Some results related to Macaulay's Theorem about Hilbert functions and applications

TL;DR

This paper provides an elementary proof of the equivalence between two versions of Macaulay's Theorem on Hilbert functions, and applies it to derive new inequalities in complex analysis related to Hermitian polynomials and holomorphic mappings.

Contribution

It establishes the equivalence of degree-dependent and degree-independent forms of Macaulay's Theorem and applies this to Hermitian polynomials in complex geometry.

Findings

Derived inequalities relating dimension, signature, and rank of Hermitian polynomials.

Extended results to norms of arbitrary signatures with uniform validity across bidegrees.

Provided an elementary proof of a classical algebraic geometry theorem.

Abstract

Let $I$ be a homogeneous ideal in the polynomial ring $R = k[z_1, \cdots, z_n]$ , where $k$ is an algebraically closed field of characteristic zero. Macaulay's Theorem provides constraints on the Hilbert function of $I$ or $R/I$ from one degree to the next. Nowadays, the standard quotation of Macaulay's theorem is $H_{R/I}(d + 1) \le H_{R/I}(d)^{\langle d\rangle}$, which is regarding the quotient $R/I$ and the combinatorial computation in the formula involves the number $d$ explicitly. However, the origin statement of Macaulay is in fact regarding the Hilbert function of $I$ itself and the relevant combinatorics explicitly involves the number of variables (i.e. $n$) and does not depend on $d$. In this paper, we provide an elementary proof of the equivalence between these two versions of Macaulay's theorem. The original degree-independent version is more suitable for problems such as those involving sums of polynomial squared norms. Motivated by the Hermitian analogue of Hilbert's 17th problem and proper holomorphic mappings between complex unit balls, some questions lead to the study of Hermitian polynomials $M(z, \bar{z}) \in \mathbb{C}[z_1, \ldots, z_n, \bar{z}_1, \ldots, \bar{z}_n]$ satisfying $M(z, \bar{z})\|z\|^{2l} = \|h\|^2$ for some $l$ and a holomorphic mapping $h = (h_1, \cdots, h_R)$ . Using Macaulay's Theorem, we derive new inequalities relating $n$, $l$, the signature $(p, q)$ of the coefficient matrix of $M(z, \bar{z})$ , and $R$ (the rank of $M(z, \bar{z})\|z\|^{2l}$ ) and extend these results to norms of arbitrary signatures, which hold uniformly for all bidegrees of $M(z, \bar{z})$.