# GB for I=ideal(x^3-3*x^2-y+1,-x^2+y^2-1) leaves out a further step.

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

I found a sample solved with GAP.

```
gap> ideal := [x^3-3*x^2-y+1, -x^2+y^2-1];;
gap> G := ReducedGroebnerBasis(ideal, lex);;
gap> Display(G);
[ 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

```
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.

Is there a way to eliminate `x`

in the first poly automatically?

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