Quantum integer-valued polynomials

TL;DR

This paper introduces a quantum deformation of integer-valued polynomials, revealing a rich combinatorial structure and positivity properties, and classifies all field maps from this quantum ring.

Contribution

It defines the quantum integer-valued polynomial ring, explores its combinatorial and positivity properties, and extends classical map classifications to the quantum setting.

Findings

Structure constants are in N[q]

The ring exhibits remarkable combinatorial structure

All maps from the quantum ring into a field are classified

Abstract

We define a $q$-deformation of the classical ring of integer-valued polynomials which we call the ring of quantum integer-valued polynomials. We show that this ring has a remarkable combinatorial structure and enjoys many positivity properties: for instance, the structure constants for this ring with respect to its basis of $q$-binomial coefficient polynomials belong to $\mathbb{N}[q]$. We then classify all maps from this ring into a field, extending a known classification in the classical case where $q=1$.