Number Theory Seminar

Marie B. LangloisCornell University
Building 3-Variable Homogeneous Integer-valued Polynomials Using Projective Planes

Friday, September 14, 2018 - 2:25pm
Malott 205

A polynomial over $\mathbb{Q}[x,y,z]$ is integer-valued if $f(x,y,z)\in \mathbb{Z}$, whenever $x,y,z$ are integers. This talk will look at the case of $f$ being homogeneous and try constructing polynomials such that the denominators are divisible by the highest power of $p=2$ possible. We will introduce projective $H$-planes, which are a generalization of finite projective planes over rings, to construct a correspondence between lines that cover $H$-planes and homogeneous IVPs that are a product of linear factors. We will illustrate this correspondence starting with the degree 8 case where we produce a polynomial with the largest possible denominator which factors as a product of linear polynomials.