will be the following lemma, which shows that one can find a vanishing polynomial of low degree, assuming each point is in many rich lines.

\[lem-findpoly\] For each $d \geq 1$ there is a constant $K_d \leq 32(2d)^d$ such that the following holds. Let $V \subset \C^d$ be a set of $n$ points and let $r\geq 4$ be an integer. Suppose that, through each point $v \in V$, there are at least $k$ $r$-rich lines where $$k \geq K_d\cdot \frac{ n}{r^{d-2}}.$$ Then, there exists a non-zero polynomial $f \in \C[x_1,\ldots,x_d]$ of degree at most $r-2$ such that $f(v)=0$ for all $v \in V$.

If we have the stronger condition that the number of $r$-rich lines through each point of $V$ is between $k$ and $8k$ then we can get the same conclusion (vanishing $f$ of degree $r-2$) under the weaker inequality $$k \geq K_d\cdot \frac{n}{r^{d-1}}.$$

Let $V = \{v_1,\ldots,v_n\}$ and let $\phi = \phi_{d,r-2} : \C^d \mapsto \C^{\m(d,r-2)}$ be the Veronese embedding with degree bound $r-2$. Let us denote $U = \{u_1,\ldots,u_n\} \subset \C^{\m(d,r-2)}$ with $u_i = \phi(v_i)$ for all $i \in [n]$.

We will prove the lemma by showing that $U$ is contained in a hyperplane and then using 