Connected levels and polynomials submersions
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Is it true that a polynomial submersion $ p: mathbb{R}^2 to mathbb{R}$ of degree $n$ has at most $n$ connected components on each level?
I think I have a proof, can someone point me out any mistakes?
Let $cin mathbb{R}$.
Since $p$ is a submersion, $p-c$ doesn't have a $(x^2+y^2-R)$ factor for any $Rin mathbb{R}^+$.
Also, each connected component of $p^{-1}(c)$ must intersect $x^2+y^2-R=0$ at least twice, for every $R$ big enough.
By Bezout theorem, the system
begin{cases} p=c \ x^2+y^2=Rend{cases}
has at most $2n$ real solutions, hence $p^{-1}(c)$ has at most $n$ connected components.
algebraic-geometry algebraic-curves
asked Sep 15 '18 at 0:49
F.FF.F
263
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add a comment |
$begingroup$
Is it true that a polynomial submersion $ p: mathbb{R}^2 to mathbb{R}$ of degree $n$ has at most $n$ connected components on each level?
I think I have a proof, can someone point me out any mistakes?
Let $cin mathbb{R}$.
Since $p$ is a submersion, $p-c$ doesn't have a $(x^2+y^2-R)$ factor for any $Rin mathbb{R}^+$.
Also, each connected component of $p^{-1}(c)$ must intersect $x^2+y^2-R=0$ at least twice, for every $R$ big enough.
By Bezout theorem, the system
begin{cases} p=c \ x^2+y^2=Rend{cases}
has at most $2n$ real solutions, hence $p^{-1}(c)$ has at most $n$ connected components.
algebraic-geometry algebraic-curves
asked Sep 15 '18 at 0:49
F.FF.F
263
$endgroup$
add a comment |
$begingroup$
Is it true that a polynomial submersion $ p: mathbb{R}^2 to mathbb{R}$ of degree $n$ has at most $n$ connected components on each level?
I think I have a proof, can someone point me out any mistakes?
Let $cin mathbb{R}$.
Since $p$ is a submersion, $p-c$ doesn't have a $(x^2+y^2-R)$ factor for any $Rin mathbb{R}^+$.
Also, each connected component of $p^{-1}(c)$ must intersect $x^2+y^2-R=0$ at least twice, for every $R$ big enough.
By Bezout theorem, the system
begin{cases} p=c \ x^2+y^2=Rend{cases}
has at most $2n$ real solutions, hence $p^{-1}(c)$ has at most $n$ connected components.
algebraic-geometry algebraic-curves
asked Sep 15 '18 at 0:49
F.FF.F
263
$endgroup$
Is it true that a polynomial submersion $ p: mathbb{R}^2 to mathbb{R}$ of degree $n$ has at most $n$ connected components on each level?
I think I have a proof, can someone point me out any mistakes?
Let $cin mathbb{R}$.
Since $p$ is a submersion, $p-c$ doesn't have a $(x^2+y^2-R)$ factor for any $Rin mathbb{R}^+$.
Also, each connected component of $p^{-1}(c)$ must intersect $x^2+y^2-R=0$ at least twice, for every $R$ big enough.
By Bezout theorem, the system
begin{cases} p=c \ x^2+y^2=Rend{cases}
has at most $2n$ real solutions, hence $p^{-1}(c)$ has at most $n$ connected components.
algebraic-geometry algebraic-curves
algebraic-geometry algebraic-curves
asked Sep 15 '18 at 0:49
F.FF.F
263
asked Sep 15 '18 at 0:49
F.FF.F
263
edited Jan 3 at 17:01
F.F
asked Sep 15 '18 at 0:49
F.FF.F
263
asked Sep 15 '18 at 0:49
F.FF.F
263
asked Sep 15 '18 at 0:49
F.FF.F
263
263
add a comment |
add a comment |
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