What You Should Know About Integer-Valued Polynomials

Abstract

The authors wish to celebrate the centenary of Pólya's paper Ueber ganzwertige ganze funktionen where first explicitly appeared the term “integer-valued polynomials.” This survey is focused on the emblematic example of the ring Int(ℤ) formed by the polynomials with rational coefficients taking integer values on the integers. This ring has surprising algebraic properties, often obtained by means of analytical properties. Yet, the article mentions also several extensions, either by considering integer-valued polynomials on a subset of ℤ, or by replacing ℤ by the ring of integers of a number field.