# Polynomial expressions of $p$-ary auction functions

## TL;DR

This paper investigates the polynomial representations of functions related to auction and voting procedures over finite fields, focusing on deriving explicit minimal polynomial expressions for these functions.

## Contribution

It provides explicit polynomial expressions for functions associated with auction and voting, expanding understanding of their algebraic structure over finite fields.

## Key findings

- Derived minimal polynomial expressions for auction-related functions
- Analyzed the algebraic structure of voting functions over finite fields
- Extended polynomial representation techniques to practical decision procedures

## Abstract

Let $\mathbb{F}_p$ be the finite field of prime order $p$. For any function $f \colon \mathbb{F}_p{}^n \to \mathbb{F}_p$, 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.