The theorem below is from page 3 in the this paper on polynomials in $\mathbb{Z}_m[x]$.
Let $F$ be a polynomial in $\mathbb{Z}_m[X]$. Then $f \equiv 0$ iff $$F \equiv F_nS_n + \sum_{k=0}^{n-1}a_k(m/(k!, m))S_k,$$ where $n = n(m)$ as given in Definition $1$; $F_n$ is an arbitrary polynomial in $\mathbb{Z}_m[X]$, which is uniquely determined by $F$; and $a_k$ is an arbitrary integer, uniquely determined $(\operatorname{mod}(k!, m))$ by $F$.
Here, $F$ denotes a polynomial in $\mathbb{Z}_m[x]$ and $f$ denotes the corresponding function $f:\mathbb{Z}_m\to \mathbb{Z}_m$ which maps $x\to F(x)$ for all $x\in \mathbb{Z}_m$ (so intuitively, one views $F$ as a mapping).
I am a bit confused with the wording of this theorem. The phrasing of the theorem seems to be a bit sloppy: is this statement saying that $F$ is uniquely determined by a unique choice of $F_n, \cdots, a_k,$ or the other way around? Also, if it is the other way around, I am having trouble understanding how $F_n, \cdots, a_k$ can be uniquely determined from $F$?
