### 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. ~~basis.

I found a sample solved with ~~GAP.
~~GAP.

`gap> ideal `~~:=[x^3-3~~*x^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^4-11~~y^3-17*y^2+9*y+17, -y^4+x*y+11*y^2-x+3*y-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^3-3~~x^2-y+1,-x^2+y^2-1);B=I.groebner~basis(); 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]
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?