minimal GB in local rings

[mikestillman]

The Mora algorithm in M2 is supposed to return a minimal Groebner/standard basis, but sometimes it doesn't. As an example:

    R=QQ{x,y,z}
    i=ideal"xyz+z5,2x2+y3+z7,3z5+y5"
    transpose gens gb i -- this is not minimal GB...  BUG!!

The problem is in e/reducedgb-field.cpp. The minimize routine used there orders the polynomials (or module elements) in an incorrect order for local orders and so returns a larger GB than desired.

[mikestillman]

See also #291

[DanGrayson]

... or should this be fixed by 2.0 instead?

[mikestillman]

This one should be relatively easy to fix, I need to sort elements in a different order. I will try for 1.9.3.

On the other hand, the other bug #568 is more significant, and is perhaps really not a bug. It will have to wait for 2.0.

[mikestillman]

Fixed in commit 4612ae32d53c7d7.

However, reduction of elements is now sometimes slower (at least over QQ). This needs to be addressed, but in the next release. Here is an example:

kk = QQ
R=kk{x,y,z}
i=ideal"xyz+z5,2x2+y3+z7,3z5+y5"
transpose gens gb i
L = gens gb i
assert(numColumns L == 5)
assert(gens gb L == L)

R1 = QQ[t,x,y,z,MonomialOrder=>Eliminate 1]
ih = ideal homogenize(sub(gens i,R1),t)
dehomog = map(R,R1, {1} | gens R)
L2 = flatten entries dehomog gens gb ih
gbTrace=3
L2_2 % i