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

Abstract

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 $r(z,\bar{z})\Vert z{\Vert }^2$ under the hypothesis that $r(z,\bar{z})\Vert z{\Vert }^2=\Vert h(z){\Vert }^2$ 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 ${\mathbb{C}}^k$. D'Angelo, 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.

17 pages. 6 figures