Claim \[cla-veronese-vanish\] to deduce the existence of the vanishing polynomial. Let $M$ be an $n \times \m(d,r-2)$ matrix whose $i$’th row is $u_i = \phi(v_i)$. To show that $U$ is contained in a hyperplane, it is enough to show that $\rank(M) < \m(d,r-2)$. This will imply that the columns of $M$ are linearly dependent, which means that all the rows lie in some hyperplane.

We will now construct a design matrix $A$ such that $A \cdot M=0$. Since $\rank(A) + \rank(M) \leq n$, we will be able to translate a lower bound on the rank of $A$ (which will be given by Theorem \[thm-rankbound\]) to the required upper bound on the rank of $M$. Each row in $A$ will correspond to some collinear $r$-tuple in $V$. We will construct $A$ in several stages. First, for each $r$-rich line $L \in \cL_r(V)$ we will construct a set of $r$-tuples $R_L \subset \binom{V}{r}$ such that

1.  Each $r$-tuple in $R_L$ is contained in $L \cap V$.

2.  Each point $v \in L \cap V$ is in at least one $r$-tuple from $R_L$.

3.  Every pair of distinct points $u,v \in L \cap V$ appear together in at most two $r$-tuples from $R_L$.

