Sum of squares conjecture: themonomial case in C3

The goal of this article is to prove the Sum of Squares Conjecture for real polynomials r (z, (z) over bar) on C-3 with diagonal coefficient matrix. This conjecture describes the possible values for the rank of r ( z, (z) over bar) parallel to z parallel to(2) under the hypothesis that r (z, (z) over bar) parallel to z parallel to(2) = parallel to h(z)parallel to(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 C-2 to the unit ball in 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.