Abstract:
A polynomial over $\mathbb{Q}[x]$ is said to be integer valued if, when evaluated at integers, it returns an integer as well. An example of an integer valued polynomial is $$f(x)=\frac{x(x-1)(x-2)}{6} .$$ We will start by introducing integer valued polynomials for a subset of the integers and introduce the greater problem of finding bases for the $\mathbb{Z}$-module they generate. Then, we will look at some work by Manjul Bhargava, and define $p$-orderings and $p$-sequences to get a generalized factorial on a subset of $\mathbb{Z}$. Finally, we will look at examples of the multivariable and homogeneous cases.