On many-to-one mappings over finite fields
<|reference_start|>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 $\mathbb{F}_{q}^{*}$ is studied by using these three constructions. A series of explicit conditions for~$f$ to be an $m$-to-$1$ mapping on $\mathbb{F}_{q}^{*}$ are found through the detailed discussion of the parameters $m$, $s$, $q$ and the polynomial $h$. These results extend many conclusions in the literature.<|reference_end|>
@article{zheng2024on,
    title={On many-to-one mappings over finite fields},
    author={Yanbin Zheng and Yanjin Ding and Meiying Zhang and Pingzhi Yuan and
  Qiang Wang},
    journal={arXiv preprint arXiv:2408.04218},
    year={2024},
    archivePrefix={arXiv},
    eprint={2408.04218},
    primaryClass={cs.IT math.IT}
}