Polynomial expressions of $p$-ary auction functions

Let ${\mathbb{F}}_p$ be the finite field of prime order $p$. For any function $f\text{\hspace{0.17em}⁣}:{\mathbb{F}}_p{}^n\to {\mathbb{F}}_p$, there exists a unique polynomial over ${\mathbb{F}}_p$ having degree at most $p-1$ with respect to each variable which coincides with $f$. We call it the minimal polynomial of $f$. It is in general a non-trivial task to find a concrete expression of the minimal polynomial of a given function, which has only been worked out for limited classes of functions in the literature. In this paper, we study minimal polynomial expressions of several functions that are closely related to some practically important procedures such as auction and voting.