For definitions of these monomial orders, see
GRevLex,
Lex,
Weights,
Eliminate,
GroupLex,
GroupRevLex,
product orders,
RevLex, and
NCLex.
Graded reverse lexicographic order
i1 : R = ZZ[a..d];
|
i2 : a+b^100+c*d
100
o2 = b + c*d + a
o2 : R
|
Lexicographic order
i3 : R = ZZ[a..d, MonomialOrder=>Lex];
|
i4 : a+b^100+c*d
100
o4 = a + b + c*d
o4 : R
|
Weight order
i5 : R = ZZ[a..d, MonomialOrder => Weights => {201,2}];
|
i6 : a+b^100+c*d
100
o6 = a + b + c*d
o6 : R
|
Graded lexicographic order
i7 : R = ZZ[a..d, MonomialOrder=>{Weights=>4:1,Lex}];
|
i8 : a+b^100+c*d
100
o8 = b + c*d + a
o8 : R
|
Elimination order
To use an elimination order, which eliminates the first 2 variables, use
i9 : R = ZZ[a..f, MonomialOrder=>Eliminate 2];
|
i10 : a+b^100+c*d
100
o10 = b + a + c*d
o10 : R
|
Alternatively, use a weight vector
i11 : R = ZZ[a..f, MonomialOrder=>Weights=>2:1];
|
i12 : a+b^100+c*d
100
o12 = b + a + c*d
o12 : R
|
Product (block) order
To make a product order where each block has the GRevLex order:
i13 : R = ZZ[a..f, MonomialOrder=>{2,4}];
|
i14 : a^2*(c+d) + b*(c^100+d^100)*(c + e + f)
2 2 101 100 100 100 100 100
o14 = a c + a d + b*c + b*c*d + b*c e + b*d e + b*c f + b*d f
o14 : R
|
The orders in each block can be other orders as well.
i15 : R = ZZ[a..f, MonomialOrder=>{Weights=>2:1,Lex}]
o15 = R
o15 : PolynomialRing
|
i16 : a^2*(c+d) + b*(c^100+d^100)*(c + e + f)
2 2 101 100 100 100 100 100
o16 = a c + a d + b*c + b*c e + b*c f + b*c*d + b*d e + b*d f
o16 : R
|
GroupLex
This order is useful for making degree rings, and allows some variables to appear with negative exponent.
i17 : R = ZZ[a..f, MonomialOrder => GroupLex => 3];
|
i18 : a^-2*(c+d) + b*(c^100+d^100)*(c + e + f)
101 100 100 100 100 100 -2 -2
o18 = b*c + b*c e + b*c f + b*c*d + b*d e + b*d f + a c + a d
o18 : R
|
GroupRevLex
This order is useful for making degree rings, and allows some variables to appear with negative exponent. Not implemented yet.
RevLex
Warning: this is a local ordering, not a global ordering.
i19 : R = ZZ[a..f, MonomialOrder=>RevLex, Global=>false];
|
i20 : a^2*(c+d) + b*(c^100+d^100)*(c + e + f)
100 100 100 100 100 101 2 2
o20 = b*d f + b*d e + b*c*d + b*c f + b*c e + b*c + a d + a c
o20 : R
|
NCLex
For non-commutative Groebner bases. Not implemented yet.