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Analytic proof that infinite level sets of complex polynomials aren't compact
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I've heard it stated that if you have a family $p_{1}, ldots, p_{ell} in mathbb{C}[x_1 , ldots , x_n]$ of $n$-variate complex polynomials, then the set $F = bigcap_{j = 1}^{ell} p_{j}^{-1} ({0}) subseteq mathbb{C}^n$ is either finite, or not compact in the standard topology on $mathbb{C}^n$. My understanding is that this claim can be demonstrated by algebraic-geometric means, though the methods are rather involved. Can it be proven by analytic methods? I assume that it won't be a simple or easy argument, but can it be reasonably done?
Thanks!
EDIT: We've figured out how to show that if $p$ is nonconstant, then the null set of $p$ is either finite (if $n = 1$) or unbounded (if $n > 1$). But that's it.
complex-analysis algebraic-geometry complex-geometry several-complex-variables
asked Jan 28 at 15:04
AJYAJY
4,27021129
$endgroup$
add a comment |
$begingroup$
I've heard it stated that if you have a family $p_{1}, ldots, p_{ell} in mathbb{C}[x_1 , ldots , x_n]$ of $n$-variate complex polynomials, then the set $F = bigcap_{j = 1}^{ell} p_{j}^{-1} ({0}) subseteq mathbb{C}^n$ is either finite, or not compact in the standard topology on $mathbb{C}^n$. My understanding is that this claim can be demonstrated by algebraic-geometric means, though the methods are rather involved. Can it be proven by analytic methods? I assume that it won't be a simple or easy argument, but can it be reasonably done?
Thanks!
EDIT: We've figured out how to show that if $p$ is nonconstant, then the null set of $p$ is either finite (if $n = 1$) or unbounded (if $n > 1$). But that's it.
complex-analysis algebraic-geometry complex-geometry several-complex-variables
asked Jan 28 at 15:04
AJYAJY
4,27021129
$endgroup$
add a comment |
$begingroup$
I've heard it stated that if you have a family $p_{1}, ldots, p_{ell} in mathbb{C}[x_1 , ldots , x_n]$ of $n$-variate complex polynomials, then the set $F = bigcap_{j = 1}^{ell} p_{j}^{-1} ({0}) subseteq mathbb{C}^n$ is either finite, or not compact in the standard topology on $mathbb{C}^n$. My understanding is that this claim can be demonstrated by algebraic-geometric means, though the methods are rather involved. Can it be proven by analytic methods? I assume that it won't be a simple or easy argument, but can it be reasonably done?
Thanks!
EDIT: We've figured out how to show that if $p$ is nonconstant, then the null set of $p$ is either finite (if $n = 1$) or unbounded (if $n > 1$). But that's it.
complex-analysis algebraic-geometry complex-geometry several-complex-variables
asked Jan 28 at 15:04
AJYAJY
4,27021129
$endgroup$
I've heard it stated that if you have a family $p_{1}, ldots, p_{ell} in mathbb{C}[x_1 , ldots , x_n]$ of $n$-variate complex polynomials, then the set $F = bigcap_{j = 1}^{ell} p_{j}^{-1} ({0}) subseteq mathbb{C}^n$ is either finite, or not compact in the standard topology on $mathbb{C}^n$. My understanding is that this claim can be demonstrated by algebraic-geometric means, though the methods are rather involved. Can it be proven by analytic methods? I assume that it won't be a simple or easy argument, but can it be reasonably done?
Thanks!
EDIT: We've figured out how to show that if $p$ is nonconstant, then the null set of $p$ is either finite (if $n = 1$) or unbounded (if $n > 1$). But that's it.
complex-analysis algebraic-geometry complex-geometry several-complex-variables
complex-analysis algebraic-geometry complex-geometry several-complex-variables
asked Jan 28 at 15:04
AJYAJY
4,27021129
asked Jan 28 at 15:04
AJYAJY
4,27021129
edited Jan 28 at 15:26
AJY
asked Jan 28 at 15:04
AJYAJY
4,27021129
asked Jan 28 at 15:04
AJYAJY
4,27021129
asked Jan 28 at 15:04
AJYAJY
4,27021129
4,27021129
add a comment |
add a comment |
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