In some contexts (for example, in the study of spherical harmonics), the connected components of the complement of the zero set of a polynomial are called *nodal domains*.

The maximum number of nodal domains of a polynomial in two real variables of degree $d$ is bounded above by $$\frac{(d-1)(d-2)}{2}+1.$$ This bound is called *[Harnack's curve theorem][1]*.


  [1]: http://en.wikipedia.org/wiki/Harnack%2527s_curve_theorem