-- Monomial Orders
--

-- You must specify the monomial order of a polynomial ring when you
-- create the ring.

-- The Lexicographic Order is called "Lex" in Macaulay2
R = QQ[x, y, z, MonomialOrder=>Lex];
describe(R)
f = x^3 + x^2*y*z^2 + x*y^3*z

-- The Graded Lexicographic Ordering is called "GLex"
R = QQ[x, y, z, MonomialOrder=>GLex];
describe(R)
f = x^3 + x^2*y*z^2 + x*y^3*z

-- Graded Reverse Lexicographical Order is called "GRevLex"
R = QQ[x, y, z, MonomialOrder=>GRevLex];
describe(R)
f = x^3 + x^2*y*z^2 + x*y^3*z

-- The default ordering is Graded Reverse Lexicographical Order
R = QQ[x, y, z];
describe(R)

-- Here are some functions that might be useful:
R = QQ[x, y, z, MonomialOrder=>Lex];
f = 5*x^3 + 2*x^2*y*z^2 - x*y^3*z

exponents(f)
leadTerm(f)
leadMonomial(f)
leadCoefficient(f)