An application of Macaulay’s estimate to sums of squares problems in several complex variables

Abstract

Several questions in CR geometry lead naturally to the study of bihomogeneous polynomials r(z,z) on C n ×C n for which r(z,z) z 2d = 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 , . . . , zn] and apply a well-known result of Macaulay to estimate some natural quantities.

On the HJY Gap Conjecture in CR geometry vs. the SOS Conjecture for polynomials

“…The collection of 1-forms (ω a α ) on S n defines the second fundamental form of the mapping f , denoted Π f : T 1,0 S n × T 1,0 S n → T 1,0 S N /f * T 1,0 S n , as described in [2]. We recall from there that (15) ω a α = ω a α β θ β , ω a α β = ω a β α .…”

“…In another recent paper [15] by Grundmeier and Halfpap, the SOS Conjecture 1.2 was established in the special case where A(z,z) is itself an SOS, i.e., (11) A…”

“…The polynomial A(z, z) is in principle computable from f , but useful properties of A seem difficult to extract directly in this way, and often it suffices to know that Ω satisfies an SOS identity of this form, for some Hermitian polynomial A. SOS identities of the form (7) appear in many different contexts, and there is an abundance of literature considering various aspects of such identities. We mention here only a few, and only ones with a connection to CR geometry and complex analysis: e.g., [27], [3], [4], [12], [7], [6], [24], [15], [16], [9], and refer the reader to these papers for further connections and references to the literature. The reader is especially referred to the paper [6] by D'Angelo, which contains an excellent discussion of SOS identities and positivity conditions.…”

Algebraic properties of Hermitian sums of squares

“…In [GK15], the first two authors prove the Sum of Squares Conjecture when r has signature (P, 0), and [BG21] shows it holds when n = 3 and the coefficient matrix of r is diagonal. The gaps in possible rank described in (2) occur in the non-negative semi-definite case, and it appears that when the coefficient matrix of r has negative eigenvalues, the inequality (1) always holds.…”

“…Algebraic Formulation. In [GK15] and [BG21], the first two authors investigate r(z, z) z 2 by translating the problem into one about homogeneous ideals in R = C[z 1 , . .…”