I have the following elementary question that I cannot quite figure out myself or find an appropriate reference:
Let $p$ be a nonzero homogeneous polynomial which we view as a function on $\mathbb R^n$. Let $S^{n-1}$ denote the unit sphere $\{x\in\mathbb R^n\colon |x|=1\}$. Prove that $$ \mathcal H^{n-2}(\{x\in S^{n-1}\colon p(x)=0\})<\infty. $$ Here $\mathcal H^d$ denotes the $d$-dimensional Hausdorff measure.
A couple of remarks: I expect a bound only in terms of the degree of $p$ and probably space dimension $n$, but I would be happy with a bound that depends on $p$ in any way. This speculation is clear if $n=2$, when $\mathcal H^{0}(\{x\in S^{n-1}\colon p(x)=0\})\leq 2\mathrm{deg\,}p$. I tried to find some sort of argument involving slicing and/or (co)area formula, but I could not make it work.
Thank you!