An Application of Macaulay's Estimate to CR Geometry

TL;DR

This paper explores the relationship between bihomogeneous polynomials in CR geometry and the growth of Hilbert functions, applying Macaulay's estimate to derive bounds on algebraic invariants.

Contribution

It reformulates CR geometric problems as algebraic questions about Hilbert functions and applies Macaulay's estimate to obtain new bounds and insights.

Findings

Established bounds on the signature and rank of bihomogeneous polynomials

Connected CR geometry problems with algebraic growth functions

Applied Macaulay's estimate to derive new inequalities

Abstract

Several questions in CR geometry lead naturally to the study of bihomogeneous polynomials $r(z,\bar{z})$ on $\C^n \times \C^n$ for which $r(z,\bar{z})\norm{z}^{2d}=\norm{h(z)}^2$ for some natural number $d$ and a holomorphic polynomial mapping $h=(h_1,..., h_K)$ from $\C^n$ to $\C^K$. When $r$ has this property for some $d$, one seeks relationships between $d$, $K$, and the signature and rank of the coefficient matrix of $r$. In this paper, we reformulate this basic question as a question about the growth of the Hilbert function of a homogeneous ideal in $\C[z_1,...,z_n]$ and apply a well-known result of Macaulay to estimate some natural quantities.