On the interpolation of integer-valued polynomials

Text. 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 Z[i] when n is large enough, thus answering the question of Hensley (1977). Also we discuss the question about minimal possible size of a set, such that if polynomial takes integer values in all points of this set then it is integer-valued. Video. For a video summary of this paper, please click here or visit http://youtu.be/hCE7M802oZ8. 2013 (C) Elsevier Inc. All rights reserved.