What is $\{ f \in F_{p^m}[x_1, \ldots, x_n] : f(a) = 0, \forall a \in A^n\}$?

Question

As the title suggests, I am interested in knowing if there is a neat description of the ideal of polynomials that vanish on affine $n$ space over a finite field with $p^m$ elements. Is there a way to describe a minimal generating set, or at least how many element are necessary to generate this ideal for a given $n$ and $m$?

Thanks.

Answer 1

Writing $q=p^m$ the required ideal is $\langle x_1^q-x_1,x_2^q-x_2,\cdots,x_n^q-x_n\rangle$.