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I am looking for an example solving multivariate polynomials using Groebner basis. I found a sample solved with GAP. ideal :=[x^3-3x^2-y+1,-x^2+y^2-1];; G := ReducedGroebnerBasis(ideal, lex);; Display(G); [ y^5+y^4-11y^3-17y^2+9y+17, -y^4+xy+11y^2-x+3y-13, x^2-y^2+1 ] In Sage, I tried and got I=ideal(x^3-3x^2-y+1,-x^2+y^2-1);B=I.groebner~basis(); B; [ y^4^-xy-11y^2+x-3y+13, xy^2-3*y^2-x-y+4,x^2-y^2+1] I could manipulate to get what GAP gives from what I got in Sage. Is there a way to eliminate x in the first poly automatically?

I am looking for an example of solving multivariate polynomials using Groebner basis. basis.

I found a sample solved with GAP. GAP.

gap> ideal :=[x^3-3x^2-y+1,-x^2+y^2-1];; := [x^3-3*x^2-y+1, -x^2+y^2-1];;
gap> G := ReducedGroebnerBasis(ideal, lex);; lex);;
gap> Display(G);
[ y^5+y^4-11y^3-17y^2+9y+17, -y^4+xy+11y^2-x+3y-13, y^5+y^4-11*y^3-17*y^2+9*y+17, -y^4+x*y+11*y^2-x+3*y-13, x^2-y^2+1 ]

In Sage, I tried and got I=ideal(x^3-3x^2-y+1,-x^2+y^2-1);B=I.groebner~basis(); B; [ y^4^-xy-11y^2+x-3y+13, xy^2-3*y^2-x-y+4,x^2-y^2+1] got

sage: R.<x, y> = QQ[]
sage: R = PolynomialRing(QQ, 'x, y')
sage: x, y = R.gens()
sage: I = ideal(x^3 - 3*x^2 - y + 1, -x^2 + y^2 - 1)
sage: I
Ideal (x^3 - 3*x^2 - y + 1, -x^2 + y^2 - 1) of Multivariate Polynomial Ring in x, y over Rational Field
sage: B = I.groebner_basis()
sage: B
[y^4 - x*y - 11*y^2 + x - 3*y + 13, x*y^2 - 3*y^2 - x - y + 4, x^2 - y^2 + 1]

I could manipulate to get what GAP gives from what I got in Sage. Sage.

Is there a way to eliminate x x in the first poly automatically?