homogeneous polynomials over finite fields

Question

Let $F$ be a finite field and $p(X_1,\dots,X_n)\neq 0$ an homogeneous polynomial with coefficients in $F$. Is it possible that $p(x_1,\dots,x_n)=0$ for every $(x_1,\dots, x_n)\in F^n$?

Answer 1

Call $f=|F|$ the number of elements of $F$. Then the polynomial $$X^fY-XY^f \in F[X,Y]$$ annihilates all points of $F^2$.

Answer 2

You usually want to consider, for example in $\mathbb{F}_{q}[x]$, the collection of polynomials in $\mathbb{F}_{q}[x]/\langle x^{q-1} -1 \rangle$, because $x^{q-1}-1$ is equivalent to the zero polynomial on $\mathbb{F}_q$ (as a mapping).

In your case, with several variables, if you restrict the exponent on each variable in each term to be strictly less than $|F|-1$, then each polynomial will represent a unique map on $F^n$ (and so the zero map is uniquely represented as the zero polynomial).