= Polynomial nonvanishing below the field size
Let $P\in\mathbb F_p[x_1,\ldots,x_n]$ have <total degree of a polynomial> less than $p$. If the associated <polynomial function> vanishes on all of $\mathbb F_p^n$, then $P$ is the zero polynomial. Induct on $n$: write
$$
P=\sum_{j=0}^{p-1}P_j(x_1,\ldots,x_{n-1})x_n^j.
$$
For each fixed $(x_1,\ldots,x_{n-1})$, the resulting univariate polynomial of degree less than $p$ has all $p$ field elements as <roots of a polynomial>[roots], so every coefficient $P_j$ vanishes everywhere. The induction hypothesis makes every $P_j$ the zero polynomial.
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