R=QQ[x,y,z,MonomialOrder=>Lex]
f=x^2*y+y^3
f1=x^2+y^2
f2=x*y
quotientRemainder(matrix{{f}},matrix{{f1,f2}})
quotientRemainder(matrix{{y^3}},matrix{{f1,f2}})
leadTerm( f1)
I=ideal(f1,f2)
leadTerm(I)
gens gb ideal(x^2+y^2,x*y)
R1=QQ[x,y,MonomialOrder=>Lex]
gens gb ideal(x^2*y-1,x*y^2-x)
gens gb ideal(x^2+y,x^4+2*x^2*y+y^2+3)
R1=QQ[x,y]
gens gb ideal(x^2*y-1,x*y^2-x)
gens gb ideal(x^2+y^2,x*y)

R=QQ[x,y,z,a,b,c,MonomialOrder=>Lex]
I=ideal(x+y+z-a,x*y+x*z+y*z-b,x*y*z-c)
leadTerm generators gb I
(x+y+z)%I
(x+y+z)^2%I
(x^2+y^2+z^2)%I
for n from 1 to 10 do print(n,(x^n+y^n+z^n)%I)
I=ideal random(R^{1:1},R^{5:0})
generators gb I
exit