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The simplest way is to do it during the Gröbner basis computation, e.g. in Buchberger's algorithm. I will repeat that it's possible to implement this in a few lines in SageMath.

As an alternative you can use the lift method on a polynomial, passing it a list of generators.

The simplest way is to do it during the Gröbner basis computation, e.g. in Buchberger's algorithm. I will repeat that it's possible to implement this in a few lines in SageMath.

As an alternative you can use the lift method on a polynomial, passing it a list of generators.generators or an ideal:

sage: R.<x,y> = QQ[] 
sage: f = x^2 - y^2
sage: f.lift(R.ideal([x, y]))
[x, -y]

The simplest way is to do it during the Gröbner basis computation, e.g. in Buchberger's algorithm. I will repeat that it's possible to implement this in a few lines in SageMath.

As an alternative you can use the lift method on a polynomial, passing it a list of generators or an ideal:

sage: R.<x,y> = QQ[] 
sage: f = x^2 - y^2
sage: f.lift(R.ideal([x, y]))
[x, -y]

The simplest way is to do it during the Gröbner basis computation, e.g. in Buchberger's algorithm. I will repeat that it's It's possible to implement this in a few lines in SageMath.

As an alternative you can use the lift method on a polynomial, passing it a list of generators or an ideal:

sage: R.<x,y> = QQ[] 
sage: f = x^2 - y^2
sage: f.lift(R.ideal([x, y]))
[x, -y]

Keep in mind that the solution is generally not unique due to the existence of syzygies between the generators.