How to specify arbitrary monomial ordering in Macaulay2?

Question

Suppose you are given an arbitrary order such as $w>x>y>z>a$ order. You can do renaming so that it becomes lex or similar errorsome renaming things to satisfy some of the more typical monomial orderings such as lex, glex and grevlex here. Is there any way to specify an arbitrary monomial ordering in Macaulay2?

Answer 1

You can use the Weights option to specify a given ordering:

i1 : R = QQ[w..z,a, MonomialOrder=>{Weights => {4,3,2,1,0}}];

i2 : toString (a+z+y+x+w)^2

o2 = w^2 + 2*w*x + x^2 + 2*w*y + 2*x*y + 2*w*z + y^2+2*x*z + 2*w*a + 2*y*z + 2*x*a + z^2+2*y*a + 2*z*a + a^2

If you want to produce a more or less arbitrary ordering, you can use Weights together with random:

i1 : R = QQ[w..z,a, MonomialOrder=>{Weights => for i in 0..4 list random 20}];

i2 : (options R).MonomialOrder

o2 = {MonomialSize => 32 }

{Weights => {14, 15, 12, 9, 1}}

{GRevLex => {1, 1, 1, 1, 1} }

{Position => Up }

o2 : VerticalList

i3 : toString (a+z+y+x+w)^2

o3 = x^2+2*w*x+w^2+2*x*y+2*w*y+y^2+2*x*z+2*w*z+2*y*z+z^2+2*x*a+2*w*a+2*y*a+2*z*a+a^2