On many-to-one mappings over finite fields

The definition of many-to-one mapping, or $m$-to-$1$ mapping for short, between two finite sets is introduced in this paper, which unifies and generalizes the definitions of $2$-to-$1$ mappings and $n$-to-$1$ mappings. A generalized local criterion is given, which is an abstract criterion for a mapping to be $m$-to-$1$. By employing the generalized local criterion, three constructions of $m$-to-$1$ mapping are proposed, which unify and generalize all the previous constructions of $2$-to-$1$ mappings and $n$-to-$1$ mappings. Then the $m$-to-$1$ property of polynomials $f(x)=x^{r}h(x^s)$ on ${\mathbb{F}}_q^{\ast }$ is studied by using these three constructions. A series of explicit conditions for~$f$ to be an $m$-to-$1$ mapping on ${\mathbb{F}}_q^{\ast }$ are found through the detailed discussion of the parameters $m$, $s$, $q$ and the polynomial $h$. These results extend many conclusions in the literature.