SONC containment of the mean forms

Determine whether the forms M_{2d,p}(X,α) are contained in the SONC cone for all admissible parameters.

Background

The paper studies the relationship between mean polynomial forms, sums of squares, and nonnegative circuit polynomials. Although the low-degree forms in Theorem M2dp-sos are shown to be SOS in several cases, their membership in the SONC cone for all parameters remains unresolved.

References

Are these forms contained in the SONC cone for all parameters?

— The Cone Generated by Positive Semidefinite Mean Polynomials  (2608.22739 - Ghasemi et al., 24 Aug 2026) in Remark rem:open, Section 4

However, the Kadison--Dubois construction in the Positivstellensatz yields a single element of M_{\mathrm{mean}, not a difference; the product depth required is not explicitly bounded. An efficient decomposition of f-λ into mean polynomial certificates (analogous to the Gram-matrix method for SOS) remains an open problem.

— The Cone Generated by Positive Semidefinite Mean Polynomials  (2608.22739 - Ghasemi et al., 24 Aug 2026) in Remark rem:efficient, Section 8.4, “Computational decomposition”