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Sorry if I'm being too general here, but here it goes. I'm trying to find out more about levels sets of polynomials of two variables of degree $d$

$$ C = \{ (x,y) \ : \sum_{1 \leq i + j \leq d} c_{i,j} x^i y^j = 1 \} $$

In particular, I want a feeling of what these curves "look like" for low $d$ ($d = 3,4,5...$), and also what kind of things can be computed "exactly" (probably need special functions), for instance:

  1. When is $C$ made up of closed curve(s)?
  2. When is $C$ connected?
  3. What is the arc length of $C$?
  4. What is the curvature of $C$?

Any tips, or paper references would be helpful too, thanks!!

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heres one result: en.wikipedia.org/wiki/Harnack%27s_curve_theorem –  yoyo Aug 2 '11 at 15:15
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