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:
- When is $C$ made up of closed curve(s)?
- When is $C$ connected?
- What is the arc length of $C$?
- What is the curvature of $C$?
Any tips, or paper references would be helpful too, thanks!!