Let $S \subset \F$ be a finite subset of an arbitrary field $\F$ and let $f \in \F[x_1,\ldots,x_d]$ be a non-zero homogeneous polynomial of degree at most $r$. Then $$\lvert \{ (1,a_2,\ldots,a_d) \in \{1\} \times S^{d-1}  \,\,|\,\, f(1,a_2,\ldots,a_d)=0 \}\rvert \leq r \cdot |S|^{d-2}.$$

Let $g(x_2,\ldots,x_d) = f(1,x_2,\ldots,x_d)$ be the polynomial one obtains from fixing $x_1=1$ in $f$. Then $g$ is a polynomial of degree at most $r$ in $d-1$ variables. If $g$ was the zero polynomial then $f$ would have been divisible by $1-x_1$ which is impossible for a homogeneous polynomial. Hence, we can use Lemma \[lem-SZ\] to bound the number of zeros of $g$ in the set $S^{d-1}$ by $r \cdot |S|^{d-2}$. This completes the proof.

Another useful claim says that if a degree one polynomial (i.e., the equation of a hyperplane) vanishes on a large subset of the product set $V^\ell$, then there is another degree one polynomial that vanishes on a large subset of $V$.

\[lem-hyperplaneproduct\] Let $V \subset \C^d$ be a set of $n$ points and let $V^\ell \subset \C^{d\ell}$ be its $\ell$-fold Cartesian product. Let $H \subset \C^{d\ell}$ be an affine hyperplane such that $|H \cap V^\ell| \geq \delta \cdot n^\ell$. Then, there exists an 