There exists a one-to-one correspondence between homogeneous polynomials on \(\mathbb{R}^d\) and there restriction on \(\mathbb{S}^{d-1}\). This is a consequence of homogeneity, since for a homogeneous polynomial we obtain
\[ p(x)=\lVert x\rVert^k p\Bigl(\tfrac{x}{\lVert x\rVert}\Bigr). \]Thus, if \(p\) vanishes on the sphere, it vanishes everywhere.