restart
QQ[x_1..x_5]
R = QQ[x,y,z] -- default is GRevLex
R = QQ[x,y,z,MonomialOrder=>Lex]
help "monomial orderings"
viewHelp "monomial orderings"
f = random(3,R) -- random homogeneous polynomial of degree 3
R1 = QQ[x,y,z,Weights=>{2,1,3}] 
R2 = QQ[x,y,z,Weights=>{{1,0,0},{0,1,0},{0,0,1}}] 
f
sub(f,R1)
sub(f,R2)
use R -- make sure x,y,z live in R again
g = x^2+y^2+z^2
f//g -- quotient
f%g -- remainder
leadMonomial f
leadTerm f
I = ideal{f,g}
leadTerm I
inI = ideal leadTerm I
mingens inI -- minimal generators of the monomial ideal