On the characterization of uniqueness polynomials: both complex and p-adic versions
Pratap Basak, Sanjay Mallick
TL;DR
The paper tackles the problem of a general characterization of $UPM$/$UPE$ polynomials for meromorphic/entire functions, extending beyond strictly CIP to include NCIP, and treats both $ ext{complex}$ and $p$-adic settings. It introduces a novel combinatorial framework based on the derivative structure of $P$, with derivative index concepts and counts $(t,t')$, and derives broad criteria guaranteeing uniqueness in both CIP and NCIP regimes. The authors establish concrete, sharp bounds for minimal degrees in complex and $p$-adic contexts, present new NCIP-based $UPM$ constructions, verify existing NCIP examples, and obtain applications to unique range sets with improved cardinalities, including explicit URSM/URSM-IM bounds. The work thereby unifies CIP and NCIP analyses under unified Nevanlinna-theoretic criteria, provides a foundation for sharp degree bounds, and expands the catalog of explicit polynomials yielding uniqueness properties across both mathematical settings.
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.