Algebraic properties of Hermitian sums of squares

ABSTRACT

We study the class of bihomogeneous polynomials $r(z,\overline{z})$ on ${\mathbb{C}}^n\times {\mathbb{C}}^n$ for which there is a positive integer d such that $r(z,\overline{z}){∥z∥}^{2d}$ can be written as a Hermitian sum of squares. We reinterpret this problem in terms of commutative algebra, and we give necessary algebraic conditions on this class of polynomials.

COMMUNICATED BY:

• H. Boas

KEYWORDS:

• Sums of squares
• real polynomials
• Betti numbers

AMS SUBJECT CLASSIFICATIONS:

• 12D15
• 14P05
• 32V99

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The first author was supported in part by NSF grant DMS 1200815.