Quantum integer-valued polynomials

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).