Sum of Squares Conjecture: the Monomial Case in $mathbb{C}^3$

The goal of this article is to prove the Sum of Squares Conjecture for real polynomials $r(z,bar{z})$ on $mathbb{C}^3$ with diagonal coefficient matrix. This conjecture describes the possible values for the rank of $r(z,bar{z}) |z|^2$ under the hypothesis that $r(z,bar{z})|z|^2=|h(z)|^2$ for some holomorphic polynomial mapping $h$. Our approach is to connect this problem to the degree estimates problem for proper holomorphic monomial mappings from the unit ball in $mathbb{C}^2$ to the unit ball in $mathbb{C}^k$. DAngelo, Kos, and Riehl proved the sharp degree estimates theorem in this setting, and we give a new proof using techniques from commutative algebra. We then complete the proof of the Sum of Squares Conjecture in this case using similar algebraic techniques.