Let $p$ be a prime and let $F=\mathbb{Z}/p\mathbb{Z}$. Can a nonzero multivariate polynomial $f\in F[x_1,….,x_n]$ such that $\mathrm{deg}_if< p$ for all $i=1,\ldots,n$ be identically zero as a polynomial function $F^n\to F$, where $\mathrm{deg}_if$ stands for the degree of $f$ in $x_i$?
Abstract Algebra – Identically Zero Multivariate Polynomial Function
abstract-algebrapolynomials
Best Answer
No. The phenomenon of polynomials over fields which are not the zero polynomial but evaluate to the zero function is studied in loving detail in $\S 2.1$ of these notes: in particular, the result you want is Theorem 8.
I also agree with Qiaochu's comment: it would be possible to make much less of a production of this by simply inducting on the number of variables, beginning with the basic fact that no polynomial with coefficients in an integral domain can have more roots than its degree (except the polynomial which has all coefficients zero, if you take the convention that this polynomial has degree $-\infty$). I have my reasons for the more elaborate treatment given in my notes: in particular, an at least somewhat careful analysis of the situation leads directly to a proof of the Chevalley-Warning Theorem.