Why is the zero set in $\mathbb{C}\times\mathbb{C}$ of a polynomial $f(x,y)$ in two complex variables always non-discrete (no zero of $f$ is isolated)?
Why are there no discrete zero sets of a polynomial in two complex variables?
4
$\begingroup$
complex-analysis
polynomials
several-complex-variables
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0Are you asking for a proof? – 2011-03-07
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3Don't you mean the zero set in $\mathbb{C}\times\mathbb{C}$? – 2011-03-07
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0consider $f(x,a)$ for a fixed value of $y$. this has $\text{deg}_xf$ zeros. varying $a$ continuously gives curves of zeros (the zeros of $f(x,a)$ depend continuously on $a$). – 2011-03-07