d0-3a3b0c0d2-4a2b2c0d0-1a1b0c2d0+7a3b0c1d0+3a1b1c0d2+4a0b3c0d0-6a1b1c1d0-3a0b0c0d4+2a3b1c0d0+8a0b0c1d2-3a2b0c0d2-2a1b2c0d0-4a0b0c2d0+3a2b0c1d0+1a0b1c0d2+1a4b0c0d0-3a0b1c1d0-4a2b1c0d0+3a1b0c0d2+4a0b2c0d0-4a1b0c1d0+3a3b0c0d0-4a1b1c0d0+3a0b0c0d2-3a0b0c1d0+1a2b0c0d0-2a1b0c0d0-1a0b0c0d0),
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(1a2b2c1d0-1a3b0c2d0-1a0b3c1d0-1a3b1c0d2+1a3b1c1d0+1a1b2c0d2+3a2b0c1d2-1a1b2c1d0+1a4b0c0d2-3a2b0c2d0-1a4b0c1d0-2a2b1c0d2+2a2b1c1d0-2a1b0c0d4+1a0b2c0d2+3a1b0c1d2-3a0b2c1d0+1a2b2c0d0-2a1b0c2d0-4a3b0c1d0+2a1b1c0d2-1a0b3c0d0+1a0b0c0d4+1a3b1c0d0-1a0b0c1d2+3a2b0c0d2-1a1b2c0d0-5a2b0c1d0-1a4b0c0d0-1a0b1c1d0+2a2b1c0d0+2a1b0c0d2-3a0b2c0d0-1a1b0c1d0-3a3b0c0d0-2a0b0c0d2+1a0b0c1d0-2a2b0c0d0-1a0b1c0d0+1a1b0c0d0+1a0b0c0d0),
(1a2b3c0d0-2a3b1c1d0-1a0b4c0d0+1a1b2c1d0+1a4b0c0d2+2a2b0c2d0-1a4b0c1d0-1a2b1c0d2-1a0b1c2d0+1a2b1c1d0+2a0b2c0d2-2a4b1c0d0-2a1b0c1d2-2a0b2c1d0-1a3b0c0d2+3a2b2c0d0+2a1b0c2d0+3a3b0c1d0-3a0b3c0d0-1a5b0c0d0+2a3b1c0d0+1a0b0c1d2-3a2b0c0d2-2a1b2c0d0-1a0b0c2d0+6a2b0c1d0+2a0b1c0d2+1a4b0c0d0-4a0b1c1d0+2a2b1c0d0+2a1b0c0d2-3a0b2c0d0-1a1b0c1d0+4a3b0c0d0-4a1b1c0d0-2a0b0c1d0-1a0b1c0d0-2a1b0c0d0),
(1a3b0c0d5-4a3b0c1d3-2a1b1c0d5+3a3b0c2d1+8a1b1c1d3-1a3b1c0d3-6a1b1c2d1+1a0b0c1d5+4a3b1c1d1-1a2b0c0d5+2a1b2c0d3-4a0b0c2d3+3a2b0c1d3-8a1b2c1d1+1a0b1c0d5+3a0b0c3d1-3a4b0c0d3-1a3b2c0d1-1a2b0c2d1-4a0b1c1d3+6a4b0c1d1+8a2b1c0d3+2a1b3c0d1+5a0b1c2d1-16a2b1c1d1-2a0b2c0d3+2a4b1c0d1-2a1b0c1d3+3a0b2c1d1-5a2b2c0d1+4a1b0c2d1+2a1b1c0d3+1a0b3c0d1+2a5b0c0d1-4a1b1c1d1-4a3b1c0d1-2a0b0c1d3+3a2b0c0d3+4a0b0c2d1-4a2b0c1d1-4a0b1c0d3+2a4b0c0d1+8a0b1c1d1-8a2b1c0d1+4a0b2c0d1+4a1b0c1d1-2a3b0c0d1+1a0b0c1d1-2a2b0c0d1+3a0b1c0d1),
(1a3b0c1d3-2a3b0c2d1-2a1b1c1d3-1a3b1c0d3+4a1b1c2d1-1a2b0c0d5+2a1b2c0d3+1a0b0c2d3+3a2b0c1d3+1a0b1c0d5-2a0b0c3d1+1a4b0c0d3+2a3b2c0d1-2a2b0c2d1-4a0b1c1d3-2a4b0c1d1-4a1b3c0d1+2a0b1c2d1+2a2b1c1d1+1a1b0c0d5-2a0b2c0d3-1a4b1c0d1-2a1b0c1d3+4a0b2c1d1+1a3b0c0d3+1a2b2c0d1-3a3b0c1d1-2a1b1c0d3+1a0b3c0d1-1a5b0c0d1+4a1b1c1d1+1a3b1c0d1-2a0b0c1d3-1a1b2c0d1+3a0b0c2d1-1a2b0c1d1-2a0b1c0d3+6a0b1c1d1+1a2b1c0d1-2a1b0c0d3+2a0b2c0d1+4a1b0c1d1+4a1b1c0d1+2a0b0c1d1-1a2b0c0d1+1a0b1c0d1+1a1b0c0d1)";
o4 : Ideal of R
i5 : C3=ideal
"(1a3b0c2d1-1a3b1c0d3-2a1b1c2d1+2a1b2c0d3-1a2b0c1d3+1a0b0c3d1+1a4b0c0d3+1a3b2c0d1+1a2b0c2d1-1a4b0c1d1-2a1b3c0d1-1a0b1c2d1+1a2b1c1d1-2a0b2c0d3+1a4b1c0d1+2a1b0c1d3+2a0b2c1d1-1a3b0c0d3-4a2b2c0d1-2a1b0c2d1+1a3b0c1d1+2a0b3c0d1-2a5b0c0d1+5a3b1c0d1-2a1b2c0d1-2a0b0c2d1-1a2b0c1d1+2a0b1c1d1-1a2b1c0d1+4a0b2c0d1-2a1b0c1d1+2a3b0c0d1-2a1b1c0d1),
(1a3b1c0d3-2a3b1c1d1-2a1b2c0d3-1a2b0c1d3+4a1b2c1d1-1a4b0c0d3-1a3b2c0d1+2a2b0c2d1+2a0b1c1d3+1a4b0c1d1+2a2b1c0d3+2a1b3c0d1-4a0b1c2d1-1a2b1c1d1+1a1b0c0d5+1a4b1c0d1-4a1b0c1d3-2a0b2c1d1+1a3b0c0d3-2a2b2c0d1+3a1b0c2d1+1a3b0c1d1-4a1b1c0d3+1a5b0c0d1+4a1b1c1d1-1a0b0c0d5-5a3b1c0d1+4a0b0c1d3-2a2b0c0d3+7a1b2c0d1-3a0b0c2d1+4a2b0c1d1+2a0b1c0d3-1a4b0c0d1-6a0b1c1d1+2a2b1c0d1-1a0b2c0d1+1a3b0c0d1+2a1b1c0d1+2a0b0c0d3-4a0b0c1d1+3a2b0c0d1-2a0b1c0d1-3a1b0c0d1-1a0b0c0d1),
(1a3b1c1d1-2a1b2c1d1-1a4b0c0d3-1a3b2c0d1-1a2b0c2d1+1a4b0c1d1+2a2b1c0d3+2a1b3c0d1+2a0b1c2d1-1a2b1c1d1+1a4b1c0d1-2a0b2c1d1+2a3b0c0d3-2a2b2c0d1-2a3b0c1d1-4a1b1c0d3+2a5b0c0d1+4a1b1c1d1-6a3b1c0d1-1a2b0c0d3+4a1b2c0d1+1a2b0c1d1+2a0b1c0d3-2a4b0c0d1-2a0b1c1d1+5a2b1c0d1-2a0b2c0d1-2a3b0c0d1+4a1b1c0d1+2a2b0c0d1-4a0b1c0d1),
(1a3b2c0d1-1a4b0c1d1-2a1b3c0d1+1a2b1c1d1-1a4b1c0d1+2a0b2c1d1+1a3b0c0d3+2a2b2c0d1-1a1b0c2d1+1a3b0c1d1-4a1b1c1d1-1a3b1c0d1-3a2b0c0d3+1a0b0c2d1+3a2b0c1d1+1a2b1c0d1+2a1b0c0d3+1a3b0c0d1-2a0b0c1d1+1a2b0c0d1-2a1b0c0d1),
(1a4b0c0d4-3a4b0c1d2-3a2b1c0d4+1a4b0c2d0+9a2b1c1d2+1a0b2c0d4-1a4b1c0d2-3a2b1c2d0+2a1b0c1d4-3a0b2c1d2+2a4b1c1d0-1a3b0c0d4+3a2b2c0d2-6a1b0c2d2+1a0b2c2d0+2a3b0c1d2-6a2b2c1d0+2a1b1c0d4+2a1b0c3d0-1a0b3c0d2-2a5b0c0d2-6a1b1c1d2+2a0b3c1d0-1a0b0c0d6+3a5b0c1d0+8a3b1c0d2+4a1b1c2d0+3a0b0c1d4-10a3b1c1d0-6a1b2c0d2-2a0b0c2d2+1a5b1c0d0-3a2b0c1d2+5a1b2c1d0+2a0b1c0d4+1a0b0c3d0-1a4b0c0d2-4a3b2c0d0+5a2b0c2d0-5a0b1c1d2+2a4b0c1d0+4a2b1c0d2+3a1b3c0d0+3a0b1c2d0+1a6b0c0d0-7a2b1c1d0+3a1b0c0d4-3a0b2c0d2-3a4b1c0d0-11a1b0c1d2+6a0b2c1d0+2a3b0c0d2+9a1b0c2d0-2a3b0c1d0-9a1b1c0d2+1a0b3c0d0+2a5b0c0d0+14a1b1c1d0+3a0b0c0d4-8a3b1c0d0-6a0b0c1d2-2a2b0c0d2+8a1b2c0d0+2a0b0c2d0+7a2b0c1d0-5a0b1c0d2+6a0b1c1d0+1a2b1c0d0-5a1b0c0d2+3a0b2c0d0+9a1b0c1d0-1a3b0c0d0+7a1b1c0d0-3a0b0c0d2+2a0b0c1d0+1a2b0c0d0+3a0b1c0d0+2a1b0c0d0+1a0b0c0d0),
(1a4b0c1d2-1a4b0c2d0-3a2b1c1d2-1a4b1c0d2+3a2b1c2d0+1a0b2c1d2-1a4b1c1d0-1a3b0c0d4+3a2b2c0d2+2a1b0c2d2-1a0b2c2d0+2a3b0c1d2+3a2b2c1d0+2a1b1c0d4-2a1b0c3d0-1a0b3c0d2+1a5b0c0d2+1a4b2c0d0-1a3b0c2d0-6a1b1c1d2-1a0b3c1d0-2a5b0c1d0-1a3b1c0d2-3a2b3c0d0-2a0b0c1d4+5a3b1c1d0+1a2b0c0d4-3a1b2c0d2+1a0b4c0d0+4a0b0c2d2+2a1b2c1d0-2a0b0c3d0+2a4b0c0d2-3a2b0c2d0+5a0b1c1d2-3a4b0c1d0-6a2b1c0d2-4a0b1c2d0-1a6b0c0d0+9a2b1c1d0-1a1b0c0d4+2a4b1c0d0+5a1b0c1d2-3a0b2c1d0-1a3b0c0d2-6a1b0c2d0-1a3b0c1d0+2a0b3c0d0-2a5b0c0d0-2a1b1c1d0+5a3b1c0d0+3a0b0c1d2+1a1b2c0d0-3a0b0c2d0-3a2b0c1d0+1a0b1c0d2-5a0b1c1d0+3a2b1c0d0+2a1b0c0d2-1a0b2c0d0-4a1b0c1d0+1a3b0c0d0-2a1b1c0d0-1a0b0c1d0-1a2b0c0d0-2a0b1c0d0-1a1b0c0d0),
(1a4b0c2d0-1a4b1c0d2-3a2b1c2d0+1a4b1c1d0+3a2b2c0d2+1a0b2c2d0-1a3b0c1d2-3a2b2c1d0+2a1b0c3d0-1a0b3c0d2+1a3b0c2d0+1a0b3c1d0+1a5b0c1d0+1a3b1c0d2-3a3b1c1d0-2a1b2c0d2-2a0b0c2d2+1a5b1c0d0+1a2b0c1d2+1a1b2c1d0+1a0b1c0d4+2a0b0c3d0-4a3b2c0d0+1a2b0c2d0-2a0b1c1d2+1a4b0c1d0+3a1b3c0d0+3a0b1c2d0-2a2b1c1d0-1a0b2c0d2+1a4b1c0d0-1a1b0c1d2+2a0b2c1d0-3a2b2c0d0+2a1b0c2d0-1a3b0c1d0-2a1b1c0d2+1a0b3c0d0+6a1b1c1d0-1a3b1c0d0+4a1b2c0d0+1a0b0c2d0-2a0b1c0d2+3a0b1c1d0+2a0b2c0d0+1a1b0c1d0+1a1b1c0d0+1a0b1c0d0),
(1a4b1c0d2-1a4b1c1d0-3a2b2c0d2-1a3b0c1d2+3a2b2c1d0+1a0b3c0d2-1a5b0c0d2-1a4b2c0d0+1a3b0c2d0+4a1b1c1d2-1a0b3c1d0+3a3b1c0d2+3a2b3c0d0-4a1b1c2d0+1a3b1c1d0+1a2b0c0d4-1a1b2c0d2-1a0b4c0d0-1a0b0c2d2-4a2b0c1d2-4a1b2c1d0-2a0b1c0d4+1a0b0c3d0+1a4b0c0d2+1a2b0c2d0+4a0b1c1d2+1a4b0c1d0-5a2b1c0d2-1a0b1c2d0+1a6b0c0d0+1a2b1c1d0+5a0b2c0d2-4a4b1c0d0+3a1b0c1d2-5a0b2c1d0-2a3b0c0d2+6a2b2c0d0-1a1b0c2d0+5a3b0c1d0+4a1b1c0d2-4a0b3c0d0-10a1b1c1d0-1a0b0c0d4+1a3b1c0d0+3a0b0c1d2-2a2b0c0d2-3a1b2c0d0-1a0b0c2d0+1a2b0c1d0+6a0b1c0d2-1a4b0c0d0-7a0b1c1d0+4a2b1c0d0+2a1b0c0d2-6a0b2c0d0-6a1b0c1d0+3a3b0c0d0-6a1b1c0d0+2a0b0c0d2-3a0b0c1d0+1a2b0c0d0-4a0b1c0d0-3a1b0c0d0-1a0b0c0d0),
(1a4b1c1d0-3a2b2c1d0-1a5b0c0d2-1a3b0c2d0+1a0b3c1d0+1a5b0c1d0+3a3b1c0d2+4a1b1c2d0-3a3b1c1d0-1a1b2c0d2-1a2b0c1d2+1a1b2c1d0-1a0b0c3d0+1a4b0c0d2+1a2b0c2d0-2a0b1c1d2-2a2b1c0d2+2a0b1c2d0+1a6b0c0d0-2a2b1c1d0+1a1b0c0d4-4a4b1c0d0-2a1b0c1d2+3a0b2c1d0+3a2b2c0d0+3a1b0c2d0-1a1b1c0d2+1a5b0c0d0+4a1b1c1d0-3a3b1c0d0-1a0b0c1d2-2a2b0c0d2+1a1b2c0d0+2a0b0c2d0+4a2b0c1d0-1a4b0c0d0+3a0b1c1d0+4a2b1c0d0-2a1b0c0d2+3a1b0c1d0+2a1b1c0d0+1a0b0c1d0+1a2b0c0d0+1a1b0c0d0),
(1a4b2c0d0-1a5b0c1d0-3a2b3c0d0+2a3b1c1d0+1a0b4c0d0+3a1b2c1d0+1a4b0c0d2-2a2b0c2d0-1a4b0c1d0-1a2b1c0d2-1a0b1c2d0-1a6b0c0d0+1a2b1c1d0-2a0b2c0d2+3a4b1c0d0+2a1b0c1d2+2a0b2c1d0-1a3b0c0d2-3a2b2c0d0-2a1b0c2d0-1a3b0c1d0+3a0b3c0d0-1a5b0c0d0+2a3b1c0d0+1a2b0c0d2+2a1b2c0d0-2a2b0c1d0-1a0b1c0d2+1a4b0c0d0+2a0b1c1d0-2a2b1c0d0+3a0b2c0d0-1a1b0c1d0-1a2b0c0d0+1a0b1c0d0),
(1a5b0c0d3-2a5b0c1d1-4a3b1c0d3+8a3b1c1d1+3a1b2c0d3-1a5b1c0d1+3a2b0c1d3-6a1b2c1d1-1a4b0c0d3+4a3b2c0d1-6a2b0c2d1-2a0b1c1d3+1a4b0c1d1+3a2b1c0d3-3a1b3c0d1+4a0b1c2d1-1a6b0c0d1-6a2b1c1d1-2a1b0c0d5-1a0b2c0d3+6a4b1c0d1+4a1b0c1d3+3a0b2c1d1-9a2b2c0d1-2a1b0c2d1-2a3b0c1d1+4a1b1c0d3+2a0b3c0d1-2a1b1c1d1+2a0b0c0d5+1a3b1c0d1-5a0b0c1d3+4a2b0c0d3-4a1b2c0d1+4a0b0c2d1-7a2b0c1d1-5a0b1c0d3+8a0b1c1d1-6a2b1c0d1+5a0b2c0d1-2a3b0c0d1+1a1b1c0d1-4a0b0c0d3+5a0b0c1d1+4a0b1c0d1+2a1b0c0d1+1a0b0c0d1),
(1a5b0c1d1-4a3b1c1d1-1a5b1c0d1+3a1b2c1d1-1a4b0c0d3+4a3b2c0d1+3a2b0c2d1+1a4b0c1d1+3a2b1c0d3-3a1b3c0d1-2a0b1c2d1-6a2b1c1d1-1a0b2c0d3+1a4b1c0d1-4a1b0c1d3+3a0b2c1d1+1a3b0c0d3-3a2b2c0d1+4a1b0c2d1-1a3b0c1d1+1a0b3c0d1+2a5b0c0d1+4a1b1c1d1+1a0b0c0d5-7a3b1c0d1+2a1b2c0d1-1a0b0c2d1+5a2b0c1d1-3a0b1c0d3+2a0b1c1d1-3a2b1c0d1-3a1b0c0d3+3a0b2c0d1+5a1b0c1d1-2a3b0c0d1+7a1b1c0d1-2a0b0c0d3-1a0b0c1d1+2a2b0c0d1+3a0b1c0d1+2a1b0c0d1+1a0b0c0d1),
(1a5b1c0d1-4a3b2c0d1-1a4b0c1d1+3a1b3c0d1-1a6b0c0d1+6a2b1c1d1+4a4b1c0d1-3a0b2c1d1+1a3b0c0d3-3a2b2c0d1-2a1b0c2d1-4a3b0c1d1-4a1b1c0d3+6a1b1c1d1-1a3b1c0d1+2a0b0c1d3+1a2b0c0d3+4a1b2c0d1-2a0b0c2d1-1a2b0c1d1+2a0b1c0d3-4a0b1c1d1-1a1b0c0d3-1a0b2c0d1+2a1b0c1d1-2a3b0c0d1+3a1b1c0d1-1a0b0c0d3-1a0b0c1d1-2a0b1c0d1+2a1b0c0d1+1a0b0c0d1),
(1a6b0c0d2-1a6b0c1d0-5a4b1c0d2+5a4b1c1d0+6a2b2c0d2-1a6b1c0d0+4a3b0c1d2-6a2b2c1d0-1a0b3c0d2-1a5b0c0d2+5a4b2c0d0-4a3b0c2d0-6a1b1c1d2+1a0b3c1d0+4a3b1c0d2-6a2b3c0d0+6a1b1c2d0-1a7b0c0d0-4a3b1c1d0-3a2b0c0d4-3a1b2c0d2+1a0b4c0d0+1a0b0c2d2+6a5b1c0d0+3a2b0c1d2+6a1b2c1d0+2a0b1c0d4-1a0b0c3d0-10a3b2c0d0-3a2b0c2d0-2a0b1c1d2-3a4b0c1d0+6a2b1c0d2+4a1b3c0d0+1a0b1c2d0+4a1b0c0d4-4a0b2c0d2+2a4b1c0d0-6a1b0c1d2+3a0b2c1d0+6a3b0c0d2-9a2b2c0d0+3a1b0c2d0-7a3b0c1d0-12a1b1c0d2+4a0b3c0d0+1a5b0c0d0+16a1b1c1d0-1a0b0c0d4-11a3b1c0d0-1a0b0c1d2-2a2b0c0d2+13a1b2c0d0+1a0b0c2d0+3a2b0c1d0-3a0b1c0d2-4a4b0c0d0+3a0b1c1d0+6a2b1c0d0-6a1b0c0d2+4a0b2c0d0+6a1b0c1d0-1a3b0c0d0+10a1b1c0d0+2a0b0c0d2+1a0b0c1d0+5a2b0c0d0+1a1b0c0d0-1a0b0c0d0),
(1a6b0c1d0-5a4b1c1d0+6a2b2c1d0-1a5b0c0d2+4a3b0c2d0-1a0b3c1d0+1a5b0c1d0+4a3b1c0d2-6a1b1c2d0-4a3b1c1d0-3a1b2c0d2-6a2b0c1d2+3a1b2c1d0+1a0b0c3d0+1a4b0c0d2+6a2b0c2d0+4a0b1c1d2-1a4b0c1d0-3a2b1c0d2-4a0b1c2d0+1a6b0c0d0+6a2b1c1d0+2a1b0c0d4+1a0b2c0d2-5a4b1c0d0-3a0b2c1d0+6a2b2c0d0-2a1b0c2d0+5a3b0c1d0-2a1b1c0d2-1a0b3c0d0+1a5b0c0d0-4a1b1c1d0-2a0b0c0d4-4a3b1c0d0+4a0b0c1d2-4a2b0c0d2+3a1b2c0d0-4a0b0c2d0+8a2b0c1d0+3a0b1c0d2-1a4b0c0d0-7a0b1c1d0+6a2b1c0d0-1a1b0c0d2-3a0b2c0d0-3a1b0c1d0+1a3b0c0d0+2a1b1c0d0+4a0b0c0d2-5a0b0c1d0+2a2b0c0d0-3a0b1c0d0-1a1b0c0d0-1a0b0c0d0),
(1a6b1c0d0-5a4b2c0d0-1a5b0c1d0+6a2b3c0d0+8a3b1c1d0-1a0b4c0d0-9a1b2c1d0+1a4b0c0d2-3a2b0c2d0-1a4b0c1d0-6a2b1c0d2+3a0b1c2d0-1a6b0c0d0+6a2b1c1d0+3a0b2c0d2+3a4b1c0d0+4a1b0c1d2-3a0b2c1d0-1a3b0c0d2+3a2b2c0d0-4a1b0c2d0-2a3b0c1d0+4a1b1c0d2-3a0b3c0d0-1a5b0c0d0-6a1b1c1d0-1a0b0c0d4+7a3b1c0d0+2a2b0c0d2-7a1b2c0d0+1a0b0c2d0-5a2b0c1d0+2a0b1c0d2+1a4b0c0d0-2a0b1c1d0-3a2b1c0d0+1a1b0c0d2-3a0b2c0d0-1a1b0c1d0-1a3b0c0d0-6a1b1c0d0+1a0b0c0d2+1a0b0c1d0-2a2b0c0d0-1a0b1c0d0+1a1b0c0d0),
(1a7b0c0d1-6a5b1c0d1+10a3b2c0d1+5a4b0c1d1-4a1b3c0d1-1a6b0c0d1-12a2b1c1d1+5a4b1c0d1+3a0b2c1d1-4a3b0c0d3-6a2b2c0d1+3a1b0c2d1+6a1b1c0d3+1a0b3c0d1+4a3b1c0d1-2a0b0c1d3+6a2b0c0d3-6a1b2c0d1+1a0b0c2d1-6a2b0c1d1-4a0b1c0d3+4a4b0c0d1+6a0b1c1d1-12a2b1c0d1-2a1b0c0d3+4a0b2c0d1+2a1b0c1d1-2a3b0c0d1+3a0b0c1d1-4a2b0c0d1+5a0b1c0d1+2a1b0c0d1),
(1a8b0c0d0-7a6b1c0d0+15a4b2c0d0+6a5b0c1d0-10a2b3c0d0-20a3b1c1d0+1a0b4c0d0+12a1b2c1d0-5a4b0c0d2+6a2b0c2d0+5a4b0c1d0+12a2b1c0d2-3a0b1c2d0-1a6b0c0d0-12a2b1c1d0-3a0b2c0d2+10a4b1c0d0-6a1b0c1d2+3a0b2c1d0+4a3b0c0d2-18a2b2c0d0+6a1b0c2d0-8a3b0c1d0-6a1b1c0d2+4a0b3c0d0+5a5b0c0d0+18a1b1c1d0+1a0b0c0d4-16a3b1c0d0+3a2b0c0d2+9a1b2c0d0-2a0b0c2d0-4a0b1c0d2+1a4b0c0d0+4a0b1c1d0-9a2b1c0d0-4a1b0c0d2+6a0b2c0d0+8a1b0c1d0-7a3b0c0d0+14a1b1c0d0-2a0b0c0d2+1a0b0c1d0+4a0b1c0d0+3a1b0c0d0+1a0b0c0d0)";
o5 : Ideal of R
i6 : C=C1+C2+C3;
o6 : Ideal of R
i7 : isSubset(C1,C)
o7 = true
i8 : isSubset(C2,C)
o8 = true
i9 : isSubset(C3,C)
o9 = false
i10 : isSubset(M6,C1)
o10 = true
i11 : isSubset(M6,C2)
o11 = true
i12 : isSubset(M6,C3)
o12 = true
i13 : isSubset(M6,C)
o13 = false
i14 : D=C1+C2+C3+M6;
o14 : Ideal of R
i15 : C==D
o15 = false
-----
Example 2:
Macaulay 2, version 1.2
with packages: Elimination, IntegralClosure, LLLBases,
PrimaryDecomposition, ReesAlgebra, SchurRings, TangentCone
i1 : R=ZZ[a,b,c,d];
i2 : M6=ideal
"(1a8b0c0d0-7a6b1c0d0+15a4b2c0d0+6a5b0c1d0-10a2b3c0d0-20a3b1c1d0+1a0b4c0d0+12a1b2c1d0-5a4b0c0d2+6a2b0c2d0+5a4b0c1d0+12a2b1c0d2-3a0b1c2d0-1a6b0c0d0-12a2b1c1d0-3a0b2c0d2+10a4b1c0d0-6a1b0c1d2+3a0b2c1d0+4a3b0c0d2-18a2b2c0d0+6a1b0c2d0-8a3b0c1d0-6a1b1c0d2+4a0b3c0d0+5a5b0c0d0+18a1b1c1d0+1a0b0c0d4-16a3b1c0d0+3a2b0c0d2+9a1b2c0d0-2a0b0c2d0-4a0b1c0d2+1a4b0c0d0+4a0b1c1d0-9a2b1c0d0-4a1b0c0d2+6a0b2c0d0+8a1b0c1d0-7a3b0c0d0+14a1b1c0d0-2a0b0c0d2+1a0b0c1d0+4a0b1c0d0+3a1b0c0d0+1a0b0c0d0)";
o2 : Ideal of R
i3 : C=ideal
"(1a0b0c0d8-7a0b0c1d6+15a0b0c2d4-1a0b1c0d6-10a0b0c3d2+10a0b1c1d4+1a0b0c4d0-18a0b1c2d2-6a1b0c0d6-4a0b2c0d4+4a0b1c3d0+27a1b0c1d4+3a0b2c1d2-33a1b0c2d2+3a0b2c2d0+11a1b1c0d4+7a1b0c3d0+5a0b3c0d2-30a1b1c1d2-2a0b3c1d0-2a0b0c0d6+15a1b1c2d0+13a0b0c1d4+9a2b0c0d4-3a1b2c0d2-1a0b4c0d0-21a0b0c2d2-27a2b0c1d2+3a1b2c1d0-3a0b1c0d4+7a0b0c3d0+13a2b0c2d0-6a0b1c1d2-13a2b1c0d2-1a1b3c0d0+11a0b1c2d0+14a2b1c1d0+14a1b0c0d4+13a0b2c0d2-38a1b0c1d2-5a0b2c1d0-3a3b0c0d2+3a2b2c0d0+22a1b0c2d0+7a3b0c1d0-16a1b1c0d2-5a0b3c0d0+18a1b1c1d0-1a0b0c0d4+3a3b1c0d0-4a0b0c1d2-17a2b0c0d2+6a0b0c2d0+19a2b0c1d0+10a0b1c0d2-2a0b1c1d0+11a2b1c0d0-7a1b0c0d2-8a0b2c0d0+11a1b0c1d0+4a3b0c0d0+3a1b1c0d0+3a0b0c0d2-1a0b0c1d0+6a2b0c0d0-5a0b1c0d0-1a0b0c0d0),
(1a0b0c1d6-5a0b0c2d4-1a0b1c0d6+6a0b0c3d2+3a0b1c1d4-1a0b0c4d0+3a0b1c2d2+1a1b0c0d6+2a0b2c0d4-3a0b1c3d0-8a1b0c1d4-9a0b2c1d2+15a1b0c2d2-6a1b0c3d0+6a1b1c1d2+3a0b3c1d0-1a0b0c0d6-8a1b1c2d0+2a0b0c1d4-2a2b0c0d4-1a1b2c0d2+3a0b0c2d2+11a2b0c1d2+2a1b2c1d0+6a0b1c0d4-5a0b0c3d0-9a2b0c2d0-12a0b1c1d2-1a0b1c2d0-7a2b1c1d0-3a1b0c0d4-8a0b2c0d2+11a1b0c1d2+10a0b2c1d0+1a3b0c0d2-11a1b0c2d0-4a3b0c1d0+1a1b1c0d2+2a0b3c0d0-4a1b1c1d0+4a0b0c0d4-7a0b0c1d2+8a2b0c0d2+2a1b2c0d0+1a0b0c2d0-10a2b0c1d0-12a0b1c0d2+9a0b1c1d0-4a2b1c0d0-1a1b0c0d2+7a0b2c0d0-2a1b0c1d0-3a3b0c0d0+2a1b1c0d0-4a0b0c0d2+4a0b0c1d0-3a2b0c0d0+6a0b1c0d0+2a1b0c0d0+1a0b0c0d0),
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(1a4b1c0d2-1a4b1c1d0-3a2b2c0d2-1a3b0c1d2+3a2b2c1d0+1a0b3c0d2-1a5b0c0d2-1a4b2c0d0+1a3b0c2d0+4a1b1c1d2-1a0b3c1d0+3a3b1c0d2+3a2b3c0d0-4a1b1c2d0+1a3b1c1d0+1a2b0c0d4-1a1b2c0d2-1a0b4c0d0-1a0b0c2d2-4a2b0c1d2-4a1b2c1d0-2a0b1c0d4+1a0b0c3d0+1a4b0c0d2+1a2b0c2d0+4a0b1c1d2+1a4b0c1d0-5a2b1c0d2-1a0b1c2d0+1a6b0c0d0+1a2b1c1d0+5a0b2c0d2-4a4b1c0d0+3a1b0c1d2-5a0b2c1d0-2a3b0c0d2+6a2b2c0d0-1a1b0c2d0+5a3b0c1d0+4a1b1c0d2-4a0b3c0d0-10a1b1c1d0-1a0b0c0d4+1a3b1c0d0+3a0b0c1d2-2a2b0c0d2-3a1b2c0d0-1a0b0c2d0+1a2b0c1d0+6a0b1c0d2-1a4b0c0d0-7a0b1c1d0+4a2b1c0d0+2a1b0c0d2-6a0b2c0d0-6a1b0c1d0+3a3b0c0d0-6a1b1c0d0+2a0b0c0d2-3a0b0c1d0+1a2b0c0d0-4a0b1c0d0-3a1b0c0d0-1a0b0c0d0),
(1a4b1c1d0-3a2b2c1d0-1a5b0c0d2-1a3b0c2d0+1a0b3c1d0+1a5b0c1d0+3a3b1c0d2+4a1b1c2d0-3a3b1c1d0-1a1b2c0d2-1a2b0c1d2+1a1b2c1d0-1a0b0c3d0+1a4b0c0d2+1a2b0c2d0-2a0b1c1d2-2a2b1c0d2+2a0b1c2d0+1a6b0c0d0-2a2b1c1d0+1a1b0c0d4-4a4b1c0d0-2a1b0c1d2+3a0b2c1d0+3a2b2c0d0+3a1b0c2d0-1a1b1c0d2+1a5b0c0d0+4a1b1c1d0-3a3b1c0d0-1a0b0c1d2-2a2b0c0d2+1a1b2c0d0+2a0b0c2d0+4a2b0c1d0-1a4b0c0d0+3a0b1c1d0+4a2b1c0d0-2a1b0c0d2+3a1b0c1d0+2a1b1c0d0+1a0b0c1d0+1a2b0c0d0+1a1b0c0d0),
(1a4b2c0d0-1a5b0c1d0-3a2b3c0d0+2a3b1c1d0+1a0b4c0d0+3a1b2c1d0+1a4b0c0d2-2a2b0c2d0-1a4b0c1d0-1a2b1c0d2-1a0b1c2d0-1a6b0c0d0+1a2b1c1d0-2a0b2c0d2+3a4b1c0d0+2a1b0c1d2+2a0b2c1d0-1a3b0c0d2-3a2b2c0d0-2a1b0c2d0-1a3b0c1d0+3a0b3c0d0-1a5b0c0d0+2a3b1c0d0+1a2b0c0d2+2a1b2c0d0-2a2b0c1d0-1a0b1c0d2+1a4b0c0d0+2a0b1c1d0-2a2b1c0d0+3a0b2c0d0-1a1b0c1d0-1a2b0c0d0+1a0b1c0d0),
(1a5b0c0d3-2a5b0c1d1-4a3b1c0d3+8a3b1c1d1+3a1b2c0d3-1a5b1c0d1+3a2b0c1d3-6a1b2c1d1-1a4b0c0d3+4a3b2c0d1-6a2b0c2d1-2a0b1c1d3+1a4b0c1d1+3a2b1c0d3-3a1b3c0d1+4a0b1c2d1-1a6b0c0d1-6a2b1c1d1-2a1b0c0d5-1a0b2c0d3+6a4b1c0d1+4a1b0c1d3+3a0b2c1d1-9a2b2c0d1-2a1b0c2d1-2a3b0c1d1+4a1b1c0d3+2a0b3c0d1-2a1b1c1d1+2a0b0c0d5+1a3b1c0d1-5a0b0c1d3+4a2b0c0d3-4a1b2c0d1+4a0b0c2d1-7a2b0c1d1-5a0b1c0d3+8a0b1c1d1-6a2b1c0d1+5a0b2c0d1-2a3b0c0d1+1a1b1c0d1-4a0b0c0d3+5a0b0c1d1+4a0b1c0d1+2a1b0c0d1+1a0b0c0d1),
(1a5b0c1d1-4a3b1c1d1-1a5b1c0d1+3a1b2c1d1-1a4b0c0d3+4a3b2c0d1+3a2b0c2d1+1a4b0c1d1+3a2b1c0d3-3a1b3c0d1-2a0b1c2d1-6a2b1c1d1-1a0b2c0d3+1a4b1c0d1-4a1b0c1d3+3a0b2c1d1+1a3b0c0d3-3a2b2c0d1+4a1b0c2d1-1a3b0c1d1+1a0b3c0d1+2a5b0c0d1+4a1b1c1d1+1a0b0c0d5-7a3b1c0d1+2a1b2c0d1-1a0b0c2d1+5a2b0c1d1-3a0b1c0d3+2a0b1c1d1-3a2b1c0d1-3a1b0c0d3+3a0b2c0d1+5a1b0c1d1-2a3b0c0d1+7a1b1c0d1-2a0b0c0d3-1a0b0c1d1+2a2b0c0d1+3a0b1c0d1+2a1b0c0d1+1a0b0c0d1),
(1a5b1c0d1-4a3b2c0d1-1a4b0c1d1+3a1b3c0d1-1a6b0c0d1+6a2b1c1d1+4a4b1c0d1-3a0b2c1d1+1a3b0c0d3-3a2b2c0d1-2a1b0c2d1-4a3b0c1d1-4a1b1c0d3+6a1b1c1d1-1a3b1c0d1+2a0b0c1d3+1a2b0c0d3+4a1b2c0d1-2a0b0c2d1-1a2b0c1d1+2a0b1c0d3-4a0b1c1d1-1a1b0c0d3-1a0b2c0d1+2a1b0c1d1-2a3b0c0d1+3a1b1c0d1-1a0b0c0d3-1a0b0c1d1-2a0b1c0d1+2a1b0c0d1+1a0b0c0d1),
(1a6b0c0d2-1a6b0c1d0-5a4b1c0d2+5a4b1c1d0+6a2b2c0d2-1a6b1c0d0+4a3b0c1d2-6a2b2c1d0-1a0b3c0d2-1a5b0c0d2+5a4b2c0d0-4a3b0c2d0-6a1b1c1d2+1a0b3c1d0+4a3b1c0d2-6a2b3c0d0+6a1b1c2d0-1a7b0c0d0-4a3b1c1d0-3a2b0c0d4-3a1b2c0d2+1a0b4c0d0+1a0b0c2d2+6a5b1c0d0+3a2b0c1d2+6a1b2c1d0+2a0b1c0d4-1a0b0c3d0-10a3b2c0d0-3a2b0c2d0-2a0b1c1d2-3a4b0c1d0+6a2b1c0d2+4a1b3c0d0+1a0b1c2d0+4a1b0c0d4-4a0b2c0d2+2a4b1c0d0-6a1b0c1d2+3a0b2c1d0+6a3b0c0d2-9a2b2c0d0+3a1b0c2d0-7a3b0c1d0-12a1b1c0d2+4a0b3c0d0+1a5b0c0d0+16a1b1c1d0-1a0b0c0d4-11a3b1c0d0-1a0b0c1d2-2a2b0c0d2+13a1b2c0d0+1a0b0c2d0+3a2b0c1d0-3a0b1c0d2-4a4b0c0d0+3a0b1c1d0+6a2b1c0d0-6a1b0c0d2+4a0b2c0d0+6a1b0c1d0-1a3b0c0d0+10a1b1c0d0+2a0b0c0d2+1a0b0c1d0+5a2b0c0d0+1a1b0c0d0-1a0b0c0d0),
(1a6b0c1d0-5a4b1c1d0+6a2b2c1d0-1a5b0c0d2+4a3b0c2d0-1a0b3c1d0+1a5b0c1d0+4a3b1c0d2-6a1b1c2d0-4a3b1c1d0-3a1b2c0d2-6a2b0c1d2+3a1b2c1d0+1a0b0c3d0+1a4b0c0d2+6a2b0c2d0+4a0b1c1d2-1a4b0c1d0-3a2b1c0d2-4a0b1c2d0+1a6b0c0d0+6a2b1c1d0+2a1b0c0d4+1a0b2c0d2-5a4b1c0d0-3a0b2c1d0+6a2b2c0d0-2a1b0c2d0+5a3b0c1d0-2a1b1c0d2-1a0b3c0d0+1a5b0c0d0-4a1b1c1d0-2a0b0c0d4-4a3b1c0d0+4a0b0c1d2-4a2b0c0d2+3a1b2c0d0-4a0b0c2d0+8a2b0c1d0+3a0b1c0d2-1a4b0c0d0-7a0b1c1d0+6a2b1c0d0-1a1b0c0d2-3a0b2c0d0-3a1b0c1d0+1a3b0c0d0+2a1b1c0d0+4a0b0c0d2-5a0b0c1d0+2a2b0c0d0-3a0b1c0d0-1a1b0c0d0-1a0b0c0d0),
(1a6b1c0d0-5a4b2c0d0-1a5b0c1d0+6a2b3c0d0+8a3b1c1d0-1a0b4c0d0-9a1b2c1d0+1a4b0c0d2-3a2b0c2d0-1a4b0c1d0-6a2b1c0d2+3a0b1c2d0-1a6b0c0d0+6a2b1c1d0+3a0b2c0d2+3a4b1c0d0+4a1b0c1d2-3a0b2c1d0-1a3b0c0d2+3a2b2c0d0-4a1b0c2d0-2a3b0c1d0+4a1b1c0d2-3a0b3c0d0-1a5b0c0d0-6a1b1c1d0-1a0b0c0d4+7a3b1c0d0+2a2b0c0d2-7a1b2c0d0+1a0b0c2d0-5a2b0c1d0+2a0b1c0d2+1a4b0c0d0-2a0b1c1d0-3a2b1c0d0+1a1b0c0d2-3a0b2c0d0-1a1b0c1d0-1a3b0c0d0-6a1b1c0d0+1a0b0c0d2+1a0b0c1d0-2a2b0c0d0-1a0b1c0d0+1a1b0c0d0),
(1a7b0c0d1-6a5b1c0d1+10a3b2c0d1+5a4b0c1d1-4a1b3c0d1-1a6b0c0d1-12a2b1c1d1+5a4b1c0d1+3a0b2c1d1-4a3b0c0d3-6a2b2c0d1+3a1b0c2d1+6a1b1c0d3+1a0b3c0d1+4a3b1c0d1-2a0b0c1d3+6a2b0c0d3-6a1b2c0d1+1a0b0c2d1-6a2b0c1d1-4a0b1c0d3+4a4b0c0d1+6a0b1c1d1-12a2b1c0d1-2a1b0c0d3+4a0b2c0d1+2a1b0c1d1-2a3b0c0d1+3a0b0c1d1-4a2b0c0d1+5a0b1c0d1+2a1b0c0d1),
(1a8b0c0d0-7a6b1c0d0+15a4b2c0d0+6a5b0c1d0-10a2b3c0d0-20a3b1c1d0+1a0b4c0d0+12a1b2c1d0-5a4b0c0d2+6a2b0c2d0+5a4b0c1d0+12a2b1c0d2-3a0b1c2d0-1a6b0c0d0-12a2b1c1d0-3a0b2c0d2+10a4b1c0d0-6a1b0c1d2+3a0b2c1d0+4a3b0c0d2-18a2b2c0d0+6a1b0c2d0-8a3b0c1d0-6a1b1c0d2+4a0b3c0d0+5a5b0c0d0+18a1b1c1d0+1a0b0c0d4-16a3b1c0d0+3a2b0c0d2+9a1b2c0d0-2a0b0c2d0-4a0b1c0d2+1a4b0c0d0+4a0b1c1d0-9a2b1c0d0-4a1b0c0d2+6a0b2c0d0+8a1b0c1d0-7a3b0c0d0+14a1b1c0d0-2a0b0c0d2+1a0b0c1d0+4a0b1c0d0+3a1b0c0d0+1a0b0c0d0)";
o3 : Ideal of R
i4 : isSubset(M6,C)
o4 = false
-----
Comment on Example 2: The polynomial M6 is exactly the last generator
of the ideal C.
-----
I am comparing ideals in polynomial rings, using the commands == and
isSubset, and am encountering a problem. It seems like the output I
am getting suggests a bug in these commands, but perhaps I have
misunderstood the basic syntax or some other issue. In short, (1) the
command "isSubset(A,A+B)" is returning "false", where A and B are
ideals in a polynomial ring, and (2) the command "isSubset(A,B+C)" is
returning false, even though "isSubset(A,B)" and "isSubset(A,C)" both
return "true". More specifically, the session is as follows, with ...
replacing various lists of polynomials:
-----
Macaulay 2, version 1.2
with packages: Elimination, IntegralClosure, LLLBases,
PrimaryDecomposition,
ReesAlgebra, SchurRings, TangentCone
i1 : R=ZZ[a,b,c,d];
i2 : M6=ideal "...";
o2 : Ideal of R
i3 : C1=ideal "...";
o3 : Ideal of R
i4 : C2=ideal "...";
o4 : Ideal of R
i5 : C3=ideal "...";
o5 : Ideal of R
i6 : C=C1+C2+C3;
o6 : Ideal of R
i7 : isSubset(C1,C)
o7 = true
i8 : isSubset(C2,C)
o8 = true
i9 : isSubset(C3,C)
o9 = false
i10 : isSubset(M6,C1)
o10 = true
i11 : isSubset(M6,C2)
o11 = true
i12 : isSubset(M6,C3)
o12 = true
i13 : isSubset(M6,C)
o13 = false
i14 : D=C1+C2+C3+M6;
o14 : Ideal of R
i15 : C==D
o15 = false
---------
The ideals in question are large, but not that large, and I hope to
work with even larger ideals. Below I attach the full session with
the ideals included. Does anyone know what might be going on or
whether it is possible to fix? If there are size limits on some of
the functions called (eg involving Groebner bases), I wouldn't mind
isSubset or == returning overflow errors or some such, but if it
returns 'false', then I'm uncertain if it really is false or if there
has been an internal error.
Note: I can produce similar examples that do not involve the "+"
command for ideals. These have the form "J=ideal "A,B,C,D"; M=ideal
"D"; isSubset(M,J)" returning "false". An example of this is included
at the end.
Thanks very much for any help!
Chris
-----
Example 1:
Macaulay 2, version 1.2
with packages: Elimination, IntegralClosure, LLLBases,
PrimaryDecomposition,
ReesAlgebra, SchurRings, TangentCone
i1 : R=ZZ[a,b,c,d];
i2 : M6=ideal
"(1a8b0c0d0-7a6b1c0d0+15a4b2c0d0+6a5b0c1d0-10a2b3c0d0-20a3b1c1d0+1a0b4c0d0+12a1b2c1d0-5a4b0c0d2+6a2b0c2d0+5a4b0c1d0+12a2b1c0d2-3a0b1c2d0-1a6b0c0d0-12a2b1c1d0-3a0b2c0d2+10a4b1c0d0-6a1b0c1d2+3a0b2c1d0+4a3b0c0d2-18a2b2c0d0+6a1b0c2d0-8a3b0c1d0-6a1b1c0d2+4a0b3c0d0+5a5b0c0d0+18a1b1c1d0+1a0b0c0d4-16a3b1c0d0+3a2b0c0d2+9a1b2c0d0-2a0b0c2d0-4a0b1c0d2+1a4b0c0d0+4a0b1c1d0-9a2b1c0d0-4a1b0c0d2+6a0b2c0d0+8a1b0c1d0-7a3b0c0d0+14a1b1c0d0-2a0b0c0d2+1a0b0c1d0+4a0b1c0d0+3a1b0c0d0+1a0b0c0d0)";
o2 : Ideal of R
i3 : C1=ideal
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(1a0b4c0d0-3a1b2c1d0+1a2b0c2d0+2a2b1c0d2+2a0b1c2d0-2a2b1c1d0-2a0b2c0d2-2a1b0c1d2+2a0b2c1d0-1a3b0c0d2-3a2b2c0d0+2a1b0c2d0+2a3b0c1d0+2a1b1c0d2+3a0b3c0d0+1a0b0c0d4-1a3b1c0d0-3a0b0c1d2-1a2b0c0d2+2a0b0c2d0+3a2b0c1d0-3a0b1c0d2+1a4b0c0d0+4a0b1c1d0-2a2b1c0d0-1a1b0c0d2+3a0b2c0d0+3a1b0c1d0+2a3b0c0d0-1a0b0c0d2+2a0b0c1d0+1a2b0c0d0+1a0b1c0d0),
(1a1b0c0d7-6a1b0c1d5+10a1b0c2d3-1a1b1c0d5-4a1b0c3d1+8a1b1c1d3-1a0b0c0d7-9a1b1c2d1+5a0b0c1d5-5a2b0c0d5-3a1b2c0d3-6a0b0c2d3+18a2b0c1d3+2a0b1c0d5+1a0b0c3d1-15a2b0c2d1-10a0b1c1d3+8a2b1c0d3+3a1b3c0d1+6a0b1c2d1-14a2b1c1d1+2a1b0c0d5+1a0b2c0d3-2a1b0c1d3+5a0b2c1d1+6a3b0c0d3-1a2b2c0d1-4a1b0c2d1-12a3b0c1d1-8a1b1c0d3-2a0b3c0d1+8a1b1c1d1+3a0b0c0d5-7a3b1c0d1-11a0b0c1d3+4a2b0c0d3+6a1b2c0d1+9a0b0c2d1-9a2b0c1d1-5a0b1c0d3-1a4b0c0d1+12a0b1c1d1-2a2b1c0d1-7a1b0c0d3-1a0b2c0d1+10a1b0c1d1-4a3b0c0d1+11a1b1c0d1-3a0b0c0d3+5a0b0c1d1+4a0b1c0d1+4a1b0c0d1+1a0b0c0d1),
(1a1b0c1d5-4a1b0c2d3-1a1b1c0d5+3a1b0c3d1+2a1b1c1d3-1a0b0c0d7+3a1b1c2d1+5a0b0c1d5+1a2b0c0d5+2a1b2c0d3-7a0b0c2d3-6a2b0c1d3-5a1b2c1d1+2a0b1c0d5+3a0b0c3d1+8a2b0c2d1-8a0b1c1d3-1a2b1c0d3-1a1b3c0d1+5a0b1c2d1+2a2b1c1d1+3a1b0c0d5-13a1b0c1d3+1a0b2c1d1-1a3b0c0d3+1a2b2c0d1+12a1b0c2d1+5a3b0c1d1+1a1b1c0d3+4a1b1c1d1+3a0b0c0d5+1a3b1c0d1-9a0b0c1d3-7a2b0c0d3-5a1b2c0d1+6a0b0c2d1+13a2b0c1d1-5a0b1c0d3-1a4b0c0d1+8a0b1c1d1+2a2b1c0d1-6a1b0c0d3+11a1b0c1d1+5a3b0c0d1+1a1b1c0d1-2a0b0c0d3+3a0b0c1d1+7a2b0c0d1+2a0b1c0d1+1a1b0c0d1)";
o3 : Ideal of R
i4 : C2=ideal
"(1a1b0c2d3-1a1b1c0d5-2a1b0c3d1+2a1b1c1d3-2a1b1c2d1-1a0b0c1d5+1a2b0c0d5+2a1b2c0d3+3a0b0c2d3-1a2b0c1d3+2a0b1c0d5-2a0b0c3d1-2a2b0c2d1-3a0b1c1d3-1a2b1c0d3-1a1b3c0d1-2a2b1c1d1-3a1b0c0d5-5a0b2c0d3+7a1b0c1d3+5a0b2c1d1-2a3b0c0d3-1a2b2c0d1-4a1b0c2d1+2a3b0c1d1+7a1b1c0d3+2a0b3c0d1-6a1b1c1d1+2a0b0c0d5+3a3b1c0d1-3a0b0c1d3+4a2b0c0d3-2a1b2c0d1-5a2b0c1d1-9a0b1c0d3+1a4b0c0d1+8a0b1c1d1-6a2b1c0d1+2a1b0c0d3+9a0b2c0d1-4a1b0c1d1-2a3b0c0d1-3a1b1c0d1-4a0b0c0d3+5a0b0c1d1-2a2b0c0d1+8a0b1c0d1+2a1b0c0d1+1a0b0c0d1),
(1a1b0c3d1-2a1b1c1d3+1a1b1c2d1+1a2b0c0d5+2a1b2c0d3-1a0b0c2d3-2a2b0c1d3+1a0b1c0d5+1a0b0c3d1+1a2b0c2d1-3a2b1c0d3-2a1b3c0d1+6a2b1c1d1-3a1b0c0d5-3a0b2c0d3+7a1b0c1d3-3a3b0c0d3-3a1b0c2d1+2a3b0c1d1+3a1b1c0d3+2a0b3c0d1-2a1b1c1d1+1a0b0c0d5+4a3b1c0d1-1a0b0c1d3+8a2b0c0d3-2a1b2c0d1-8a2b0c1d1-5a0b1c0d3+2a4b0c0d1+2a0b1c1d1-4a2b1c0d1+2a1b0c0d3+5a0b2c0d1-6a1b0c1d1-4a3b0c0d1+2a1b1c0d1-2a0b0c0d3+2a0b0c1d1-7a2b0c0d1+4a0b1c0d1+2a1b0c0d1+1a0b0c0d1),
(1a1b1c0d5-4a1b1c1d3+3a1b1c2d1-1a0b0c1d5-1a2b0c0d5-1a1b2c0d3+4a0b0c2d3+3a2b0c1d3+4a1b2c1d1-3a0b0c3d1-1a2b0c2d1+1a0b1c1d3-1a2b1c0d3-1a1b3c0d1-4a0b1c2d1+2a2b1c1d1+1a1b0c0d5+1a1b0c1d3+1a0b2c1d1+2a3b0c0d3+1a2b2c0d1-6a1b0c2d1-4a3b0c1d1-5a1b1c0d3+4a1b1c1d1+1a3b1c0d1+1a0b0c1d3+2a2b0c0d3+4a1b2c0d1-3a0b0c2d1-5a2b0c1d1+1a0b1c0d3-1a4b0c0d1+2a2b1c0d1-3a1b0c0d3-1a0b2c0d1+2a1b0c1d1-4a3b0c0d1+3a1b1c0d1-1a0b0c0d3+1a0b0c1d1+4a1b0c0d1+1a0b0c0d1),
(1a1b1c1d3-2a1b1c2d1-1a2b0c0d5-1a1b2c0d3-1a0b0c2d3+3a2b0c1d3+2a0b0c3d1-2a2b0c2d1+1a0b1c1d3+3a2b1c0d3+2a1b3c0d1-4a2b1c1d1+2a1b0c0d5-6a1b0c1d3-2a0b2c1d1+2a3b0c0d3-2a2b2c0d1+4a1b0c2d1-4a3b0c1d1-6a1b1c0d3+8a1b1c1d1-1a0b0c0d5-4a3b1c0d1+3a0b0c1d3-2a2b0c0d3+4a1b2c0d1-2a0b0c2d1+4a2b0c1d1+3a0b1c0d3-4a0b1c1d1+4a2b1c0d1-2a1b0c0d3-2a0b2c0d1+4a1b0c1d1+4a1b1c0d1+2a0b0c0d3-4a0b0c1d1-4a0b1c0d1),
(1a1b1c2d1-1a1b2c0d3-1a2b0c1d3-1a0b0c3d1+1a2b0c2d1+1a0b1c1d3+3a2b1c0d3+1a1b3c0d1-2a2b1c1d1+1a1b0c0d5-3a1b0c1d3-1a0b2c1d1-1a3b0c0d3-1a2b2c0d1+2a1b0c2d1+2a3b0c1d1-3a1b1c0d3-1a0b0c0d5-3a3b1c0d1+2a0b0c1d3-3a2b0c0d3+4a1b2c0d1+1a0b0c2d1+7a2b0c1d1+2a0b1c0d3+1a4b0c0d1-4a0b1c1d1+2a1b0c0d3-1a0b2c0d1+4a3b0c0d1-1a1b1c0d1+2a0b0c0d3-3a0b0c1d1-2a0b1c0d1-4a1b0c0d1-1a0b0c0d1),
(1a1b2c0d3-1a2b0c1d3-2a1b2c1d1+2a2b0c2d1-1a0b1c1d3-1a2b1c0d3-1a1b3c0d1+2a0b1c2d1+2a2b1c1d1+1a1b0c0d5-1a1b0c1d3+1a0b2c1d1+1a2b2c0d1+2a3b0c1d1-1a1b1c0d3-2a1b1c1d1-1a0b0c0d5+1a3b1c0d1+1a0b0c1d3-2a2b0c0d3-2a1b2c0d1+1a2b0c1d1+3a0b1c0d3-1a4b0c0d1+2a2b1c0d1-1a0b2c0d1+2a3b0c0d1-1a1b1c0d1+2a0b0c0d3-1a0b0c1d1+2a2b0c0d1-4a0b1c0d1-2a1b0c0d1-1a0b0c0d1),
(1a1b2c1d1-1a2b0c2d1-1a2b1c0d3-1a1b3c0d1-1a0b1c2d1+2a2b1c1d1+3a1b0c1d3+1a0b2c1d1+1a3b0c0d3+1a2b2c0d1-3a1b0c2d1-1a3b0c1d1-1a1b1c0d3-2a1b1c1d1-1a0b0c0d5+1a3b1c0d1+1a0b0c1d3-1a2b0c0d3+1a1b2c0d1-5a2b0c1d1+2a0b1c0d3-1a4b0c0d1+2a2b1c0d1+3a1b0c0d3-1a0b2c0d1-3a1b0c1d1-1a3b0c0d1-1a1b1c0d1+2a0b0c0d3-1a0b0c1d1-2a0b1c0d1-3a1b0c0d1-1a0b0c0d1),
(1a1b3c0d1-2a2b1c1d1-1a0b2c1d1+1a3b0c0d3-1a2b2c0d1+2a1b0c2d1+2a1b1c1d1-1a3b1c0d1-1a0b0c1d3-3a2b0c0d3+3a2b0c1d1+1a0b1c0d3-1a4b0c0d1+4a2b1c0d1+1a1b0c0d3-1a0b2c0d1-2a1b0c1d1+2a3b0c0d1-3a1b1c0d1+1a0b0c0d3+1a0b0c1d1+2a2b0c0d1-2a0b1c0d1-2a1b0c0d1-1a0b0c0d1),
(1a2b0c0d6-5a2b0c1d4-1a0b1c0d6+6a2b0c2d2+5a0b1c1d4-1a2b1c0d4-1a2b0c3d0-6a0b1c2d2+6a2b1c1d2-1a1b0c0d6+1a0b2c0d4+1a0b1c3d0-3a2b1c2d0+4a1b0c1d4-6a0b2c1d2-4a3b0c0d4-2a2b2c0d2-3a1b0c2d2+3a0b2c2d0+11a3b0c1d2-1a2b2c1d0+6a1b1c0d4+2a0b3c0d2-5a3b0c2d0-18a1b1c1d2+1a0b3c1d0+5a3b1c0d2+1a2b3c0d0+6a1b1c2d0+1a0b0c1d4-6a3b1c1d0+1a2b0c0d4-5a1b2c0d2-1a0b4c0d0-3a0b0c2d2+8a1b2c1d0+2a0b1c0d4+1a0b0c3d0+4a4b0c0d2-4a2b0c2d0-9a0b1c1d2-5a4b0c1d0-6a2b1c0d2+8a0b1c2d0+5a2b1c1d0+3a1b0c0d4-3a4b1c0d0-9a1b0c1d2+5a0b2c1d0+3a3b0c0d2+5a2b2c0d0+6a1b0c2d0-6a3b0c1d0-13a1b1c0d2-3a0b3c0d0-1a5b0c0d0+18a1b1c1d0-2a3b1c0d0-2a0b0c1d2-4a2b0c0d2+5a1b2c0d0+4a0b0c2d0+5a2b0c1d0-2a0b1c0d2-2a4b0c0d0+5a0b1c1d0+8a2b1c0d0-3a1b0c0d2-1a0b2c0d0+6a1b0c1d0+8a1b1c0d0+1a0b0c1d0+2a2b0c0d0+1a0b1c0d0+1a1b0c0d0),
(1a2b0c1d4-3a2b0c2d2-1a0b1c1d4-1a2b1c0d4+1a2b0c3d0+3a0b1c2d2+1a2b1c1d2-1a1b0c0d6+1a0b2c0d4-1a0b1c3d0+2a2b1c2d0+4a1b0c1d4-1a0b2c1d2+1a3b0c0d4+2a2b2c0d2-4a1b0c2d2-2a0b2c2d0-4a3b0c1d2-1a2b2c1d0+1a1b1c0d4+1a1b0c3d0-2a0b3c0d2+4a3b0c2d0-1a1b1c1d2+1a0b3c1d0+1a0b0c0d6-2a3b1c0d2-1a2b3c0d0-2a1b1c2d0-4a0b0c1d4+2a3b1c1d0+2a2b0c0d4+1a1b2c0d2+1a0b4c0d0+4a0b0c2d2-9a2b0c1d2-2a1b2c1d0-1a0b1c0d4-1a0b0c3d0-1a4b0c0d2+7a2b0c2d0+6a0b1c1d2+3a4b0c1d0+2a2b1c0d2-6a0b1c2d0-1a2b1c1d0-4a0b2c0d2+2a4b1c0d0+1a1b0c1d2+1a0b2c1d0-3a3b0c0d2-4a2b2c0d0-1a1b0c2d0+7a3b0c1d0+3a1b1c0d2+4a0b3c0d0-6a1b1c1d0-3a0b0c0d4+2a3b1c0d0+8a0b0c1d2-3a2b0c0d2-2a1b2c0d0-4a0b0c2d0+3a2b0c1d0+1a0b1c0d2+1a4b0c0d0-3a0b1c1d0-4a2b1c0d0+3a1b0c0d2+4a0b2c0d0-4a1b0c1d0+3a3b0c0d0-4a1b1c0d0+3a0b0c0d2-3a0b0c1d0+1a2b0c0d0-2a1b0c0d0-1a0b0c0d0),
(1a2b0c2d2-1a2b1c0d4-1a2b0c3d0-1a0b1c2d2+1a2b1c1d2+1a0b2c0d4+1a0b1c3d0-2a2b1c2d0-1a1b0c1d4-1a0b2c1d2+1a3b0c0d4+2a2b2c0d2+2a1b0c2d2+2a0b2c2d0+1a1b1c0d4-1a1b0c3d0-2a0b3c0d2-3a3b0c2d0-1a1b1c1d2+2a1b1c2d0+1a0b0c1d4-2a3b1c1d0-3a2b0c0d4-4a1b2c0d2-2a0b0c2d2+7a2b0c1d2+4a1b2c1d0+1a0b0c3d0-2a4b0c0d2-1a3b2c0d0-5a2b0c2d0-4a0b1c1d2-1a4b0c1d0+5a2b1c0d2+2a1b3c0d0+5a0b1c2d0-1a2b1c1d0+2a1b0c0d4-1a0b2c0d2-3a1b0c1d2+2a0b2c1d0+5a3b0c0d2+2a1b0c2d0-5a3b0c1d0-7a1b1c0d2+1a5b0c0d0+8a1b1c1d0-4a3b1c0d0-3a0b0c1d2+1a2b0c0d2+6a1b2c0d0+3a0b0c2d0-3a2b0c1d0+1a0b1c0d2-1a4b0c0d0+4a0b1c1d0+1a2b1c0d0-4a1b0c0d2+4a1b0c1d0-4a3b0c0d0+6a1b1c0d0+2a0b0c1d0+1a2b0c0d0+2a1b0c0d0),
(1a2b0c3d0-2a2b1c1d2-1a0b1c3d0+2a2b1c2d0+2a0b2c1d2+1a3b0c0d4+1a2b2c0d2-1a1b0c2d2-2a0b2c2d0-3a3b0c1d2+1a1b0c3d0-1a0b3c0d2+3a3b0c2d0+2a1b1c1d2-1a3b1c0d2-1a2b3c0d0-2a1b1c2d0+4a3b1c1d0-1a2b0c0d4+1a0b4c0d0+1a0b0c2d2-5a1b2c1d0-1a0b1c0d4-1a0b0c3d0-3a4b0c0d2+3a2b0c2d0+3a0b1c1d2+3a4b0c1d0+3a2b1c0d2-4a0b1c2d0-1a2b1c1d0+2a4b1c0d0+3a1b0c1d2-2a0b2c1d0+1a3b0c0d2-3a2b2c0d0-3a1b0c2d0+3a3b0c1d0+1a1b1c0d2+3a0b3c0d0+1a5b0c0d0-8a1b1c1d0+2a2b0c0d2-2a1b2c0d0-2a0b0c2d0-3a2b0c1d0+2a0b1c0d2+1a4b0c0d0-2a0b1c1d0-4a2b1c0d0+1a0b2c0d0-3a1b0c1d0-1a3b0c0d0-2a1b1c0d0-1a2b0c0d0-1a0b1c0d0),
(1a2b1c0d4-3a2b1c1d2-1a0b2c0d4+1a2b1c2d0-1a1b0c1d4+3a0b2c1d2-1a3b0c0d4-1a2b2c0d2+3a1b0c2d2-1a0b2c2d0+2a3b0c1d2+2a2b2c1d0+1a1b1c0d4-1a1b0c3d0+1a0b3c0d2-1a1b1c1d2-2a0b3c1d0+1a0b0c0d6-2a1b1c2d0-4a0b0c1d4+2a3b1c1d0+1a2b0c0d4+4a0b0c2d2+1a2b0c1d2-2a1b2c1d0-3a0b1c0d4-1a0b0c3d0+1a4b0c0d2-3a2b0c2d0+8a0b1c1d2-1a4b0c1d0-6a2b1c0d2-4a0b1c2d0+3a2b1c1d0-3a1b0c0d4+6a0b2c0d2+9a1b0c1d2-7a0b2c1d0+1a3b0c0d2+2a2b2c0d0-7a1b0c2d0-3a3b0c1d0+3a1b1c0d2-2a0b3c0d0-6a1b1c1d0-3a0b0c0d4+2a3b1c0d0+8a0b0c1d2+3a2b0c0d2-2a1b2c0d0-4a0b0c2d0-5a2b0c1d0+7a0b1c0d2-1a4b0c0d0-8a0b1c1d0+2a2b1c0d0+4a1b0c0d2-6a0b2c0d0-6a1b0c1d0-3a3b0c0d0-4a1b1c0d0+2a0b0c0d2-3a0b0c1d0-2a2b0c0d0-4a0b1c0d0),
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(1a2b1c2d0-1a2b2c0d2-1a0b2c2d0-1a3b0c1d2+1a2b2c1d0-1a1b0c3d0+1a0b3c0d2+1a3b0c2d0+2a1b1c1d2-1a0b3c1d0+1a3b1c0d2-2a1b1c2d0+1a3b1c1d0+1a2b0c0d4-1a1b2c0d2+1a0b0c2d2-2a2b0c1d2-1a1b2c1d0-2a0b1c0d4-1a0b0c3d0+5a0b1c1d2+1a4b0c1d0-1a2b1c0d2-4a0b1c2d0-2a2b1c1d0-1a1b0c0d4+2a0b2c0d2+3a1b0c1d2-3a0b2c1d0-3a3b0c0d2+1a2b2c0d0-3a1b0c2d0+2a3b0c1d0+6a1b1c0d2-1a0b3c0d0-8a1b1c1d0+1a3b1c0d0+1a0b0c1d2+1a2b0c0d2-1a1b2c0d0-2a0b0c2d0-1a2b0c1d0+3a0b1c0d2+1a4b0c0d0-5a0b1c1d0-3a2b1c0d0+2a1b0c0d2-2a0b2c0d0-3a1b0c1d0+1a3b0c0d0-6a1b1c0d0-1a0b0c1d0-1a2b0c0d0-1a0b1c0d0-1a1b0c0d0),
(1a2b2c0d2-1a3b0c1d2-1a2b2c1d0-1a0b3c0d2+1a3b0c2d0+1a0b3c1d0-1a3b1c0d2-1a2b3c0d0+1a3b1c1d0+1a2b0c0d4+1a1b2c0d2+1a0b4c0d0+1a0b1c0d4+1a2b0c2d0-3a0b1c1d2+2a4b0c1d0-1a2b1c0d2+1a0b1c2d0-3a2b1c1d0-3a1b0c0d4-3a0b2c0d2+2a4b1c0d0+3a1b0c1d2+5a0b2c1d0-2a3b0c0d2-4a2b2c0d0+1a3b0c1d0+4a1b1c0d2+4a0b3c0d0+2a0b0c0d4+1a3b1c0d0-3a0b0c1d2+4a2b0c0d2+1a0b0c2d0-4a2b0c1d0-6a0b1c0d2+7a0b1c1d0-4a2b1c0d0+2a1b0c0d2+6a0b2c0d0-4a0b0c0d2+3a0b0c1d0-1a2b0c0d0+4a0b1c0d0+1a0b0c0d0),
(1a2b2c1d0-1a3b0c2d0-1a0b3c1d0-1a3b1c0d2+1a3b1c1d0+1a1b2c0d2+3a2b0c1d2-1a1b2c1d0+1a4b0c0d2-3a2b0c2d0-1a4b0c1d0-2a2b1c0d2+2a2b1c1d0-2a1b0c0d4+1a0b2c0d2+3a1b0c1d2-3a0b2c1d0+1a2b2c0d0-2a1b0c2d0-4a3b0c1d0+2a1b1c0d2-1a0b3c0d0+1a0b0c0d4+1a3b1c0d0-1a0b0c1d2+3a2b0c0d2-1a1b2c0d0-5a2b0c1d0-1a4b0c0d0-1a0b1c1d0+2a2b1c0d0+2a1b0c0d2-3a0b2c0d0-1a1b0c1d0-3a3b0c0d0-2a0b0c0d2+1a0b0c1d0-2a2b0c0d0-1a0b1c0d0+1a1b0c0d0+1a0b0c0d0),
(1a2b3c0d0-2a3b1c1d0-1a0b4c0d0+1a1b2c1d0+1a4b0c0d2+2a2b0c2d0-1a4b0c1d0-1a2b1c0d2-1a0b1c2d0+1a2b1c1d0+2a0b2c0d2-2a4b1c0d0-2a1b0c1d2-2a0b2c1d0-1a3b0c0d2+3a2b2c0d0+2a1b0c2d0+3a3b0c1d0-3a0b3c0d0-1a5b0c0d0+2a3b1c0d0+1a0b0c1d2-3a2b0c0d2-2a1b2c0d0-1a0b0c2d0+6a2b0c1d0+2a0b1c0d2+1a4b0c0d0-4a0b1c1d0+2a2b1c0d0+2a1b0c0d2-3a0b2c0d0-1a1b0c1d0+4a3b0c0d0-4a1b1c0d0-2a0b0c1d0-1a0b1c0d0-2a1b0c0d0),
(1a3b0c0d5-4a3b0c1d3-2a1b1c0d5+3a3b0c2d1+8a1b1c1d3-1a3b1c0d3-6a1b1c2d1+1a0b0c1d5+4a3b1c1d1-1a2b0c0d5+2a1b2c0d3-4a0b0c2d3+3a2b0c1d3-8a1b2c1d1+1a0b1c0d5+3a0b0c3d1-3a4b0c0d3-1a3b2c0d1-1a2b0c2d1-4a0b1c1d3+6a4b0c1d1+8a2b1c0d3+2a1b3c0d1+5a0b1c2d1-16a2b1c1d1-2a0b2c0d3+2a4b1c0d1-2a1b0c1d3+3a0b2c1d1-5a2b2c0d1+4a1b0c2d1+2a1b1c0d3+1a0b3c0d1+2a5b0c0d1-4a1b1c1d1-4a3b1c0d1-2a0b0c1d3+3a2b0c0d3+4a0b0c2d1-4a2b0c1d1-4a0b1c0d3+2a4b0c0d1+8a0b1c1d1-8a2b1c0d1+4a0b2c0d1+4a1b0c1d1-2a3b0c0d1+1a0b0c1d1-2a2b0c0d1+3a0b1c0d1),
(1a3b0c1d3-2a3b0c2d1-2a1b1c1d3-1a3b1c0d3+4a1b1c2d1-1a2b0c0d5+2a1b2c0d3+1a0b0c2d3+3a2b0c1d3+1a0b1c0d5-2a0b0c3d1+1a4b0c0d3+2a3b2c0d1-2a2b0c2d1-4a0b1c1d3-2a4b0c1d1-4a1b3c0d1+2a0b1c2d1+2a2b1c1d1+1a1b0c0d5-2a0b2c0d3-1a4b1c0d1-2a1b0c1d3+4a0b2c1d1+1a3b0c0d3+1a2b2c0d1-3a3b0c1d1-2a1b1c0d3+1a0b3c0d1-1a5b0c0d1+4a1b1c1d1+1a3b1c0d1-2a0b0c1d3-1a1b2c0d1+3a0b0c2d1-1a2b0c1d1-2a0b1c0d3+6a0b1c1d1+1a2b1c0d1-2a1b0c0d3+2a0b2c0d1+4a1b0c1d1+4a1b1c0d1+2a0b0c1d1-1a2b0c0d1+1a0b1c0d1+1a1b0c0d1)";
o4 : Ideal of R
i5 : C3=ideal
"(1a3b0c2d1-1a3b1c0d3-2a1b1c2d1+2a1b2c0d3-1a2b0c1d3+1a0b0c3d1+1a4b0c0d3+1a3b2c0d1+1a2b0c2d1-1a4b0c1d1-2a1b3c0d1-1a0b1c2d1+1a2b1c1d1-2a0b2c0d3+1a4b1c0d1+2a1b0c1d3+2a0b2c1d1-1a3b0c0d3-4a2b2c0d1-2a1b0c2d1+1a3b0c1d1+2a0b3c0d1-2a5b0c0d1+5a3b1c0d1-2a1b2c0d1-2a0b0c2d1-1a2b0c1d1+2a0b1c1d1-1a2b1c0d1+4a0b2c0d1-2a1b0c1d1+2a3b0c0d1-2a1b1c0d1),
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(1a3b1c1d1-2a1b2c1d1-1a4b0c0d3-1a3b2c0d1-1a2b0c2d1+1a4b0c1d1+2a2b1c0d3+2a1b3c0d1+2a0b1c2d1-1a2b1c1d1+1a4b1c0d1-2a0b2c1d1+2a3b0c0d3-2a2b2c0d1-2a3b0c1d1-4a1b1c0d3+2a5b0c0d1+4a1b1c1d1-6a3b1c0d1-1a2b0c0d3+4a1b2c0d1+1a2b0c1d1+2a0b1c0d3-2a4b0c0d1-2a0b1c1d1+5a2b1c0d1-2a0b2c0d1-2a3b0c0d1+4a1b1c0d1+2a2b0c0d1-4a0b1c0d1),
(1a3b2c0d1-1a4b0c1d1-2a1b3c0d1+1a2b1c1d1-1a4b1c0d1+2a0b2c1d1+1a3b0c0d3+2a2b2c0d1-1a1b0c2d1+1a3b0c1d1-4a1b1c1d1-1a3b1c0d1-3a2b0c0d3+1a0b0c2d1+3a2b0c1d1+1a2b1c0d1+2a1b0c0d3+1a3b0c0d1-2a0b0c1d1+1a2b0c0d1-2a1b0c0d1),
(1a4b0c0d4-3a4b0c1d2-3a2b1c0d4+1a4b0c2d0+9a2b1c1d2+1a0b2c0d4-1a4b1c0d2-3a2b1c2d0+2a1b0c1d4-3a0b2c1d2+2a4b1c1d0-1a3b0c0d4+3a2b2c0d2-6a1b0c2d2+1a0b2c2d0+2a3b0c1d2-6a2b2c1d0+2a1b1c0d4+2a1b0c3d0-1a0b3c0d2-2a5b0c0d2-6a1b1c1d2+2a0b3c1d0-1a0b0c0d6+3a5b0c1d0+8a3b1c0d2+4a1b1c2d0+3a0b0c1d4-10a3b1c1d0-6a1b2c0d2-2a0b0c2d2+1a5b1c0d0-3a2b0c1d2+5a1b2c1d0+2a0b1c0d4+1a0b0c3d0-1a4b0c0d2-4a3b2c0d0+5a2b0c2d0-5a0b1c1d2+2a4b0c1d0+4a2b1c0d2+3a1b3c0d0+3a0b1c2d0+1a6b0c0d0-7a2b1c1d0+3a1b0c0d4-3a0b2c0d2-3a4b1c0d0-11a1b0c1d2+6a0b2c1d0+2a3b0c0d2+9a1b0c2d0-2a3b0c1d0-9a1b1c0d2+1a0b3c0d0+2a5b0c0d0+14a1b1c1d0+3a0b0c0d4-8a3b1c0d0-6a0b0c1d2-2a2b0c0d2+8a1b2c0d0+2a0b0c2d0+7a2b0c1d0-5a0b1c0d2+6a0b1c1d0+1a2b1c0d0-5a1b0c0d2+3a0b2c0d0+9a1b0c1d0-1a3b0c0d0+7a1b1c0d0-3a0b0c0d2+2a0b0c1d0+1a2b0c0d0+3a0b1c0d0+2a1b0c0d0+1a0b0c0d0),
(1a4b0c1d2-1a4b0c2d0-3a2b1c1d2-1a4b1c0d2+3a2b1c2d0+1a0b2c1d2-1a4b1c1d0-1a3b0c0d4+3a2b2c0d2+2a1b0c2d2-1a0b2c2d0+2a3b0c1d2+3a2b2c1d0+2a1b1c0d4-2a1b0c3d0-1a0b3c0d2+1a5b0c0d2+1a4b2c0d0-1a3b0c2d0-6a1b1c1d2-1a0b3c1d0-2a5b0c1d0-1a3b1c0d2-3a2b3c0d0-2a0b0c1d4+5a3b1c1d0+1a2b0c0d4-3a1b2c0d2+1a0b4c0d0+4a0b0c2d2+2a1b2c1d0-2a0b0c3d0+2a4b0c0d2-3a2b0c2d0+5a0b1c1d2-3a4b0c1d0-6a2b1c0d2-4a0b1c2d0-1a6b0c0d0+9a2b1c1d0-1a1b0c0d4+2a4b1c0d0+5a1b0c1d2-3a0b2c1d0-1a3b0c0d2-6a1b0c2d0-1a3b0c1d0+2a0b3c0d0-2a5b0c0d0-2a1b1c1d0+5a3b1c0d0+3a0b0c1d2+1a1b2c0d0-3a0b0c2d0-3a2b0c1d0+1a0b1c0d2-5a0b1c1d0+3a2b1c0d0+2a1b0c0d2-1a0b2c0d0-4a1b0c1d0+1a3b0c0d0-2a1b1c0d0-1a0b0c1d0-1a2b0c0d0-2a0b1c0d0-1a1b0c0d0),
(1a4b0c2d0-1a4b1c0d2-3a2b1c2d0+1a4b1c1d0+3a2b2c0d2+1a0b2c2d0-1a3b0c1d2-3a2b2c1d0+2a1b0c3d0-1a0b3c0d2+1a3b0c2d0+1a0b3c1d0+1a5b0c1d0+1a3b1c0d2-3a3b1c1d0-2a1b2c0d2-2a0b0c2d2+1a5b1c0d0+1a2b0c1d2+1a1b2c1d0+1a0b1c0d4+2a0b0c3d0-4a3b2c0d0+1a2b0c2d0-2a0b1c1d2+1a4b0c1d0+3a1b3c0d0+3a0b1c2d0-2a2b1c1d0-1a0b2c0d2+1a4b1c0d0-1a1b0c1d2+2a0b2c1d0-3a2b2c0d0+2a1b0c2d0-1a3b0c1d0-2a1b1c0d2+1a0b3c0d0+6a1b1c1d0-1a3b1c0d0+4a1b2c0d0+1a0b0c2d0-2a0b1c0d2+3a0b1c1d0+2a0b2c0d0+1a1b0c1d0+1a1b1c0d0+1a0b1c0d0),
(1a4b1c0d2-1a4b1c1d0-3a2b2c0d2-1a3b0c1d2+3a2b2c1d0+1a0b3c0d2-1a5b0c0d2-1a4b2c0d0+1a3b0c2d0+4a1b1c1d2-1a0b3c1d0+3a3b1c0d2+3a2b3c0d0-4a1b1c2d0+1a3b1c1d0+1a2b0c0d4-1a1b2c0d2-1a0b4c0d0-1a0b0c2d2-4a2b0c1d2-4a1b2c1d0-2a0b1c0d4+1a0b0c3d0+1a4b0c0d2+1a2b0c2d0+4a0b1c1d2+1a4b0c1d0-5a2b1c0d2-1a0b1c2d0+1a6b0c0d0+1a2b1c1d0+5a0b2c0d2-4a4b1c0d0+3a1b0c1d2-5a0b2c1d0-2a3b0c0d2+6a2b2c0d0-1a1b0c2d0+5a3b0c1d0+4a1b1c0d2-4a0b3c0d0-10a1b1c1d0-1a0b0c0d4+1a3b1c0d0+3a0b0c1d2-2a2b0c0d2-3a1b2c0d0-1a0b0c2d0+1a2b0c1d0+6a0b1c0d2-1a4b0c0d0-7a0b1c1d0+4a2b1c0d0+2a1b0c0d2-6a0b2c0d0-6a1b0c1d0+3a3b0c0d0-6a1b1c0d0+2a0b0c0d2-3a0b0c1d0+1a2b0c0d0-4a0b1c0d0-3a1b0c0d0-1a0b0c0d0),
(1a4b1c1d0-3a2b2c1d0-1a5b0c0d2-1a3b0c2d0+1a0b3c1d0+1a5b0c1d0+3a3b1c0d2+4a1b1c2d0-3a3b1c1d0-1a1b2c0d2-1a2b0c1d2+1a1b2c1d0-1a0b0c3d0+1a4b0c0d2+1a2b0c2d0-2a0b1c1d2-2a2b1c0d2+2a0b1c2d0+1a6b0c0d0-2a2b1c1d0+1a1b0c0d4-4a4b1c0d0-2a1b0c1d2+3a0b2c1d0+3a2b2c0d0+3a1b0c2d0-1a1b1c0d2+1a5b0c0d0+4a1b1c1d0-3a3b1c0d0-1a0b0c1d2-2a2b0c0d2+1a1b2c0d0+2a0b0c2d0+4a2b0c1d0-1a4b0c0d0+3a0b1c1d0+4a2b1c0d0-2a1b0c0d2+3a1b0c1d0+2a1b1c0d0+1a0b0c1d0+1a2b0c0d0+1a1b0c0d0),
(1a4b2c0d0-1a5b0c1d0-3a2b3c0d0+2a3b1c1d0+1a0b4c0d0+3a1b2c1d0+1a4b0c0d2-2a2b0c2d0-1a4b0c1d0-1a2b1c0d2-1a0b1c2d0-1a6b0c0d0+1a2b1c1d0-2a0b2c0d2+3a4b1c0d0+2a1b0c1d2+2a0b2c1d0-1a3b0c0d2-3a2b2c0d0-2a1b0c2d0-1a3b0c1d0+3a0b3c0d0-1a5b0c0d0+2a3b1c0d0+1a2b0c0d2+2a1b2c0d0-2a2b0c1d0-1a0b1c0d2+1a4b0c0d0+2a0b1c1d0-2a2b1c0d0+3a0b2c0d0-1a1b0c1d0-1a2b0c0d0+1a0b1c0d0),
(1a5b0c0d3-2a5b0c1d1-4a3b1c0d3+8a3b1c1d1+3a1b2c0d3-1a5b1c0d1+3a2b0c1d3-6a1b2c1d1-1a4b0c0d3+4a3b2c0d1-6a2b0c2d1-2a0b1c1d3+1a4b0c1d1+3a2b1c0d3-3a1b3c0d1+4a0b1c2d1-1a6b0c0d1-6a2b1c1d1-2a1b0c0d5-1a0b2c0d3+6a4b1c0d1+4a1b0c1d3+3a0b2c1d1-9a2b2c0d1-2a1b0c2d1-2a3b0c1d1+4a1b1c0d3+2a0b3c0d1-2a1b1c1d1+2a0b0c0d5+1a3b1c0d1-5a0b0c1d3+4a2b0c0d3-4a1b2c0d1+4a0b0c2d1-7a2b0c1d1-5a0b1c0d3+8a0b1c1d1-6a2b1c0d1+5a0b2c0d1-2a3b0c0d1+1a1b1c0d1-4a0b0c0d3+5a0b0c1d1+4a0b1c0d1+2a1b0c0d1+1a0b0c0d1),
(1a5b0c1d1-4a3b1c1d1-1a5b1c0d1+3a1b2c1d1-1a4b0c0d3+4a3b2c0d1+3a2b0c2d1+1a4b0c1d1+3a2b1c0d3-3a1b3c0d1-2a0b1c2d1-6a2b1c1d1-1a0b2c0d3+1a4b1c0d1-4a1b0c1d3+3a0b2c1d1+1a3b0c0d3-3a2b2c0d1+4a1b0c2d1-1a3b0c1d1+1a0b3c0d1+2a5b0c0d1+4a1b1c1d1+1a0b0c0d5-7a3b1c0d1+2a1b2c0d1-1a0b0c2d1+5a2b0c1d1-3a0b1c0d3+2a0b1c1d1-3a2b1c0d1-3a1b0c0d3+3a0b2c0d1+5a1b0c1d1-2a3b0c0d1+7a1b1c0d1-2a0b0c0d3-1a0b0c1d1+2a2b0c0d1+3a0b1c0d1+2a1b0c0d1+1a0b0c0d1),
(1a5b1c0d1-4a3b2c0d1-1a4b0c1d1+3a1b3c0d1-1a6b0c0d1+6a2b1c1d1+4a4b1c0d1-3a0b2c1d1+1a3b0c0d3-3a2b2c0d1-2a1b0c2d1-4a3b0c1d1-4a1b1c0d3+6a1b1c1d1-1a3b1c0d1+2a0b0c1d3+1a2b0c0d3+4a1b2c0d1-2a0b0c2d1-1a2b0c1d1+2a0b1c0d3-4a0b1c1d1-1a1b0c0d3-1a0b2c0d1+2a1b0c1d1-2a3b0c0d1+3a1b1c0d1-1a0b0c0d3-1a0b0c1d1-2a0b1c0d1+2a1b0c0d1+1a0b0c0d1),
(1a6b0c0d2-1a6b0c1d0-5a4b1c0d2+5a4b1c1d0+6a2b2c0d2-1a6b1c0d0+4a3b0c1d2-6a2b2c1d0-1a0b3c0d2-1a5b0c0d2+5a4b2c0d0-4a3b0c2d0-6a1b1c1d2+1a0b3c1d0+4a3b1c0d2-6a2b3c0d0+6a1b1c2d0-1a7b0c0d0-4a3b1c1d0-3a2b0c0d4-3a1b2c0d2+1a0b4c0d0+1a0b0c2d2+6a5b1c0d0+3a2b0c1d2+6a1b2c1d0+2a0b1c0d4-1a0b0c3d0-10a3b2c0d0-3a2b0c2d0-2a0b1c1d2-3a4b0c1d0+6a2b1c0d2+4a1b3c0d0+1a0b1c2d0+4a1b0c0d4-4a0b2c0d2+2a4b1c0d0-6a1b0c1d2+3a0b2c1d0+6a3b0c0d2-9a2b2c0d0+3a1b0c2d0-7a3b0c1d0-12a1b1c0d2+4a0b3c0d0+1a5b0c0d0+16a1b1c1d0-1a0b0c0d4-11a3b1c0d0-1a0b0c1d2-2a2b0c0d2+13a1b2c0d0+1a0b0c2d0+3a2b0c1d0-3a0b1c0d2-4a4b0c0d0+3a0b1c1d0+6a2b1c0d0-6a1b0c0d2+4a0b2c0d0+6a1b0c1d0-1a3b0c0d0+10a1b1c0d0+2a0b0c0d2+1a0b0c1d0+5a2b0c0d0+1a1b0c0d0-1a0b0c0d0),
(1a6b0c1d0-5a4b1c1d0+6a2b2c1d0-1a5b0c0d2+4a3b0c2d0-1a0b3c1d0+1a5b0c1d0+4a3b1c0d2-6a1b1c2d0-4a3b1c1d0-3a1b2c0d2-6a2b0c1d2+3a1b2c1d0+1a0b0c3d0+1a4b0c0d2+6a2b0c2d0+4a0b1c1d2-1a4b0c1d0-3a2b1c0d2-4a0b1c2d0+1a6b0c0d0+6a2b1c1d0+2a1b0c0d4+1a0b2c0d2-5a4b1c0d0-3a0b2c1d0+6a2b2c0d0-2a1b0c2d0+5a3b0c1d0-2a1b1c0d2-1a0b3c0d0+1a5b0c0d0-4a1b1c1d0-2a0b0c0d4-4a3b1c0d0+4a0b0c1d2-4a2b0c0d2+3a1b2c0d0-4a0b0c2d0+8a2b0c1d0+3a0b1c0d2-1a4b0c0d0-7a0b1c1d0+6a2b1c0d0-1a1b0c0d2-3a0b2c0d0-3a1b0c1d0+1a3b0c0d0+2a1b1c0d0+4a0b0c0d2-5a0b0c1d0+2a2b0c0d0-3a0b1c0d0-1a1b0c0d0-1a0b0c0d0),
(1a6b1c0d0-5a4b2c0d0-1a5b0c1d0+6a2b3c0d0+8a3b1c1d0-1a0b4c0d0-9a1b2c1d0+1a4b0c0d2-3a2b0c2d0-1a4b0c1d0-6a2b1c0d2+3a0b1c2d0-1a6b0c0d0+6a2b1c1d0+3a0b2c0d2+3a4b1c0d0+4a1b0c1d2-3a0b2c1d0-1a3b0c0d2+3a2b2c0d0-4a1b0c2d0-2a3b0c1d0+4a1b1c0d2-3a0b3c0d0-1a5b0c0d0-6a1b1c1d0-1a0b0c0d4+7a3b1c0d0+2a2b0c0d2-7a1b2c0d0+1a0b0c2d0-5a2b0c1d0+2a0b1c0d2+1a4b0c0d0-2a0b1c1d0-3a2b1c0d0+1a1b0c0d2-3a0b2c0d0-1a1b0c1d0-1a3b0c0d0-6a1b1c0d0+1a0b0c0d2+1a0b0c1d0-2a2b0c0d0-1a0b1c0d0+1a1b0c0d0),
(1a7b0c0d1-6a5b1c0d1+10a3b2c0d1+5a4b0c1d1-4a1b3c0d1-1a6b0c0d1-12a2b1c1d1+5a4b1c0d1+3a0b2c1d1-4a3b0c0d3-6a2b2c0d1+3a1b0c2d1+6a1b1c0d3+1a0b3c0d1+4a3b1c0d1-2a0b0c1d3+6a2b0c0d3-6a1b2c0d1+1a0b0c2d1-6a2b0c1d1-4a0b1c0d3+4a4b0c0d1+6a0b1c1d1-12a2b1c0d1-2a1b0c0d3+4a0b2c0d1+2a1b0c1d1-2a3b0c0d1+3a0b0c1d1-4a2b0c0d1+5a0b1c0d1+2a1b0c0d1),
(1a8b0c0d0-7a6b1c0d0+15a4b2c0d0+6a5b0c1d0-10a2b3c0d0-20a3b1c1d0+1a0b4c0d0+12a1b2c1d0-5a4b0c0d2+6a2b0c2d0+5a4b0c1d0+12a2b1c0d2-3a0b1c2d0-1a6b0c0d0-12a2b1c1d0-3a0b2c0d2+10a4b1c0d0-6a1b0c1d2+3a0b2c1d0+4a3b0c0d2-18a2b2c0d0+6a1b0c2d0-8a3b0c1d0-6a1b1c0d2+4a0b3c0d0+5a5b0c0d0+18a1b1c1d0+1a0b0c0d4-16a3b1c0d0+3a2b0c0d2+9a1b2c0d0-2a0b0c2d0-4a0b1c0d2+1a4b0c0d0+4a0b1c1d0-9a2b1c0d0-4a1b0c0d2+6a0b2c0d0+8a1b0c1d0-7a3b0c0d0+14a1b1c0d0-2a0b0c0d2+1a0b0c1d0+4a0b1c0d0+3a1b0c0d0+1a0b0c0d0)";
o5 : Ideal of R
i6 : C=C1+C2+C3;
o6 : Ideal of R
i7 : isSubset(C1,C)
o7 = true
i8 : isSubset(C2,C)
o8 = true
i9 : isSubset(C3,C)
o9 = false
i10 : isSubset(M6,C1)
o10 = true
i11 : isSubset(M6,C2)
o11 = true
i12 : isSubset(M6,C3)
o12 = true
i13 : isSubset(M6,C)
o13 = false
i14 : D=C1+C2+C3+M6;
o14 : Ideal of R
i15 : C==D
o15 = false
-----
Example 2:
Macaulay 2, version 1.2
with packages: Elimination, IntegralClosure, LLLBases,
PrimaryDecomposition, ReesAlgebra, SchurRings, TangentCone
i1 : R=ZZ[a,b,c,d];
i2 : M6=ideal
"(1a8b0c0d0-7a6b1c0d0+15a4b2c0d0+6a5b0c1d0-10a2b3c0d0-20a3b1c1d0+1a0b4c0d0+12a1b2c1d0-5a4b0c0d2+6a2b0c2d0+5a4b0c1d0+12a2b1c0d2-3a0b1c2d0-1a6b0c0d0-12a2b1c1d0-3a0b2c0d2+10a4b1c0d0-6a1b0c1d2+3a0b2c1d0+4a3b0c0d2-18a2b2c0d0+6a1b0c2d0-8a3b0c1d0-6a1b1c0d2+4a0b3c0d0+5a5b0c0d0+18a1b1c1d0+1a0b0c0d4-16a3b1c0d0+3a2b0c0d2+9a1b2c0d0-2a0b0c2d0-4a0b1c0d2+1a4b0c0d0+4a0b1c1d0-9a2b1c0d0-4a1b0c0d2+6a0b2c0d0+8a1b0c1d0-7a3b0c0d0+14a1b1c0d0-2a0b0c0d2+1a0b0c1d0+4a0b1c0d0+3a1b0c0d0+1a0b0c0d0)";
o2 : Ideal of R
i3 : C=ideal
"(1a0b0c0d8-7a0b0c1d6+15a0b0c2d4-1a0b1c0d6-10a0b0c3d2+10a0b1c1d4+1a0b0c4d0-18a0b1c2d2-6a1b0c0d6-4a0b2c0d4+4a0b1c3d0+27a1b0c1d4+3a0b2c1d2-33a1b0c2d2+3a0b2c2d0+11a1b1c0d4+7a1b0c3d0+5a0b3c0d2-30a1b1c1d2-2a0b3c1d0-2a0b0c0d6+15a1b1c2d0+13a0b0c1d4+9a2b0c0d4-3a1b2c0d2-1a0b4c0d0-21a0b0c2d2-27a2b0c1d2+3a1b2c1d0-3a0b1c0d4+7a0b0c3d0+13a2b0c2d0-6a0b1c1d2-13a2b1c0d2-1a1b3c0d0+11a0b1c2d0+14a2b1c1d0+14a1b0c0d4+13a0b2c0d2-38a1b0c1d2-5a0b2c1d0-3a3b0c0d2+3a2b2c0d0+22a1b0c2d0+7a3b0c1d0-16a1b1c0d2-5a0b3c0d0+18a1b1c1d0-1a0b0c0d4+3a3b1c0d0-4a0b0c1d2-17a2b0c0d2+6a0b0c2d0+19a2b0c1d0+10a0b1c0d2-2a0b1c1d0+11a2b1c0d0-7a1b0c0d2-8a0b2c0d0+11a1b0c1d0+4a3b0c0d0+3a1b1c0d0+3a0b0c0d2-1a0b0c1d0+6a2b0c0d0-5a0b1c0d0-1a0b0c0d0),
(1a0b0c1d6-5a0b0c2d4-1a0b1c0d6+6a0b0c3d2+3a0b1c1d4-1a0b0c4d0+3a0b1c2d2+1a1b0c0d6+2a0b2c0d4-3a0b1c3d0-8a1b0c1d4-9a0b2c1d2+15a1b0c2d2-6a1b0c3d0+6a1b1c1d2+3a0b3c1d0-1a0b0c0d6-8a1b1c2d0+2a0b0c1d4-2a2b0c0d4-1a1b2c0d2+3a0b0c2d2+11a2b0c1d2+2a1b2c1d0+6a0b1c0d4-5a0b0c3d0-9a2b0c2d0-12a0b1c1d2-1a0b1c2d0-7a2b1c1d0-3a1b0c0d4-8a0b2c0d2+11a1b0c1d2+10a0b2c1d0+1a3b0c0d2-11a1b0c2d0-4a3b0c1d0+1a1b1c0d2+2a0b3c0d0-4a1b1c1d0+4a0b0c0d4-7a0b0c1d2+8a2b0c0d2+2a1b2c0d0+1a0b0c2d0-10a2b0c1d0-12a0b1c0d2+9a0b1c1d0-4a2b1c0d0-1a1b0c0d2+7a0b2c0d0-2a1b0c1d0-3a3b0c0d0+2a1b1c0d0-4a0b0c0d2+4a0b0c1d0-3a2b0c0d0+6a0b1c0d0+2a1b0c0d0+1a0b0c0d0),
(1a0b0c2d4-1a0b1c0d6-3a0b0c3d2+3a0b1c1d4+1a0b0c4d0-3a0b1c2d2+1a1b0c0d6+2a0b2c0d4+3a0b1c3d0-2a1b0c1d4-3a0b2c1d2-3a1b0c2d2+1a0b2c2d0+5a1b0c3d0-1a0b3c0d2-6a1b1c1d2-1a0b3c1d0-1a0b0c0d6+7a1b1c2d0+2a0b0c1d4-2a2b0c0d4-2a0b0c2d2+3a2b0c1d2+2a1b2c1d0+7a0b1c0d4+4a0b0c3d0+5a2b0c2d0-11a0b1c1d2+1a2b1c0d2+3a0b1c2d0+5a2b1c1d0-1a1b0c0d4-6a0b2c0d2-2a1b0c1d2+6a1b0c2d0-6a1b1c0d2+10a1b1c1d0+3a0b0c0d4-3a0b0c1d2+3a2b0c0d2+2a1b2c0d0+2a2b0c1d0-9a0b1c0d2+3a0b1c1d0+3a2b1c0d0+1a0b2c0d0+2a1b0c1d0+4a1b1c0d0-3a0b0c0d2-1a2b0c0d0+2a0b1c0d0+1a0b0c0d0),
(1a0b0c3d2-2a0b1c1d4-1a0b0c4d0+3a0b1c2d2+1a1b0c0d6+2a0b2c0d4-3a0b1c3d0-3a1b0c1d4+1a0b2c1d2+5a1b0c2d2-1a0b2c2d0-5a1b1c0d4-5a1b0c3d0-4a0b3c0d2+12a1b1c1d2+2a0b3c1d0-1a0b0c0d6-9a1b1c2d0+2a0b0c1d4-3a2b0c0d4+2a1b2c0d2+1a0b4c0d0+1a0b0c2d2+7a2b0c1d2-3a1b2c1d0+4a0b1c0d4-3a0b0c3d0-7a2b0c2d0-3a0b1c1d2+10a2b1c0d2+1a1b3c0d0-2a0b1c2d0-10a2b1c1d0+1a1b0c0d4-9a0b2c0d2+2a1b0c1d2+5a0b2c1d0+2a3b0c0d2-3a2b2c0d0-7a1b0c2d0-3a3b0c1d0+2a1b1c0d2+5a0b3c0d0-8a1b1c1d0+2a0b0c0d4-3a3b1c0d0-5a0b0c1d2+1a2b0c0d2+1a1b2c0d0+1a0b0c2d0-6a2b0c1d0-6a0b1c0d2+6a0b1c1d0-7a2b1c0d0-2a1b0c0d2+7a0b2c0d0+1a1b0c1d0-1a3b0c0d0+1a1b1c0d0-1a0b0c0d2+3a0b0c1d0+3a0b1c0d0+1a1b0c0d0),
(1a0b0c4d0-3a0b1c2d2+1a0b2c0d4+3a0b1c3d0+2a1b0c1d4-6a1b0c2d2+1a0b2c2d0-2a1b1c0d4+5a1b0c3d0-1a0b3c0d2-2a0b3c1d0-1a0b0c0d6+9a1b1c2d0+5a0b0c1d4+1a2b0c0d4+1a1b2c0d2-8a0b0c2d2-8a2b0c1d2+1a0b1c0d4+3a0b0c3d0+8a2b0c2d0-2a0b1c1d2+1a2b1c0d2+4a0b1c2d0+6a2b1c1d0+4a1b0c0d4-1a0b2c0d2-13a1b0c1d2-4a0b2c1d0-1a3b0c0d2+11a1b0c2d0+4a3b0c1d0-1a1b1c0d2-1a0b3c0d0+6a1b1c1d0+1a0b0c0d4-5a0b0c1d2-7a2b0c0d2+3a0b0c2d0+11a2b0c1d0+2a0b1c0d2-2a0b1c1d0+4a2b1c0d0-5a1b0c0d2-3a0b2c0d0+6a1b0c1d0+3a3b0c0d0+2a0b0c0d2+4a2b0c0d0-3a0b1c0d0-1a0b0c0d0),
(1a0b1c0d6-5a0b1c1d4+6a0b1c2d2-1a1b0c0d6-1a0b2c0d4-1a0b1c3d0+4a1b0c1d4+6a0b2c1d2-3a1b0c2d2-3a0b2c2d0-2a1b1c0d4-2a0b3c0d2+4a1b1c1d2-1a0b3c1d0+1a0b0c0d6-4a1b1c2d0-4a0b0c1d4+3a2b0c0d4+5a1b2c0d2+1a0b4c0d0+4a0b0c2d2-7a2b0c1d2-4a1b2c1d0-5a0b1c0d4-1a0b0c3d0+1a2b0c2d0+15a0b1c1d2+1a2b1c0d2-8a0b1c2d0-1a2b1c1d0+1a1b0c0d4+3a0b2c0d2-6a0b2c1d0-3a3b0c0d2-3a2b2c0d0-2a1b0c2d0+2a3b0c1d0+4a1b1c0d2+2a0b3c0d0-8a1b1c1d0-4a0b0c0d4+9a0b0c1d2-4a2b0c0d2-4a1b2c0d0-4a0b0c2d0+2a2b0c1d0+9a0b1c0d2+1a4b0c0d0-9a0b1c1d0-1a2b1c0d0+3a1b0c0d2-2a0b2c0d0-4a1b0c1d0+2a3b0c0d0-4a1b1c0d0+4a0b0c0d2-4a0b0c1d0-4a0b1c0d0-2a1b0c0d0-1a0b0c0d0),
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(1a4b1c0d2-1a4b1c1d0-3a2b2c0d2-1a3b0c1d2+3a2b2c1d0+1a0b3c0d2-1a5b0c0d2-1a4b2c0d0+1a3b0c2d0+4a1b1c1d2-1a0b3c1d0+3a3b1c0d2+3a2b3c0d0-4a1b1c2d0+1a3b1c1d0+1a2b0c0d4-1a1b2c0d2-1a0b4c0d0-1a0b0c2d2-4a2b0c1d2-4a1b2c1d0-2a0b1c0d4+1a0b0c3d0+1a4b0c0d2+1a2b0c2d0+4a0b1c1d2+1a4b0c1d0-5a2b1c0d2-1a0b1c2d0+1a6b0c0d0+1a2b1c1d0+5a0b2c0d2-4a4b1c0d0+3a1b0c1d2-5a0b2c1d0-2a3b0c0d2+6a2b2c0d0-1a1b0c2d0+5a3b0c1d0+4a1b1c0d2-4a0b3c0d0-10a1b1c1d0-1a0b0c0d4+1a3b1c0d0+3a0b0c1d2-2a2b0c0d2-3a1b2c0d0-1a0b0c2d0+1a2b0c1d0+6a0b1c0d2-1a4b0c0d0-7a0b1c1d0+4a2b1c0d0+2a1b0c0d2-6a0b2c0d0-6a1b0c1d0+3a3b0c0d0-6a1b1c0d0+2a0b0c0d2-3a0b0c1d0+1a2b0c0d0-4a0b1c0d0-3a1b0c0d0-1a0b0c0d0),
(1a4b1c1d0-3a2b2c1d0-1a5b0c0d2-1a3b0c2d0+1a0b3c1d0+1a5b0c1d0+3a3b1c0d2+4a1b1c2d0-3a3b1c1d0-1a1b2c0d2-1a2b0c1d2+1a1b2c1d0-1a0b0c3d0+1a4b0c0d2+1a2b0c2d0-2a0b1c1d2-2a2b1c0d2+2a0b1c2d0+1a6b0c0d0-2a2b1c1d0+1a1b0c0d4-4a4b1c0d0-2a1b0c1d2+3a0b2c1d0+3a2b2c0d0+3a1b0c2d0-1a1b1c0d2+1a5b0c0d0+4a1b1c1d0-3a3b1c0d0-1a0b0c1d2-2a2b0c0d2+1a1b2c0d0+2a0b0c2d0+4a2b0c1d0-1a4b0c0d0+3a0b1c1d0+4a2b1c0d0-2a1b0c0d2+3a1b0c1d0+2a1b1c0d0+1a0b0c1d0+1a2b0c0d0+1a1b0c0d0),
(1a4b2c0d0-1a5b0c1d0-3a2b3c0d0+2a3b1c1d0+1a0b4c0d0+3a1b2c1d0+1a4b0c0d2-2a2b0c2d0-1a4b0c1d0-1a2b1c0d2-1a0b1c2d0-1a6b0c0d0+1a2b1c1d0-2a0b2c0d2+3a4b1c0d0+2a1b0c1d2+2a0b2c1d0-1a3b0c0d2-3a2b2c0d0-2a1b0c2d0-1a3b0c1d0+3a0b3c0d0-1a5b0c0d0+2a3b1c0d0+1a2b0c0d2+2a1b2c0d0-2a2b0c1d0-1a0b1c0d2+1a4b0c0d0+2a0b1c1d0-2a2b1c0d0+3a0b2c0d0-1a1b0c1d0-1a2b0c0d0+1a0b1c0d0),
(1a5b0c0d3-2a5b0c1d1-4a3b1c0d3+8a3b1c1d1+3a1b2c0d3-1a5b1c0d1+3a2b0c1d3-6a1b2c1d1-1a4b0c0d3+4a3b2c0d1-6a2b0c2d1-2a0b1c1d3+1a4b0c1d1+3a2b1c0d3-3a1b3c0d1+4a0b1c2d1-1a6b0c0d1-6a2b1c1d1-2a1b0c0d5-1a0b2c0d3+6a4b1c0d1+4a1b0c1d3+3a0b2c1d1-9a2b2c0d1-2a1b0c2d1-2a3b0c1d1+4a1b1c0d3+2a0b3c0d1-2a1b1c1d1+2a0b0c0d5+1a3b1c0d1-5a0b0c1d3+4a2b0c0d3-4a1b2c0d1+4a0b0c2d1-7a2b0c1d1-5a0b1c0d3+8a0b1c1d1-6a2b1c0d1+5a0b2c0d1-2a3b0c0d1+1a1b1c0d1-4a0b0c0d3+5a0b0c1d1+4a0b1c0d1+2a1b0c0d1+1a0b0c0d1),
(1a5b0c1d1-4a3b1c1d1-1a5b1c0d1+3a1b2c1d1-1a4b0c0d3+4a3b2c0d1+3a2b0c2d1+1a4b0c1d1+3a2b1c0d3-3a1b3c0d1-2a0b1c2d1-6a2b1c1d1-1a0b2c0d3+1a4b1c0d1-4a1b0c1d3+3a0b2c1d1+1a3b0c0d3-3a2b2c0d1+4a1b0c2d1-1a3b0c1d1+1a0b3c0d1+2a5b0c0d1+4a1b1c1d1+1a0b0c0d5-7a3b1c0d1+2a1b2c0d1-1a0b0c2d1+5a2b0c1d1-3a0b1c0d3+2a0b1c1d1-3a2b1c0d1-3a1b0c0d3+3a0b2c0d1+5a1b0c1d1-2a3b0c0d1+7a1b1c0d1-2a0b0c0d3-1a0b0c1d1+2a2b0c0d1+3a0b1c0d1+2a1b0c0d1+1a0b0c0d1),
(1a5b1c0d1-4a3b2c0d1-1a4b0c1d1+3a1b3c0d1-1a6b0c0d1+6a2b1c1d1+4a4b1c0d1-3a0b2c1d1+1a3b0c0d3-3a2b2c0d1-2a1b0c2d1-4a3b0c1d1-4a1b1c0d3+6a1b1c1d1-1a3b1c0d1+2a0b0c1d3+1a2b0c0d3+4a1b2c0d1-2a0b0c2d1-1a2b0c1d1+2a0b1c0d3-4a0b1c1d1-1a1b0c0d3-1a0b2c0d1+2a1b0c1d1-2a3b0c0d1+3a1b1c0d1-1a0b0c0d3-1a0b0c1d1-2a0b1c0d1+2a1b0c0d1+1a0b0c0d1),
(1a6b0c0d2-1a6b0c1d0-5a4b1c0d2+5a4b1c1d0+6a2b2c0d2-1a6b1c0d0+4a3b0c1d2-6a2b2c1d0-1a0b3c0d2-1a5b0c0d2+5a4b2c0d0-4a3b0c2d0-6a1b1c1d2+1a0b3c1d0+4a3b1c0d2-6a2b3c0d0+6a1b1c2d0-1a7b0c0d0-4a3b1c1d0-3a2b0c0d4-3a1b2c0d2+1a0b4c0d0+1a0b0c2d2+6a5b1c0d0+3a2b0c1d2+6a1b2c1d0+2a0b1c0d4-1a0b0c3d0-10a3b2c0d0-3a2b0c2d0-2a0b1c1d2-3a4b0c1d0+6a2b1c0d2+4a1b3c0d0+1a0b1c2d0+4a1b0c0d4-4a0b2c0d2+2a4b1c0d0-6a1b0c1d2+3a0b2c1d0+6a3b0c0d2-9a2b2c0d0+3a1b0c2d0-7a3b0c1d0-12a1b1c0d2+4a0b3c0d0+1a5b0c0d0+16a1b1c1d0-1a0b0c0d4-11a3b1c0d0-1a0b0c1d2-2a2b0c0d2+13a1b2c0d0+1a0b0c2d0+3a2b0c1d0-3a0b1c0d2-4a4b0c0d0+3a0b1c1d0+6a2b1c0d0-6a1b0c0d2+4a0b2c0d0+6a1b0c1d0-1a3b0c0d0+10a1b1c0d0+2a0b0c0d2+1a0b0c1d0+5a2b0c0d0+1a1b0c0d0-1a0b0c0d0),
(1a6b0c1d0-5a4b1c1d0+6a2b2c1d0-1a5b0c0d2+4a3b0c2d0-1a0b3c1d0+1a5b0c1d0+4a3b1c0d2-6a1b1c2d0-4a3b1c1d0-3a1b2c0d2-6a2b0c1d2+3a1b2c1d0+1a0b0c3d0+1a4b0c0d2+6a2b0c2d0+4a0b1c1d2-1a4b0c1d0-3a2b1c0d2-4a0b1c2d0+1a6b0c0d0+6a2b1c1d0+2a1b0c0d4+1a0b2c0d2-5a4b1c0d0-3a0b2c1d0+6a2b2c0d0-2a1b0c2d0+5a3b0c1d0-2a1b1c0d2-1a0b3c0d0+1a5b0c0d0-4a1b1c1d0-2a0b0c0d4-4a3b1c0d0+4a0b0c1d2-4a2b0c0d2+3a1b2c0d0-4a0b0c2d0+8a2b0c1d0+3a0b1c0d2-1a4b0c0d0-7a0b1c1d0+6a2b1c0d0-1a1b0c0d2-3a0b2c0d0-3a1b0c1d0+1a3b0c0d0+2a1b1c0d0+4a0b0c0d2-5a0b0c1d0+2a2b0c0d0-3a0b1c0d0-1a1b0c0d0-1a0b0c0d0),
(1a6b1c0d0-5a4b2c0d0-1a5b0c1d0+6a2b3c0d0+8a3b1c1d0-1a0b4c0d0-9a1b2c1d0+1a4b0c0d2-3a2b0c2d0-1a4b0c1d0-6a2b1c0d2+3a0b1c2d0-1a6b0c0d0+6a2b1c1d0+3a0b2c0d2+3a4b1c0d0+4a1b0c1d2-3a0b2c1d0-1a3b0c0d2+3a2b2c0d0-4a1b0c2d0-2a3b0c1d0+4a1b1c0d2-3a0b3c0d0-1a5b0c0d0-6a1b1c1d0-1a0b0c0d4+7a3b1c0d0+2a2b0c0d2-7a1b2c0d0+1a0b0c2d0-5a2b0c1d0+2a0b1c0d2+1a4b0c0d0-2a0b1c1d0-3a2b1c0d0+1a1b0c0d2-3a0b2c0d0-1a1b0c1d0-1a3b0c0d0-6a1b1c0d0+1a0b0c0d2+1a0b0c1d0-2a2b0c0d0-1a0b1c0d0+1a1b0c0d0),
(1a7b0c0d1-6a5b1c0d1+10a3b2c0d1+5a4b0c1d1-4a1b3c0d1-1a6b0c0d1-12a2b1c1d1+5a4b1c0d1+3a0b2c1d1-4a3b0c0d3-6a2b2c0d1+3a1b0c2d1+6a1b1c0d3+1a0b3c0d1+4a3b1c0d1-2a0b0c1d3+6a2b0c0d3-6a1b2c0d1+1a0b0c2d1-6a2b0c1d1-4a0b1c0d3+4a4b0c0d1+6a0b1c1d1-12a2b1c0d1-2a1b0c0d3+4a0b2c0d1+2a1b0c1d1-2a3b0c0d1+3a0b0c1d1-4a2b0c0d1+5a0b1c0d1+2a1b0c0d1),
(1a8b0c0d0-7a6b1c0d0+15a4b2c0d0+6a5b0c1d0-10a2b3c0d0-20a3b1c1d0+1a0b4c0d0+12a1b2c1d0-5a4b0c0d2+6a2b0c2d0+5a4b0c1d0+12a2b1c0d2-3a0b1c2d0-1a6b0c0d0-12a2b1c1d0-3a0b2c0d2+10a4b1c0d0-6a1b0c1d2+3a0b2c1d0+4a3b0c0d2-18a2b2c0d0+6a1b0c2d0-8a3b0c1d0-6a1b1c0d2+4a0b3c0d0+5a5b0c0d0+18a1b1c1d0+1a0b0c0d4-16a3b1c0d0+3a2b0c0d2+9a1b2c0d0-2a0b0c2d0-4a0b1c0d2+1a4b0c0d0+4a0b1c1d0-9a2b1c0d0-4a1b0c0d2+6a0b2c0d0+8a1b0c1d0-7a3b0c0d0+14a1b1c0d0-2a0b0c0d2+1a0b0c1d0+4a0b1c0d0+3a1b0c0d0+1a0b0c0d0)";
o3 : Ideal of R
i4 : isSubset(M6,C)
o4 = false
-----
Comment on Example 2: The polynomial M6 is exactly the last generator
of the ideal C.
-----
Daniel R. Grayson
Apr 30, 2009, 10:31:27 PM4/30/09
to maca...@googlegroups.com
That looks like a real bug ...
Thomas Kahle
May 1, 2009, 5:17:37 AM5/1/09
to maca...@googlegroups.com
Christopher Douglas wrote:
> Dear all,
>
> I am comparing ideals in polynomial rings, using the commands == and
> isSubset, and am encountering a problem. It seems like the output I
> am getting suggests a bug in these commands, but perhaps I have
> misunderstood the basic syntax or some other issue. In short, (1) the
> command "isSubset(A,A+B)" is returning "false", where A and B are
> ideals in a polynomial ring, and (2) the command "isSubset(A,B+C)" is
> returning false, even though "isSubset(A,B)" and "isSubset(A,C)" both
> return "true". More specifically, the session is as follows, with ...
> replacing various lists of polynomials:
> Dear all,
>
> I am comparing ideals in polynomial rings, using the commands == and
> isSubset, and am encountering a problem. It seems like the output I
> am getting suggests a bug in these commands, but perhaps I have
> misunderstood the basic syntax or some other issue. In short, (1) the
> command "isSubset(A,A+B)" is returning "false", where A and B are
> ideals in a polynomial ring, and (2) the command "isSubset(A,B+C)" is
> returning false, even though "isSubset(A,B)" and "isSubset(A,C)" both
> return "true". More specifically, the session is as follows, with ...
> replacing various lists of polynomials:
I played around with your example 2 and here is an observation. The
(potentially wrong) Groebner Basis of C has 67 generators as printed by
i1 : gens gb C
If you however compute a Groebner basis again, then there is an
additional 68th generator as seen in:
i2 : gens gb ideal gens gb C
So, there is something wrong with the GB engine and coefficients from
ZZ, as in QQ[a,b,c,d] everything seems to be fine.
cheers
Thomas
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--
Thomas Kahle
The fundamental theorem of algebra is open source. Like any other
mathematical theorem it can be applied free of charge and everybody
has access to its proof and can convince himself how it works. Why
should software be any different?
Christopher Douglas
May 1, 2009, 9:31:14 PM5/1/09
to Macaulay2
Thanks for looking into this and confirming that there is a bug in the
Groebner basis engine. Doesn't this seem like a serious-ish issue,
since presumably the GB engine is called inside many other functions?
Q: Any suggestions for how to get around this inside Macaulay? (In
particular, I don't always need a true/false answer, but I would
definitely need to know when the computation fails, rather than having
to guess what's too big to try.) Barring that, though I'm a big fan
of Macaulay and am loathe to leave it, any recommendations for other
computer algebra programs that could handle this kind of computation
with ideals in polynomial rings over the integers?
Thanks again!
Chris
> ...
>
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>
> signature.asc
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Groebner basis engine. Doesn't this seem like a serious-ish issue,
since presumably the GB engine is called inside many other functions?
Q: Any suggestions for how to get around this inside Macaulay? (In
particular, I don't always need a true/false answer, but I would
definitely need to know when the computation fails, rather than having
to guess what's too big to try.) Barring that, though I'm a big fan
of Macaulay and am loathe to leave it, any recommendations for other
computer algebra programs that could handle this kind of computation
with ideals in polynomial rings over the integers?
Thanks again!
Chris
>
> read more »
>
> signature.asc
> < 1KViewDownload
Thomas Kahle
May 2, 2009, 5:09:08 AM5/2/09
to maca...@googlegroups.com
Christopher Douglas wrote:
> Thanks for looking into this and confirming that there is a bug in the
> Groebner basis engine. Doesn't this seem like a serious-ish issue,
> since presumably the GB engine is called inside many other functions?
> Thanks for looking into this and confirming that there is a bug in the
> Groebner basis engine. Doesn't this seem like a serious-ish issue,
> since presumably the GB engine is called inside many other functions?
Depends on how many people work with non-invertible coefficients. I
didn't knew that it was possible to define ZZ[x] before your example.
> Q: Any suggestions for how to get around this inside Macaulay? (In
> particular, I don't always need a true/false answer, but I would
> definitely need to know when the computation fails, rather than having
> to guess what's too big to try.) Barring that, though I'm a big fan
> of Macaulay and am loathe to leave it, any recommendations for other
> computer algebra programs that could handle this kind of computation
> with ideals in polynomial rings over the integers?
May I ask if Groebner Bases over ZZ[x] are supposed to work at all in M2?
Googling a little I found that here
http://www.msri.org/about/computing/docs/magma/html/text1158.htm#11132
it is claimed that Magma computes Groebner Bases for Polynomials over
Euclidean Rings.
cheers
Thomas
Alex Ghitza
May 2, 2009, 5:24:45 AM5/2/09
to maca...@googlegroups.com
On Sat, May 2, 2009 at 7:09 PM, Thomas Kahle <tom...@gmx.de> wrote:
> Christopher Douglas wrote:
>> Thanks for looking into this and confirming that there is a bug in the
>> Groebner basis engine. Doesn't this seem like a serious-ish issue,
>> since presumably the GB engine is called inside many other functions?
>
> Depends on how many people work with non-invertible coefficients. I
> didn't knew that it was possible to define ZZ[x] before your example.
>
>> Q: Any suggestions for how to get around this inside Macaulay? (In
>> particular, I don't always need a true/false answer, but I would
>> definitely need to know when the computation fails, rather than having
>> to guess what's too big to try.) Barring that, though I'm a big fan
>> of Macaulay and am loathe to leave it, any recommendations for other
>> computer algebra programs that could handle this kind of computation
>> with ideals in polynomial rings over the integers?
>
> May I ask if Groebner Bases over ZZ[x] are supposed to work at all in M2?
> Googling a little I found that here
> http://www.msri.org/about/computing/docs/magma/html/text1158.htm#11132
> it is claimed that Magma computes Groebner Bases for Polynomials over
> Euclidean Rings.
>
> Christopher Douglas wrote:
>> Thanks for looking into this and confirming that there is a bug in the
>> Groebner basis engine. Doesn't this seem like a serious-ish issue,
>> since presumably the GB engine is called inside many other functions?
>
> Depends on how many people work with non-invertible coefficients. I
> didn't knew that it was possible to define ZZ[x] before your example.
>
>> Q: Any suggestions for how to get around this inside Macaulay? (In
>> particular, I don't always need a true/false answer, but I would
>> definitely need to know when the computation fails, rather than having
>> to guess what's too big to try.) Barring that, though I'm a big fan
>> of Macaulay and am loathe to leave it, any recommendations for other
>> computer algebra programs that could handle this kind of computation
>> with ideals in polynomial rings over the integers?
>
> May I ask if Groebner Bases over ZZ[x] are supposed to work at all in M2?
> Googling a little I found that here
> http://www.msri.org/about/computing/docs/magma/html/text1158.htm#11132
> it is claimed that Magma computes Groebner Bases for Polynomials over
> Euclidean Rings.
>
Apparently, the new version (3.1.0) of Singular should also handle
Groebner bases over ZZ.
Best,
Alex
--
Alex Ghitza -- Lecturer in Mathematics -- The University of Melbourne
-- Australia -- http://www.ms.unimelb.edu.au/~aghitza/
Dan Grayson
May 2, 2009, 11:25:54 AM5/2/09
to Macaulay2
Groebner bases are definitely supposed to work over ZZ in M2, and
we'll get this bug repaired as soon as possible and put a new version
out for you to use.
On May 2, 4:09 am, Thomas Kahle <tom...@gmx.de> wrote:
...
we'll get this bug repaired as soon as possible and put a new version
out for you to use.
On May 2, 4:09 am, Thomas Kahle <tom...@gmx.de> wrote:
...
> May I ask if Groebner Bases over ZZ[x] are supposed to work at all in M2?
...
Michael Stillman
Oct 21, 2009, 11:07:38 PM10/21/09
to maca...@googlegroups.com
Thanks very much for reporting this bug! It was fixed on the main
trunk in late May, but then I forgot to post that the fix had been
made. Thankfully, Dan reminded me of this!
In particular, this bug is fixed in version 1.3, which is to be
released in the next few days. Please check it out. If any problems
remain, please let us know!
The bug involved an optimization I had in the algorithm.
Unfortunately, the logic for the optimization didn't cover all cases,
and so it was possible (and as you found, it happens) that under
certain circumstances, a GB element of the form c x^a + ... (c in ZZ),
would get swapped for d x^a + ... (d dividing c) (as it should be),
yet it would sometimes not be tagged as a minimal GB element.
Mike
trunk in late May, but then I forgot to post that the fix had been
made. Thankfully, Dan reminded me of this!
In particular, this bug is fixed in version 1.3, which is to be
released in the next few days. Please check it out. If any problems
remain, please let us know!
The bug involved an optimization I had in the algorithm.
Unfortunately, the logic for the optimization didn't cover all cases,
and so it was possible (and as you found, it happens) that under
certain circumstances, a GB element of the form c x^a + ... (c in ZZ),
would get swapped for d x^a + ... (d dividing c) (as it should be),
yet it would sometimes not be tagged as a minimal GB element.
Mike
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