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Harnack's curve theorem tells us that the maximum number of connected components of an algebraic curve of degree $d$ in the real projective plane is $1 + (d-1)(d-2)/2$ (and this bound is sharp).

What happens if "projective" is replaced by "affine"? What is the maximum number of connected components then?

The only relevant reference I have been able to find so far is a paper by Korchagin and Weinberg on the isotopy classification of affine quartic curves, where they calculate that a compact affine quartic curve has no more than 4 connected components and a noncompact affine quartic curve has no more than 7 connected components. I think their argument can be extended to higher degree to give an upper bound, but are these bounds sharp?

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  • $\begingroup$ One possible strategy is to find hyperplanes of projective space cutting a given curve $C$ in as few components as possible (thus cutting those up into many pieces). I have no idea how to do this. $\endgroup$ Nov 16, 2018 at 16:38
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    $\begingroup$ It seems the answer is here mathoverflow.net/questions/68954/… $\endgroup$
    – Vlad Matei
    Nov 16, 2018 at 16:43
  • $\begingroup$ @VladMatei : I saw that question before I posted my question but I don't see how it answers my question. $\endgroup$ Nov 16, 2018 at 16:56
  • $\begingroup$ You are right it does not seem quite that. Let's try this on page 8: math.caltech.edu/~2014-15/3term/ma191c-sec2/… $\endgroup$
    – Vlad Matei
    Nov 16, 2018 at 17:04
  • $\begingroup$ @VladMatei : That argument gives an upper bound but still does not answer the question. $\endgroup$ Nov 16, 2018 at 19:45

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