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For all complex polynomials in 2 variables I know, the set of zeros looks like a union of curves.

Wrong: I can get a circle with $x^2+y^2-1$ or two vertical lines with $(x-1)(x-2)$?.

Can I get isolated sets of point? In particular, can I find a polynomial in 2 variables which is zero only at a single point?

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    Perhaps someone would like to write up an answer as an answer? Derek, you can do it if you now understand the situation.2012-07-08

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More generally, the zero set of a holomorphic function of $n>1$ variables does not have isolated points. This follows from the Hartogs theorem.