# General questions about level sets of polynomials of two variables

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!!

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