# Singular Book 1.7.10 -- standard bases

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.
 i1 : A = QQ[x,y]; i2 : I = ideal "x10+x9y2,y8-x2y7"; o2 : Ideal of A i3 : transpose gens gb I o3 = {-9} | x2y7-y8 | {-11} | x9y2+x10 | {-13} | x12y+xy11 | {-13} | x13-xy12 | {-14} | y14+xy12 | {-14} | xy13+y12 | 6 1 o3 : Matrix A <--- A

Lexicographic order:

 i4 : A1 = QQ[x,y,MonomialOrder=>Lex]; i5 : I = substitute(I,A1) 10 9 2 2 7 8 o5 = ideal (x + x y , - x y + y ) o5 : Ideal of A1 i6 : transpose gens gb I o6 = {-15} | y15-y12 | {-14} | xy12+y14 | {-9} | x2y7-y8 | {-11} | x10+x9y2 | 4 1 o6 : Matrix A1 <--- A1

Now we change to a local order

 i7 : B = QQ[x,y,MonomialOrder=>{Weights=>{-1,-1},2},Global=>false]; i8 : I = substitute(I,B) 10 9 2 8 2 7 o8 = ideal (x + x y , y - x y ) o8 : Ideal of B i9 : transpose gens gb I o9 = {-9} | y8-x2y7 | {-11} | x10+x9y2 | 2 1 o9 : Matrix B <--- B

Another local order: negative lexicographic.

 i10 : B = QQ[x,y,MonomialOrder=>{Weights=>{-1,0},Weights=>{0,-1}},Global=>false]; i11 : I = substitute(I,B) 9 2 10 8 2 7 o11 = ideal (x y + x , y - x y ) o11 : Ideal of B i12 : transpose gens gb I o12 = {-9} | y8-x2y7 | {-11} | x9y2+x10 | {-16} | x13-x13y3 | 3 1 o12 : Matrix B <--- B

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 i14 : mo = apply(entries M, e -> Weights => e) o14 = {Weights => {1, 1, 1}, Weights => {0, -1, -1}, Weights => {0, 0, -1}} o14 : List i15 : C = QQ[t,x,y,MonomialOrder=>mo]; 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 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 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
The first two elements form a standard basis.