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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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$x^2+y^2\phantom{}$? – J. M. Jul 7 '12 at 15:37
@J.M. Thanks. I missed that. What about 2 isolated points? – Derek Jul 7 '12 at 15:38
Oh, probably something like $(x^2+y^2)((x-1)^2+y^2)$ – Derek Jul 7 '12 at 15:42
Yes, in general, if you want something that touches the $x$-$y$ plane in some set of isolated points, you try assembling such a polynomial from sums of squares... – J. M. Jul 7 '12 at 15:44
@J.M. Actually, I was trying to think if there are pairs of polynomials for which the set of common roots is not the set of roots of any single polynomial. – Derek Jul 7 '12 at 15:45

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

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