Homogeneous Integer-Valued Polynomials of Three Variables

Abstract

A polynomial f in Q[x,y,z] is integer-valued if f(x,y,z)is an integer, whenever x, y, z are integers. This work will look at the case where f is homogeneous and construct polynomials such that the denominators are divisible by the highest prime power possible and find bases for the modules of homogeneous integer-valued polynomials (IVPs). We will present computational methods for constructing such bases and an algebraic method to construct these. 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 in both cases. In order to obtain polynomials written as a product of linear factors, we will look into extending the construction of finite projective planes to rings and explain a connection between line coverings of those planes and homogeneous IVPs.