restart
needsPackage "NumericalAlgebraicGeometry"
R=QQ[x,y,z,MonomialOrder=>Lex]
I=ideal(x^2+y+z-1,x+z+y^2-1,x+z^2+y-1)
I=ideal groebnerBasis I
--In the first eq we have eliminated y and x and have things
--only in k[z]... we can think of this as intersecting I with k[z]
S=CC[z]
f=sub(I_0,S)
s = solveSystem {f}
-- 0,1, −1 ± sqrt(2).
--we then solve this simpler equation and "extend" to get the other
--coordinates on our vareity
restart
R=QQ[x,y,z,MonomialOrder=>Lex]
I=ideal(x*y-1,x*z-1)
ideal groebnerBasis I
ideal groebnerBasis ideal(x^2-y,x^2-z)