Hermitian symmetric polynomials and CR complexity

TL;DR

This paper explores the properties of Hermitian symmetric polynomials and forms, revealing how their signature pairs behave under multiplication and applying these findings to CR geometry and rational mapping complexity.

Contribution

It introduces new results on signature pairs of Hermitian forms under polynomial products and applies these to CR geometry and rational mapping complexity theory.

Findings

Most signature pairs can be obtained from indefinite forms through multiplication

A stability result for the existence of rational mappings with specific signatures

New applications to CR geometry and hyperquadric mappings

Abstract

Properties of Hermitian forms are used to investigate several natural questions from CR Geometry. To each Hermitian symmetric polynomial we assign a Hermitian form. We study how the signature pairs of two Hermitian forms behave under the polynomial product. We show, except for three trivial cases, that every signature pair can be obtained from the product of two indefinite forms. We provide several new applications to the complexity theory of rational mappings between hyperquadrics, including a stability result about the existence of non-trivial rational mappings from a sphere to a hyperquadric with a given signature pair.