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

Abstract

Let [Formula: see text] be a homogeneous ideal in the polynomial ring [Formula: see text], where [Formula: see text] is an algebraically closed field of characteristic zero. Macaulay’s Theorem provides constraints on the Hilbert function of [Formula: see text] or [Formula: see text] from one degree to the next. Nowadays, the standard quotation of Macaulay’s theorem is [Formula: see text], which is regarding the quotient [Formula: see text] and the combinatorial computation in the formula involves the number [Formula: see text] explicitly. However, the origin statement of Macaulay is in fact regarding the Hilbert function of [Formula: see text] itself and the relevant combinatorics explicitly involves the number of variables (i.e. [Formula: see text]) and does not depend on [Formula: see text]. 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, SOS conjecture and proper holomorphic mappings between complex unit balls, some questions lead to the study of Hermitian polynomials [Formula: see text] satisfying [Formula: see text] for some [Formula: see text] and a holomorphic mapping [Formula: see text]. Using Macaulay’s Theorem, we derive new inequalities relating [Formula: see text], [Formula: see text], the signature [Formula: see text] of the coefficient matrix of [Formula: see text], and [Formula: see text] (the rank of [Formula: see text] ) and extend these results to norms of arbitrary signatures, which hold uniformly for all bidegrees of [Formula: see text].