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I am trying to show that some given simple affine algebraic varieties are rational (i.e. birationally equivalent to some A^k).

Are there any tricks or even nice algorithms for finding the birational maps and their inverses? Examples are the curve x^2 + y^2 = 1 and the surface x^2 + y^2 + z^2 = 1?

I have tried to tackle the first one by assuming that the map from A^1 to the curve takes the form of a pair of quotients of linear polynomials, and then try to work out suitable coeffiecients of these polynomials but it gets very messy very quickly and offers no insight into making the thing invertible.

Any help would be greatly appreciated. Thanks.

Are there any tricks or even nice algorithms for finding the birational maps and their inverses? Examples are the curve x^2 + y^2 = 1 and the surface x^2 + y^2 + z^2 = 1?

I have tried to tackle the first one by assuming that the map from A^1 to the curve takes the form of a pair of quotients of linear polynomials, and then try to work out suitable coeffiecients of these polynomials but it gets very messy very quickly and offers no insight into making the thing invertible.

Any help would be greatly appreciated. Thanks.

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