Algebraic properties of Hermitian sums of squares, II

TL;DR

This paper investigates algebraic properties of certain bihomogeneous polynomials related to the Sum of Squares Conjecture, providing algebraic constraints on their signatures and ranks, with implications for holomorphic mappings between complex balls.

Contribution

It applies algebraic methods to derive constraints on the signatures and ranks of bihomogeneous polynomials relevant to the Sum of Squares Conjecture.

Findings

Derived algebraic constraints on polynomial signatures.

Connected polynomial properties to ideal theory.

Implications for proper holomorphic mappings.

Abstract

We study real bihomogeneous polynomials $r(z,\bar{z})$ in $n$ complex variables for which $r(z,\bar{z}) \|z\|^2$ is the squared norm of a holomorphic polynomial mapping. Such polynomials are the focus of the Sum of Squares Conjecture, which describes the possible ranks for the squared norm $r(z,\bar{z}) \|z\|^2$ and has important implications for the study of proper holomorphic mappings between balls in complex Euclidean spaces of different dimension. Questions about the possible signatures for $r(z,\bar{z})$ and the rank of $r(z,\bar{z}) \|z\|^2$ can be reformulated as questions about polynomial ideals. We take this approach and apply purely algebraic tools to obtain constraints on the signature of $r$.