Problem:
Solution:
With the previous problem, this is relatively easy now. Fix and arbitrary $ y > 0 $, Any polynomial $ f(x, y) $ becomes $ g(x) $ and it vanishes everywhere, so it must be the zero polynomial. That proves all polynomials defining the affine variety is the zero polynomial, but then the affine variety must also vanish in the lower half plane as well, the contradiction shows the upper half plane is not an affine variety!
Solution:
With the previous problem, this is relatively easy now. Fix and arbitrary $ y > 0 $, Any polynomial $ f(x, y) $ becomes $ g(x) $ and it vanishes everywhere, so it must be the zero polynomial. That proves all polynomials defining the affine variety is the zero polynomial, but then the affine variety must also vanish in the lower half plane as well, the contradiction shows the upper half plane is not an affine variety!
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