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<head><title>Singular Book 1.7.10 -- standard bases</title>
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<div><a href="index.html" title="">Macaulay2Doc</a> > <a href="_basic_spcommutative_spalgebra.html" title="">basic commutative algebra</a> > <a href="___M2__Singular__Book.html" title="Macaulay2 examples for the Singular book">M2SingularBook</a> > <a href="___Singular_sp__Book_sp1.7.10.html" title="standard bases">Singular Book 1.7.10</a></div>
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<div><h1>Singular Book 1.7.10 -- standard bases</h1>
<div>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.<table class="examples"><tr><td><pre>i1 : A = QQ[x,y];</pre>
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<tr><td><pre>i2 : I = ideal "x10+x9y2,y8-x2y7";
o2 : Ideal of A</pre>
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<tr><td><pre>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</pre>
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Lexicographic order:<table class="examples"><tr><td><pre>i4 : A1 = QQ[x,y,MonomialOrder=>Lex];</pre>
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<tr><td><pre>i5 : I = substitute(I,A1)
10 9 2 2 7 8
o5 = ideal (x + x y , - x y + y )
o5 : Ideal of A1</pre>
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<tr><td><pre>i6 : transpose gens gb I
o6 = {-15} | y15-y12 |
{-14} | xy12+y14 |
{-9} | x2y7-y8 |
{-11} | x10+x9y2 |
4 1
o6 : Matrix A1 <--- A1</pre>
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Now we change to a local order<table class="examples"><tr><td><pre>i7 : B = QQ[x,y,MonomialOrder=>{Weights=>{-1,-1},2},Global=>false];</pre>
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<tr><td><pre>i8 : I = substitute(I,B)
10 9 2 8 2 7
o8 = ideal (x + x y , y - x y )
o8 : Ideal of B</pre>
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<tr><td><pre>i9 : transpose gens gb I
o9 = {-11} | x10+x9y2 |
{-9} | y8-x2y7 |
2 1
o9 : Matrix B <--- B</pre>
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Another local order: negative lexicographic.<table class="examples"><tr><td><pre>i10 : B = QQ[x,y,MonomialOrder=>{Weights=>{-1,0},Weights=>{0,-1}},Global=>false];</pre>
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<tr><td><pre>i11 : I = substitute(I,B)
9 2 10 8 2 7
o11 = ideal (x y + x , y - x y )
o11 : Ideal of B</pre>
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<tr><td><pre>i12 : transpose gens gb I
o12 = {-16} | x13-x13y3 |
{-11} | x9y2+x10 |
{-9} | y8-x2y7 |
3 1
o12 : Matrix B <--- B</pre>
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One method to compute a standard basis is via homogenization. The example below does this, obtaining a standard basis which is not minimal.<table class="examples"><tr><td><pre>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</pre>
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<tr><td><pre>i14 : mo = apply(entries M, e -> Weights => e)
o14 = {Weights => {1, 1, 1}, Weights => {0, -1, -1}, Weights => {0, 0, -1}}
o14 : List</pre>
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<tr><td><pre>i15 : C = QQ[t,x,y,MonomialOrder=>mo];</pre>
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<tr><td><pre>i16 : I = homogenize(substitute(I,C),t)
10 9 2 8 2 7
o16 = ideal (t*x + x y , t*y - x y )
o16 : Ideal of C</pre>
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<tr><td><pre>i17 : transpose gens gb I
o17 = {-9} | ty8-x2y7 |
{-11} | tx10+x9y2 |
{-19} | x12y7+x9y10 |
3 1
o17 : Matrix C <--- C</pre>
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<tr><td><pre>i18 : substitute(transpose gens gb I, {t=>1})
o18 = {-9} | -x2y7+y8 |
{-11} | x9y2+x10 |
{-19} | x12y7+x9y10 |
3 1
o18 : Matrix C <--- C</pre>
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The first two elements form a standard basis.</div>
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