The 2-adic valuation of the general degree-2 polynomial in 2 variables

Abstract

We define the $p$-adic valuation tree of a polynomial $f(x,y)$ $\in$ $\mathbb{Z}[x,y]$ by which we can find its $p$-adic valuation at any point. This work includes diverse $2$-adic valuation trees of certain degree-two polynomials in two variables. Among these, the $2$-adic valuation tree of $x^2+y^2$ is most interesting. We use the observations from these trees to study the $2$-adic valuation tree of the general degree-two polynomial in $2$ variables. We also study the $2$-adic valuation tree of the polynomial $x^2y+5$.