On the interpolation of integer-valued polynomials

TL;DR

This paper investigates the properties of integer-valued polynomials over Gaussian integers, demonstrating limitations of interpolation sets and exploring minimal conditions for integer-valuedness in complex domains.

Contribution

It proves that for large degrees, no set of n+1 points in Gaussian integers guarantees integer-valued polynomials, and discusses minimal sets for integer-valuedness.

Findings

No n+1 point set in Gaussian integers guarantees integer-valued polynomials for large n.

Characterization of minimal sets ensuring integer-valuedness.

Extension of classical interpolation results to Gaussian integers.

Abstract

It is well known, that if polynomial with rational coefficients of degree $n$ takes integer values in points $0,1,...,n$ then it takes integer values in all integer points. Are there sets of $n+1$ points with the same property in other integral domains? We show that answer is negative for the ring of Gaussian integers $\mathbb{Z}[i]$ when $n$ is large enough. Also we discuss the question about minimal possible size of set, such that if polynomial takes integer values in all points of this set then it is integer-valued.