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<div><h1>Singular Book 1.2.13 -- monomial orderings</h1>
<div>Monomial orderings are specified when defining a polynomial ring.<h2>global orderings</h2>
The default order is the graded (degree) reverse lexicographic order.<table class="examples"><tr><td><pre>i1 : A2 = QQ[x,y,z];</pre>
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<tr><td><pre>i2 : A2 = QQ[x,y,z,MonomialOrder=>GRevLex];</pre>
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<tr><td><pre>i3 : f = x^3*y*z+y^5+z^4+x^3+x*y^2
5 3 4 3 2
o3 = y + x y*z + z + x + x*y
o3 : A2</pre>
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Lexicographic order.<table class="examples"><tr><td><pre>i4 : A1 = QQ[x,y,z,MonomialOrder=>Lex];</pre>
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<tr><td><pre>i5 : substitute(f,A1)
3 3 2 5 4
o5 = x y*z + x + x*y + y + z
o5 : A1</pre>
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Graded (degree) lexicographic order.<table class="examples"><tr><td><pre>i6 : A3 = QQ[x,y,z,MonomialOrder=>{Weights=>{1,1,1},Lex}];</pre>
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<tr><td><pre>i7 : substitute(f,A3)
3 5 4 3 2
o7 = x y*z + y + z + x + x*y
o7 : A3</pre>
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Graded (degree) lexicographic order, with nonstandard weights.<table class="examples"><tr><td><pre>i8 : A4 = QQ[x,y,z,MonomialOrder=>{Weights=>{5,3,2},Lex}];</pre>
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<tr><td><pre>i9 : substitute(f,A4)
3 3 5 2 4
o9 = x y*z + x + y + x*y + z
o9 : A4</pre>
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A product order, with each block being GRevLex.<table class="examples"><tr><td><pre>i10 : A = QQ[x,y,z,MonomialOrder=>{1,2}];</pre>
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<tr><td><pre>i11 : substitute(f,A)
3 3 2 5 4
o11 = x y*z + x + x*y + y + z
o11 : A</pre>
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<h2>local orderings</h2>
Negative lexicographic order.<table class="examples"><tr><td><pre>i12 : A = QQ[x,y,z,MonomialOrder=>{Weights=>{-1,0,0},Weights=>{0,-1,0},Weights=>{0,0,-1}},Global=>false];</pre>
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<tr><td><pre>i13 : substitute(f,A)
4 5 2 3 3
o13 = z + y + x*y + x + x y*z
o13 : A</pre>
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Negative graded reverse lexicographic order.<table class="examples"><tr><td><pre>i14 : A = QQ[x,y,z,MonomialOrder=>{Weights=>{-1,-1,-1},GRevLex},Global=>false];</pre>
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<tr><td><pre>i15 : substitute(f,A)
3 2 4 5 3
o15 = x + x*y + z + y + x y*z
o15 : A</pre>
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