Weak (Alternative) SOS Conjecture

Establish that if a real-valued Hermitian polynomial A(z, \bar{z}) on C^n is not a sum of squares but its first prolongation A(z, \bar{z})\|z\|^2 is a sum of squares, then the sum-of-squares rank R of A(z, \bar{z})\|z\|^2 satisfies R ≥ (κ0 + 1)n − ((κ0 + 1)κ0)/2 − 1, where κ0 is the largest integer satisfying κ(κ + 1)/2 < n.

Background

Ebenfelt suggested a strengthened scenario in which rank gaps predicted by the SOS conjecture occur only when A is itself an SOS, and when A is not SOS but A|z|2 is SOS, one always has the conjectured lower bound on the rank. He termed this the weak (alternative) SOS conjecture.

If true, this conjecture would imply the full SOS conjecture in view of the Grundmeier–Halfpap result for the case when A is SOS. The present paper tackles this conjecture in the diagonal case, confirming the bound for 2 ≤ n ≤ 6 and giving partial estimates for larger n by introducing a prolongation map with a Macaulay-type representation and deriving rank lower bounds.

References

Based on the Grundmeier--Halfpap result, Ebenfelt pointed out that an optimistic view of the situation in the conjecture would be to hope that the "gaps" in linear ranks predicted in (1.2) can only occur when A(z,\bar{z}) is itself an SOS. Furthermore, if A(z,\bar{z}) is not an SOS but A(z,\bar{z})||z||2 is still an SOS, then the lower bound (1.1) always holds. This is named the weak (alternative) sum-of-squares conjecture. If true, it implies the SOS conjecture in view of the Grundmeier-Halfpap result. If A(z,\bar z) is not a sum of squares but A(z,\bar z)||z||2 is a sum of squares, then (1.1) holds.

— Macaulay representation of the prolongation matrix and the SOS conjecture  (2509.04314 - Wang et al., 4 Sep 2025) in Conjecture 1.5, Section 1 (Introduction)