Express Groebner basis in terms of original basis

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The simplest way is to do it during the Gröbner basis computation, e.g. in Buchberger's algorithm. 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.

I think the quick way for this is to carry around marker variables, say b1,b2 and then compute the groebner basis of the ideal (b1-f1,b2-f2), making sure you have a block order that prioritizes the original variables (that occur in f1,f2). You can then see in the resulting basis from the part of the polynomials in the b1,b2 how your original generators were combined. It's comparable to the trick of augmenting a matrix for Gaussian elimination to find the transformation that gets you there (i.e., the inverse of the matrix).