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 \(r(z,\bar{z}) \left\Vert {z} \right\Vert ^2\) under the hypothesis that \(r(z,\bar{z})\left\Vert {z} \right\Vert ^2=\left\Vert {h(z)} \right\Vert ^2\) 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.