We show the Groebner and standard bases of an ideal under several different orders and localizations. First, the default order is graded (degree) reverse lexicographic.
Another local order: negative lexicographic.
One method to compute a standard basis is via homogenization. The example below does this, obtaining a standard basis which is not minimal.
i13 : M = matrix{{1,1,1},{0,-1,-1},{0,0,-1}}
o13 = | 1 1 1 |
| 0 -1 -1 |
| 0 0 -1 |
3 3
o13 : Matrix ZZ <-- ZZ
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i14 : mo = apply(entries M, e -> Weights => e)
o14 = {Weights => {1, 1, 1}, Weights => {0, -1, -1}, Weights => {0, 0, -1}}
o14 : List
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i15 : C = QQ[t,x,y,MonomialOrder=>mo];
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i16 : I = homogenize(substitute(I,C),t)
8 2 7 10 9 2 11 12 13 12
o16 = ideal (- t*y + x y , t*x + x y , t*x*y + x y, x - x*y ,
-----------------------------------------------------------------------
12 14 2 12 13
t*x*y + y , t y + x*y )
o16 : Ideal of C
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i17 : transpose gens gb I
o17 = {-9} | ty8-x2y7 |
{-11} | tx10+x9y2 |
{-13} | x12y+x3y10 |
{-13} | x13-xy12 |
{-14} | x3y11+y14 |
{-14} | x4y10+xy13 |
{-14} | x11y3-x5y9 |
{-15} | x6y9-y15 |
{-15} | x10y5+x7y8 |
{-16} | x8y8-x2y14 |
10 1
o17 : Matrix C <-- C
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i18 : substitute(transpose gens gb I, {t=>1})
o18 = {-9} | -x2y7+y8 |
{-11} | x9y2+x10 |
{-13} | x12y+x3y10 |
{-13} | x13-xy12 |
{-14} | x3y11+y14 |
{-14} | x4y10+xy13 |
{-14} | x11y3-x5y9 |
{-15} | x6y9-y15 |
{-15} | x10y5+x7y8 |
{-16} | x8y8-x2y14 |
10 1
o18 : Matrix C <-- C
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The first two elements form a standard basis.