Difference between a function and polynomial

Question

This question might be silly and simple, but I'm confused. Author mentions, let $f(t)\in \mathbb{F}_{q}[t]$ be a function and $h(t)$, a generic polynomial of the form $h(t)=\sum_{i=0}^{i=m}a_it^i$, where $a_i$ variables over $\mathbb{F}_q$. What does this mean? When $a_i$, takes values from $F_q$ does it become a function in $F_q[t]$? Please help

Answer 1

Given field $\mathbb F$ and a symbol $t$, a member of $\mathbb F[t]$ is an expression of the form $\sum_{i=0}^m a_i t^i$, where $a_i \in \mathbb F$. This is a polynomial over $\mathbb F$, i.e. with coefficients in $\mathbb F$, in the indeterminate $t$, but it is not a function. You get a function associated to the polynomial by specifying a domain $D$ (perhaps $\mathbb F$, or some algebra over $\mathbb F$), and mapping each $x \in D$ to $\sum_{i=0}^m a_i x^i$.

The distinction between polynomials and polynomial functions may seem artificial when dealing with the real or complex fields, but with finite fields it is important. For example, the function $x \mapsto x^p - x$ on $\mathbb F_p$ is identically $0$, i.e. $x^p - x = 0$ for every $x \in \mathbb F_p$, but the polynomial $x^p - x$ in $\mathbb F_p[x]$ is not the $0$ polynomial.