# If $P \in V$ is a polynomial with maximal support $\Sigma$, then $|\Sigma| \geq dim(V)$

I was wondering if anyone could help me understand a statement in the proof of Theorem 4 of this paper: https://www.jstor.org/stable/pdf/24906443.pdf. I'm aiming to prove the theorem for $$q=3$$ and $$\alpha= \beta = \gamma = 1$$.

$$F_3$$ is a field of size $$3$$. $$A$$ is a subset of $$F_3^n$$ such that no three elements (with at least 2 distinct) sum to $$0$$ in $$A$$.

$$S_n^d$$ is defined to be the span of monomials in $$x_1, ... , x_n$$ with degree at most $$d$$, and $$m_d$$ is the dimension of $$S_n^d$$. Here $$d$$ is a real number between $$0$$ and $$2n$$.

$$V$$ is the space of polynomials in $$S_n^d$$ that vanish on the complement of $$-A$$. I say the support of a polynomial $$P$$ to mean the elements $$x$$ such that $$P(x) \neq 0$$.

Then if $$P \in V$$ has support $$\Sigma$$ of maximal size and $$|\Sigma| < dim(V)$$, then there exists another polynomial $$Q \in V$$ that vanishes on $$\Sigma$$.

However I am struggling to see why this is the case or to come up with some polynomial to demonstrate this. Any help would be appreciated!

The condition that a polynomial vanish at a particular point is a single linear equation in the coefficients. You then have $$|\Sigma|$$ linear equations in $$m_d$$ variables (the coefficients), which automatically has a solution if $$|\Sigma|.
Alternatively, more basis freely: The map that sends a polynomial in $$V$$ to its list of values on $$\Sigma$$ is a linear map. If the dimension on $$V$$ is larger than $$|\Sigma|$$ this linear map automatically has non-trivial kernel.