Ebenfelt’s SOS Conjecture: Rank gaps after first prolongation

Determine whether, for all integers n ≥ 2, every real-valued Hermitian polynomial A(z, \bar{z}) on C^n that becomes positive semi-definite after the first prolongation (i.e., A(z, \bar{z})\|z\|^2 is a sum of squares) has the sum-of-squares rank R of A(z, \bar{z})\|z\|^2 constrained as follows: either R ≥ (κ0 + 1)n − ((κ0 + 1)κ0)/2 − 1, where κ0 is the largest integer satisfying κ(κ + 1)/2 < n, or there exists κ ∈ {0, 1, …, κ0} such that κn − κ(κ − 1)/2 ≤ R ≤ κn. Here rank means the number of squared norms in an SOS representation of A(z, \bar{z})\|z\|^2.

Background

The paper studies ranks of sums of squares arising from Hermitian polynomials under prolongation by the Euclidean norm. Motivated by Hilbert’s 17th problem and Quillen’s theorem on positive definiteness after sufficient prolongation, Ebenfelt formulated the SOS conjecture to predict specific linear-rank gaps for the first prolongation.

The conjecture is closely connected to geometric problems in several complex variables, notably Huang’s lemma and the Huang–Ji–Yin Gap Conjecture concerning proper rational maps between unit balls; Ebenfelt showed that the Gap Conjecture follows from the SOS conjecture under a CR Gauss equation framework. Prior evidence includes results when A is already SOS (Grundmeier–Halfpap), for n = 2 (Huang), for n = 3 diagonal cases (Brooks–Grundmeier), and more recent geometric advances by Gao–Ng.

This paper confirms the conjecture for diagonal (not necessarily bihomogeneous) Hermitian polynomials when 2 ≤ n ≤ 6, and provides partial results for n ≥ 7, by developing Macaulay-type representations and rank estimates for prolongation matrices.