Elimination theory

[Neda]

Hello
Could you pleas tell me how can I solve the system of equations x^2 + y + z - 1 =0 x+ y^2 + z - 1 =0 x + y + z^ 2 - 1 =0 over C[x,y] with a given ideal I =< x^2 +y+z-1,x+y^2+z-1,x+y+z^2-1 > and Groebner basis g1=x+y+z^2-1
g2= y^2-y-z^2+z
g3=2yz^2 +z^4 -z^2
g4=z6-4z^4+4z^3-z^2
with Elimination theory in sage? I know how to solve it but I can't solve it in sage.

[Volker Braun]

Your question is not about elimination theory.

Your lex order groebner basis is the solution. g4 determines z. Then g3 and g2 determine y. Then g1 determines x.

[Neda]

 so Is there any way for writing Elimination theory in sage? 

Let I c k [ x₁,...,x_n] be an ideal and let G be a groebner basis of I with respect to lex order wherex₁> x₂ > ... >x_n , then for every 0< l <n , the set  G_l =G  k[x_l+1,...,x_n ] is the groebner basis of the l-th elimination ideal.

[Volker Braun]

See http://www.sagemath.org/doc/reference/polynomial_rings/sage/rings/polynomial/multi_polynomial_ideal.html#sage.rings.polynomial.multi_polynomial_ideal.MPolynomialIdeal_singular_repr.elimination_ideal