MonomialOrder -- monomial ordering

In Macaulay 2, each polynomial ring (and also each monoid) is equipped with a monomial order, which is used for display of polynomials (terms are listed in descending monomial order), and also for Groebner basis computations.

In the most general setting, a monomial ordering is given by a list of permissible elements, listed and described below. Monomials are compared using the first element of the list. If they are indistinguishable using this first element, they are compared using the second element, and so on. At the end, if necessary, the graded reverse lexicographic order is used to compare the monomials. For examples, see below, or see monomial orderings.

Permissible elements:

• GRevLex => n -- A graded reverse lexicographic block of variables
• Lex => n
• Weights => {...}
• Position => Up or Position => Down
• RevLex => n
• GroupLex => n
• GroupRevLex => n
• MonomialSize => n, n being 8,16,32, or 64. Set the packing size for exponents for further variables

Some examples of monomial orders. Note that if only one item is in the list, we can dispense with the list.

• MonomialOrder => {GRevLex=>2, GRevLex=>3} -- a product order
• MonomialOrder => {2, 3} -- same
• MonomialOrder => {Weights=>{1,13,6,2}} -- a weight order
• MonomialOrder => Weights=>{1,13,6,2} -- same

i1 : QQ[x,y, Weights => {-1,1}, Global => false]

o1 = QQ [x, y]

o1 : PolynomialRing

i2 : x<1

o2 = true

i3 : y<1

o3 = false