R = QQ[a..d, MonomialOrder=>GRevLex]
f= 3 + 8*a^2*b + 7*b*c^2
leadTerm (3 + 8*a^2*b + 7*b*c^2)
leadTerm(f)
leadCoefficient f
leadMonomial f

g= 3 + 5*a - 8*a^2*b + 7*b*c^2
leadTerm g
leadTerm(f+g)
leadTerm(f*g)





----------------------------------------
R=ZZ/101[x,y, MonomialOrder=>GLex]
S=ZZ/101[x,y, MonomialOrder=>GRevLex]

i=ideal vars R
j=ideal vars S

i2=i^2
j2=j^2

leadTerm(i2)
leadTerm(j2)


i3=i^3
j3=j^3

leadTerm(i3)
leadTerm(j3)

-- There is in fact only one order in two vars that refines 
-- the order by degree and the convention that x > y


---------------------------------------
R=ZZ/101[x,y,z, MonomialOrder=>GLex]
S=ZZ/101[x,y,z, MonomialOrder=>GRevLex]

i=ideal vars R
j=ideal vars S

i2=i^2
j2=j^2

leadTerm(i2)
leadTerm(j2)

-- Quadratic monomials in 3 vars in the first time 
-- the two orders (GLex and GRevLex)  are different



R=ZZ/101[a,b,c,x,y,z,MonomialOrder=>ProductOrder{3,3}]
f= 3 + 8*a^2*b + 7*b*c^2
g= 3 + 8*x^2*y*z + 7*b*c^2
h= z + x

leadTerm(f)
leadTerm(g)

leadTerm(f+g)
leadTerm(f*g)

leadTerm(h)
leadTerm(f*h)
leadTerm(f+h)