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

TL;DR

This paper proves the Sum of Squares Conjecture for certain real polynomials in three complex variables with diagonal coefficients, linking it to degree estimates of holomorphic monomial mappings and employing algebraic methods.

Contribution

It provides the first proof of the conjecture in the monomial case in CA3^3, using novel algebraic techniques and connecting to degree estimates in holomorphic mappings.

Findings

Confirmed the conjecture for the monomial case in CA3^3

Established a new algebraic proof for degree estimates in holomorphic mappings

Extended the understanding of polynomial rank conditions in complex analysis

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 $r(z,\bar{z}) \|z\|^2$ 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 $\mathbb{C}^k$. 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.