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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 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 $(d-1)(d-2)/2+1$.
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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.
show/hide this revision's text 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.