Connected levels and polynomials submersions












4














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.










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    4














    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.










    share|cite|improve this question



























      4












      4








      4


      2





      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.










      share|cite|improve this question















      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






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      share|cite|improve this question













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      edited 2 days ago

























      asked Sep 15 '18 at 0:49









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