Hermitian rank in ideal powers

Abstract

We prove that the (hermitian) rank of $Q{P}^d$ is bounded from below by the rank of $P^d$ whenever $Q$ is not identically zero and real-analytic in a neighborhood of some point on the zero set of $P$ in ${\mathbb{C}}^n$ and $P$ is a polynomial of bidegree at most $(1,1)$. This result generalizes the theorem of D'Angelo and the second author for bihomogeneous $P$ and Huang's Lemma. The inequality demonstrates that real-analytic multipliers cannot reduce the algebraic complexity of ideal powers. Examples show that no hypothesis can be dropped. As an application, we give a lower bound on the target dimension for a CR map of hyperquadrics depending on the order of contact.

arXiv comment16 pages; Proof simplified; Application added; Comments are welcome!