The nonexistence of a polynomial

Question

I'm studying algebraic geometry. To illustrate a nonalgebraic set, it is given that a unit circle except for a point on it in cartesian product or whole plane except for one point. Why doesn't a polynomial whose zeros are a unit circle except for a point or whole plane except for one point exist? I'll be glad if one explains this situation.

Answer 1

Polynomials are continuous functions. A function that is identically zero on the plane except at one point is not continuous. The same goes for the unit circle.