small degree polys don't intersect hyperplanes everywhere

Statement

Let $g \in R [x_{1}, \dots, x_{n}]$ be a polynomial in $n$ variables over some ring $R$ . If

$\deg g < n$ ,
$g |_{x_{i} = 0}$ is identically zero for all $i$
Then $g$ is identically zero.

Proof

Suppose $g (x_{1}, \dots, x_{n}) \neq 0$ , so there is a term of the form $c x_{1}^{a_{1}} x_{2}^{a_{2}} \dots x_{n}^{a_{n}}$ in $g$ . In particular, $a_{1} + \dots + a_{n} < n$ , so there is some $i$ such that $a_{i} = 0$ . Now consider setting $x_{i} = 0$ . This term would still survive, so $g |_{x_{i} = 0}$ is some nonzero polynomial in $n - 1$ variables and thus not identically zero, a contradiction.