Holomorphic factorization of matrices of polynomials
This paper considers some work done by the author and Catlin [CD1,CD2,CD3] concerning positivity conditions for bihomogeneous polynomials and metrics on bundles over certain complex manifolds. It presents a simpler proof of a special case of the main result in [CD3], providing also a self-contained proof of a generalization of the main result from [CD1]. Some new examples and applications appear here as well. The idea is to use the Bergman kernel function and some operator theory to prove purely algebraic theorems about matrices of polynomials. Theorem 1. [Catlin-D'Angelo]. Suppose that $f$ is a bihomogeneous real-valued polynomial on ${\bf C^n}$ of degree $2m$. Then $f$ is positive away from the origin if and only there is an integer $d$ and a holomorphic homogeneous polynomial mapping $A$, whose components span the space of homogeneous polynomials of degree $m+d$, such that $$ ||z||^{2d} f(z,{\overline z}) = ||A(z)||^2.$$ Suppose that $F(z,{\overline z})$ is an $r$ by $r$ matrix whose entries are bihomogeneous polynomials of degree $2m$. Then $F(z,{\overline z})$ is positive-definite at each point $z \ne 0$ if and only if there is an integer $d$ and a holomorphic homogeneous polynomial matrix $A$, whose row vectors span the space of $r$-tuples of homogeneous polynomials of degree $m+d$, such that $$||z||^{2d} F(z,{\overline z}) = A(z)^* A(z).$$