Corollary

Let $V$ be a vector space of finite dimension $n$ over $K$ and $I$ a finite set, and for each $i \in I$, let $F_i : V \to K$ be a homogeneous polynomial mapping of degree $d_i > 0$. If we have $\sum_{i \in I} d_i < n$, then there exists a nonzero element $x$ of $V$ such that $F_i(x) = 0$ for every $i \in I$.

Let $(e_1, \ldots, e_n)$ be a basis of $V$ over $K$. By the definition of homogeneous polynomial mappings (IV, §5, No. 9, p. 55, Definition 3), for every $i \in I$, there exists a homogeneous polynomial $f_i$ in $K[X_1, \ldots, X_n]$ of degree $d_i$ such that we have $F_i(\xi_1 e_1, \ldots, \xi_n e_n) = f_i(\xi_1, \ldots, \xi_n)$. Let $Z$ be the set of elements $x$ of $V$ such that $F_i(x) = 0$ for every $i \in I$. By Theorem 1, the cardinal of $Z$ is divisible by $p$, and since $0$ belongs to $S$, we have $\operatorname{Card}(Z) \geq p > 1$.