Algorithms for computing bases for 3-variable homogeneous integer-valued polynomials

Abstract

A polynomial f in Q[x,y,z] is integer-valued if f(x,y,z)∈Z, whenever x,y,z are integers. This work will look at the case where f is homogeneous and will present algorithms for constructing polynomials fpk with f∈Z[x,y,z] such that k is as large as possible for a given degree. From this we will find bases for the modules of homogeneous integer-valued polynomials (IVPs) in a range of degrees. IVPs have been studied for their topological applications, including the homogeneous ones. We explain the connection between 3-variable homogeneous IVPs of degree m and 3-variable IVPs of degree m, as well as with 2-variable IVPs of degree m evaluated at odd values only, then use linear algebra to calculate bases for both cases in a range of degrees.