On the characterization of uniqueness polynomials: both complex and p-adic versions

TL;DR

This paper advances the understanding of uniqueness polynomials by providing a comprehensive characterization for both complex and p-adic cases, including new examples, bounds, and verification of existing cases.

Contribution

It introduces a new approach to characterize uniqueness polynomials, including critically injective ones, for both complex and p-adic settings, addressing a long-standing open problem.

Findings

Characterization of uniqueness polynomials for complex and p-adic cases

Examples of unique range sets with minimal cardinalities

Sharp bounds for least degree uniqueness polynomials

Abstract

The problem "A general characterization of uniqueness polynomial for non-critically injective polynomials" has been remained open since the last two decades. In this paper, we explore this open problem. To this end, we initiate a new approach that also includes critically injective polynomials. We provide this characterization for both the complex and p-adic cases. We also provide various examples as an application of our results along with the verification of the existing examples. Consequently, we find examples of unique range sets generated by non-critically injective polynomials with least cardinalities achieved so far and one of these results is sharp with respect to all the available formulas in the literature. Furthermore, we cover the part of least degree uniqueness polynomials. In this part, we also provide some sharp bounds.