The maximum number of nodal domains in the real projective plane of a polynomial of two variables of degree $d$ (i.e. a homogeneous polynomial in $\mathbb{R}^3$) is bounded above by $$\frac{(d-1)(d-2)}{2}+1.$$ This bound d(d-1)+2.$A nice exposition of this result can be found Leydold's paper On the number of nodal domains of spherical harmonics. A related result is called Harnack's curve theorem. It says that the number of connected components of the zero set of a polynomial in the real projective plane is bounded by$(d-1)(d-2)/2+1$. 2 added 24 characters in body; edited body; deleted 4 characters in body In some contexts (for example, in the study of spherical harmonics), the connected components of the complement of the zero set of a polynomial are called nodal domains. The maximum number of nodal domains in the real projective plane of a polynomial in of two real variables of degree$d$is bounded above by $$\frac{(d-1)(d-2)}{2}+1.$$ This bound is called Harnack's curve theorem. 1 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.