CoCoA » CoCoA-5

Here is the same example as in issue #973

use P::=QQ[x,y,z];
I := ideal(x^5-3*y^3*z+x*y*z-2, y^3-2*z+5, z^2-12*x+7*y);
GF := GroebnerFanIdeals(I);
OrderMats := [OrdMat(RingOf(J)) | J in GF];
[M in OrderMats | max(flatten(GetRows(M))) > 1000];  --> two matrices

As far as I can tell the order of the ideals in GF is the same from run to run, so I may refer to certain ideals just by specifying their indices. Anyway, I wanted to see which ideals corresponded to "lex" orderings: on my computer these have indices [36, 38, 94].

I had naively hoped to see that the matrices were just permutations of the identity matrix; instead they have larger entries:

>>> OrdMat(RingOf(GF[36]));
matrix(ZZ,
 [[29, 30, 1],
  [842, 870, 0],
  [0, 0, -1]])

>>> OrdMat(RingOf(GF[38]));
matrix(ZZ,
 [[6, 1, 4],
  [37, 0, 24],
  [0, 0, -1]])

>>> OrdMat(RingOf(GF[94]));
matrix(ZZ,
 [[1, 29, 58],
  [0, 842, 1682],
  [0, 0, -1]])

Can something be done to make the matrix entries smaller?

For instance, it seems to me that the second row of the last matrix could be divided by 2 without affecting the ordering; similarly for second row of first matrix.

I noticed that in this example the order matrices always have strictly positive entries in the first row. Are zeroes not allowed?

It seems that GFan usually produces the first two rows, and then the last row is just [0,0,-1] with the sole exception of the very first ideal whose order matrix corresponds to StdDegRevLex.

One option could be to determine the "best" order matrix for each ideal once the supports of its reduced G-basis elements have been computed. This appears to be an integer linear programming problem.

It seems that the first two rows are "almost parallel" -- why should that be?
Anyway, it should be easy to reduce the size of entries in row 2.