Say you have a finite field $F$ of order $p^k$. Suppose that $f,g\in F[X_1,\dots,X_m]$, such that the degree of each $X_i$ is strictly less than $p^k$ in both $f$ and $g$. I'm putting this condition to avoid things like $f=X_1X_2X_3^{p^k}$ and $g=X_1X_2X_3$ which technically define the same polynomial function over $F$ since $X_i^{p^k}-X_i$ is in the kernel of the evaluation homomorphism, but are not equal in the polynomial ring.
Under this condition, if $f$ and $g$ define the same polynomial function over $F$, are they equal as polynomials? By equality of polynomial functions, I mean they are equal as sets of ordered pairs. I feel like restricting the degree of each indeterminate should force this to be so, but how can it actually be proven?